1. Introduction

In this paper, we consider the initial value problem of the fractional Navier-Stokes equations with the Coriolis force in \(\mathbb{R}^{3}\), where \(u=u(t,x) = (u_{1} (t,x), u_{2}(t,x),u_{3}(t,x))\) represents the unknown velocity vector, the scalar function \(\pi=\pi(t,x)\) denotes the unknown scalar pressure and \(u_{0}\) is a divergence free vector field. The constant \(\mu>0\) indicates the viscosity coefficient of the fluid, \(\Omega\in \mathbb{R}\) represents the speed of rotation around the vertical unit vector \(e_{3} = (0, 0,1)\), which is called the Coriolis parameter, and \(\times\) represents the outer product, hence, \(-\Omega e_{3}\times u=(\Omega u_{2},-\Omega u_{1},0).\) We recall that the Coriolis term has an another expression \(-\Omega e_{3}\times u=-\Omega \mathrm{J}u\), where the skew-symmetric matrix \(\mathrm{J}\) defined by \[\mathrm{J}=\left(\begin{array}{rlc} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{array}\right).\] The operator \((-\Delta)^{\alpha}\) is the Fourier multiplier with symbol \(|\xi|^{2\alpha}\). When \(\alpha= 1\), the equation (1) corresponds to the usual Navier-Stokes equation with Coriolis force, which receives some attention for its importance in geophysical flow applications. In particular, large scale atmospheric and oceanic flows are dominated by rotational effects, see ([1, 2]). When \(\alpha = 1\) and \(\Omega\neq0\), Hieber and Shibata [3] obtained the uniform global well-posedness for the Navier-Stokes equations with Coriolis force for small initial data in the Sobolev space \(H^{\frac{1}{2}}(\mathbb{R}^{3})\). Chemin et al. [2, 4] established that for any initial data \(u_{0}\) belonging to \(L^{2}(\mathbb{R}^{2}) + H^{\frac{1}{2}}(\mathbb{R}^{3})\) there exists a unique solution to the Navier-Stokes equations with Coriolis force when \(|\Omega|> \Omega_{0}>0\). Iwabuchi and Takada [5] proved the existence of global solutions for the Navier-Stokes equations with Coriolis force in Sobolev spaces \(\dot{H}^{s}(\mathbb{R}^{3})\) with \(1/2< s< 3/4\) if the speed of rotation \(\Omega\) is large enough compared with the norm of initial data \(\|u_{0}\|_{\dot{H}^{s}}\); they also obtained the global existence and the uniqueness of the mild solution for small initial data in the Fourier-Besov spaces \(\mathrm{F\dot{B}}_{1,2}^{-1}\), and proved the ill-posedness in the space \(\mathrm{F\dot{B}}_{1,q}^{-1}\), \(2< q\leq\infty\) for all \(\Omega\in \mathbb{R}\) (see [6]). For the local existence of solutions of these equations, we quote the results of Giga et al. [7, 8] and Sawada [9]. Recently, W. Wang and G. Wu [10] established the global well-posedness of mild solution to the three-dimensional incompressible generalized Navier-Stokes equations with Coriolis force if the initial data are in the Lei-Lin's space \(\chi^{1-2\alpha}\), they also gave Gevrey class regularity of the solution. In the case \(\Omega=0\) and \(\alpha=1\), the problem (1) corresponds to the usual Navier-Stokes equations: \begin{equation*} \left\{ \begin{array}{l} u_{t}-\mu\Delta u+(u.\nabla)u+\nabla \pi=0\;\;\;\;(t,x)\in \mathbb{R}^{+}\times \mathbb{R}^{3},\\ \nabla.u = 0,\\ u(0,x) = u_{0}(x)\;\;x\in \mathbb{R}^{3}\,. \end{array} \right. \end{equation*} Throughout this paper, we use \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) to denote the homogenous Fourier Besov-Morrey spaces, \(C\) will denote constants which can be different at different places, \({\mathsf U}\lesssim{\mathsf V}\) means that there exists a constant \(C>0\) such that \({\mathsf U}\leq C{\mathsf V}\), and \(p'\) is the conjugate of \(p\) satisfying \(\frac{1}{p}+\frac{1}{p'}= 1\) for \(1\leq p\leq\infty\). Inspired by the works [6, 10, 11, 12, 13, 14], the aim of this paper is to prove the global existence and the decay property and the stability of the global solutions of the fractional Navier-Stokes equations with Coriolis force (1) in the Fourier-Besov-Morrey space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}(\mathbb{R}^{3})\).

2. Preliminaries and main results

The results presented in this work are based on homogeneous Littlewood-Paley decomposition in the Fourier variables. We evoke briefly this construction below. We begin by dyadic decomposition of \(\mathcal {\mathbb{R}}^n\). Choose two nonnegative smooth radial functions \(\chi ,\;\varphi\) satisfying \begin{gather*} \operatorname{supp}\varphi \subset \{\xi\in {\mathbb{R}}^n:\frac 34\leq|\xi|\leq \frac 83\},\quad \sum_{j\in \mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n\backslash\{0\},\\ \operatorname{supp}\chi \subset \{\xi\in {\mathbb{R}}^n:|\xi|\leq \frac 43\},\quad \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n\,. \end{gather*} We denote \(\varphi_{j}(\xi)=\varphi(2^{-j}\xi)\) and \(\mathcal{P}\) the set of all polynomials. The space of tempered distributions is denoted by \(S'\). The homogeneous dyadic blocks \(\dot{\Delta}_{j}\) and \(\dot{S}_{j}\) are defined for all \(j\in\mathbb{Z}\) by \begin{equation*} \label{e2.1} \begin{gathered} \dot{\Delta}_ju=\varphi(2^{-j}D)u=2^{jn}\int h(2^jy)u(x-y)\,dy, \\ \dot{S}_ju=\sum_{k\leq j-1}\dot{\Delta}_ku=\chi(2^{-j}D)u=2^{jn}\int \tilde{h}(2^jy)u(x-y)\,dy, \end{gathered} \end{equation*} where \(h=\mathcal{F}^{-1}\varphi\) and \(\tilde{h}=\mathcal{F}^{-1}\chi\). First, we give the definition of the Morrey spaces which are a complement to the \(L^{p}\) spaces.

Definition 1.[14, 15] For \(1\leq p< \infty\), \(0\leq\lambda< n\), the Morrey spaces \(\mathrm{M}_{p}^{\lambda}=\mathrm{M}_{p}^{\lambda}(\mathbb{R}^{n})\) is defined as the set of functions \(f\in L_{loc}^{p}(\mathbb{R}^{n})\) such that

where \(B(x_{0},r)\) denotes the ball in \(\mathbb{R}^{n}\) with center \(x_{0}\) and radius \(r\). It is easy to observe that the relation \(\mathrm{M}_{p_{1}}^{\lambda}\hookrightarrow \mathrm{M}_{p_{2}}^{\mu}\) provided \(\frac{n-\mu}{p_{2}}\geq\frac{n-\lambda}{p_{1}}\) and \( p_{2}\leq p_{1}\), and \(\mathrm{M}_{p}^{0}=L^{p}\).
If \(1\leq p_{1},p_{2},p_{3}< \infty\) and \( 0\leq\lambda_{1},\;\lambda_{2},\,\lambda_{3}< n\) with \( \frac{1}{p_{3}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\) and \( \frac{\lambda_{3}}{p_{3}}=\frac{\lambda_{1}}{p_{1}}+\frac{\lambda_{2}}{p_{2}}\), then we have the Hölder inequality \begin{equation*} \|fg\|_{\mathrm{M}_{p_{3}}^{\lambda_{3}}}\leq\|f\|_{\mathrm{M}_{p_{1}}^{\lambda_{1}}} \|g\|_{\mathrm{M}_{p_{2}}^{\lambda_{2}}}\,. \end{equation*} Also, for \(1\leq p< \infty\) and \(0\leq\lambda< n,\) for all \(\varphi\in L^{1}\) and \(g\in\mathrm{M}_{p}^{\lambda}\).

Definition 2.(Homogeneous Besov-Morrey spaces ) Let \(s\in\mathbb{R}\), \(1\leq p< +\infty\), \(1\leq q\leq+\infty\), and \(0\leq\lambda< n\), the space \(\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) is defined by \begin{equation*} \mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})=\Big\{u\in \mathcal{Z}'(\mathbb{R}^{n});\;\;\;\| u\| _{\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})}< \infty\Big\}\,. \end{equation*} Here\[ \|u\|_{\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})} = \left\{ \begin{array}{l l}\label{nc} \Big\{\underset{j\in\mathbb{Z}}{\sum}2^{jqs}\|\dot{\Delta}_{j}u\| _{\mathrm{M}_{p}^{\lambda}} ^q \Big\}^{1/q} & \quad for\;\;q< \infty,\\ \underset{j\in\mathbb{Z}}{\sup}2^{js}\|\dot{\Delta}_{j}u\| _{\mathrm{M}_{p}^{\lambda}}& \quad for \;\;q=\infty\,.\\ \end{array}\right.\] The space \(\mathcal{Z}'(\mathbb{R}^{n})\) denotes the topological dual of the space \(\mathcal{Z}(\mathbb{R}^{n})=\big\{f\in\mathcal{S}(\mathbb{R}^{n});\partial^{\alpha }\widehat{f}(0)=0\text{ for every multi-index }\alpha\big\},\) and can be identified to the quotient space \(\mathcal{S'}(\mathbb{R}^{n})/\mathcal{P}\), where \(\mathcal{P}\) represents the set of all polynomials on \(\mathbb{R}^{n}.\) We refer to [16, chap.8] for more details.

Definition 3. (Homogeneous Fourier-Besov-Morrey spaces ) Let \(s\in\mathbb{R}, \;0\leq\lambda< n\), \(1\leq p< +\infty\) and \(1\leq q\leq+\infty\). The space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) denotes the set of all \(u\in \mathcal{Z'}(\mathbb{R}^{n}) \) such that

with appropriate modifications made when \(q = \infty\).
Note that the space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) equipped with the norm (4) is a Banach space. Since \(\mathrm{M}_{p}^{0}=L^{p}\), we have \(\mathcal{F} \dot{\mathcal{N}}_{p, 0, q}^{s}=F \dot{B}_{p, q}^{s}, \, \mathcal{F} \dot{\mathcal{N}}_{1, 0, q}^{s}=F \dot{B}_{1, q}^{s}=\dot{\mathcal{B}}_{q}^{s}\) and \(\mathcal{F} \dot{\mathcal{N}}_{1, 0, 1}^{-1}=\chi^{-1}\) where \(\dot{\mathcal{B}}_{q}^{s}\) is the Fourier-Herz space and \(\chi^{-1}\) is the Lei-Lin space [17, 18].

Now, we give the definition of the mixed space-time spaces.

Definition 4. Let \(s\in\mathbb{ R},\;1\leq p< \infty,\; 1\leq q,\rho\leq\infty, \;0\leq\lambda< n\), and \(I=[0,T),\;T\in(0,\infty]\). The space-time norm is defined on \(u(t,x)\) by \begin{eqnarray*} \|u(t,x)\|_{\mathcal{L}^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}= \Big\{\sum_{j\in \mathbb{Z}}2^{jqs}\| \widehat{\dot{\Delta}_{j}u}\| _{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} ^q \Big\}^{1/q}, \end{eqnarray*} and denote by \(\mathcal{L}^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\) the set of distributions in \(S'(\mathbb{R}\times\mathbb{R}^{n})/\mathcal{P}\) with finite \(\|.\|_{\mathcal{L}^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}\) norm.
According to Minkowski's inequality, we have

where \(\|u(t,x)\|_{L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})} :=\Big(\int_I\|u(\tau,\cdot)\|^\rho_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}}d\tau\Big)^{1/\rho}\,.\)

Our main result is the following theorem.

Theorem 5. Let \(\Omega\in\mathbb{ R},\;0\leq\lambda< 3\) and \(1\leq q\leq 2\).
For \(\max\{1,\frac{3-\lambda}{2}\}\leq p< \infty\) and \(\frac{2}{3}< \alpha\leq\frac{2}{3}+\frac{1}{p'}+\frac{\lambda}{3p}\), there exists a positive time \(T\) such that for any \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) and \(\nabla.u_{0}=0\), the equation (1) admits a unique local solution \(u\) in \(\mathcal{L}^{4}\Big([0,T),\mathcal{F\dot{N}}_{p, \lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big).\)
Furthermore, for all \(1\leq p< \infty\) and \(\frac{1}{2}< \alpha\leq1+\frac{3}{2p'}+\frac{\lambda}{2p}\) there exists a constant \(C_{0}(p,q)\) such that for any \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) satisfying \(\nabla.u_{0}=0\) and \( \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}< C_{0}\mu\), the equation (1) admits a unique global solution \begin{eqnarray*} u\in \mathcal{C}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\,, \end{eqnarray*} and it satisfies \begin{equation*} \|u\|_{\mathcal{L}^{\infty}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} +\mu\|u\|_{\mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \leq2C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,, \end{equation*} where \(C\) is a positive constant.

Now, we give some remarks about this result.

Remark 1. When \(\alpha=1\), there are a different results which investigate the existence of a unique global solution to the Navier-stokes equations with Coriolis forces, especially in Fourier-Herz spaces \(\mathcal{\dot{B}}_{2}^{-1}\) [6, 19] , in Lei-lin spaces \(\chi^{-1}\) [20] and in Fourier-Besov spaces \(\mathrm{F\dot{B}}_{p,\infty}^{2-\frac{3}{p}}\) [13]. Theorem 5 is an extension and an improvement of these works to the Fourier-Besov-Morrey space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1 -2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}(\mathbb{R}^{3})\).
We note that the fractional Navier-Stokes-Coriolis system is well-posed uniformly in the sense that the smallness condition is independent of \(\Omega\).

Theorem 6. Let \(\Omega\in\mathbb{R},\,1\leq p,q\leq2,\,0\leq\lambda\leq3-\frac{3}{2}p\) and \(\frac{5}{6}< \alpha\leq1\). Assume that \(u\in \mathcal{C}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\) is a global solution of the system (1) given by Theorem 5, then

Remark Theorem 6 specifies the asymptotic behavior of a given global solution for (1) in the space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\). The long time decay property is also valid in the case \(\Omega=0\) where the equation (1) is reduced to the fractional Navier-Stokes equation. At present, we are unable to establish that (6) still holds true for \(\frac{1}{2}< \alpha\leq \frac{5}{6}\). The principal reason is that the proof is largely based on the lemma 14.

Theorem 7. Let \(T^*\) denote the maximal time of existence of a solution \(u\) in
\(\mathcal{L}^{\infty}\Big([0,T^*);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\,.\) If \(T^*< \infty\), then \begin{align*} \|u\|_{\mathcal{L}^{1}\Big([0,T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}=\infty. \end{align*} Besides; if \(u\in C\Big(\mathbb{R}^{+},\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big) \) is a global solution of (1), and for all \(v_{0}\in\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) satisfying

for some constant \(C_{0}\) sufficiently small, then the fractional Navier-Stokes-Coriolis system starting by \(v_{0}\) has a global solution \(v\) fulfilling the inequality \begin{equation*} \begin{aligned} & \|v(t)-u(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\frac{\mu}{2} \|v(s)-u(s)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\ & < C\|v_{0}-u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\exp\Big\{\int_{0}^{\infty}C\Big(|\Omega|+\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}}\Big)\Big\} \end{aligned} \end{equation*} where \(C\) is a positive constant.

Remark 3. In the case \(\Omega=0\) and \(\alpha=1\), the result of stability of global solutions for the usual Navier-Stokes equations is developed by several researchers in different function spaces such as \(\mathrm{H^{1}},\,\mathrm{L^{3}},\,\mathrm{\dot{B}}_{p,q}^{\frac{3}{p}-1}\) and \(\chi^{-1}\) [11, 21, 22, 23]. To show Theorem 7, we adapt the method of the above works to our problem (1) in the spaces \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\). Theorem 7 extends the works of [11, 21, 23] to a more general frame.

We finish this section with a Bernstein type lemma in Fourier variables in Morrey spaces.

Lemma 8.[24] Let \(1\leq q\leq p< \infty,\, 0\leq\lambda_{1},\lambda_{2}< n,\;\frac{n-\lambda_{1}}{p}\leq\frac{n-\lambda_{2}}{q}\), and let \(\gamma\) be a multiindex. If \(supp(\widehat{f})\subset\{|\xi|\leq A2^{j}\}\) then there is a constant \(C>0\) independent of \(f\) and \(j\) such that

3. Well-posedness

In order to solve (1), we consider the following integral equation: \begin{align*}u(t)=T_{\Omega,\alpha}(t)u_{0}- \int_{0}^{t}T_{\Omega,\alpha}(t-\tau)\mathbb{P}\nabla\cdot(u\otimes u)d\tau, \end{align*} where \(\mathbb{P}=(\delta_{ij}+R_{i}R_{j})_{1\leq i,j\leq 3}\) denotes the Helmholtz projection onto the divergence-free vector fields, which is a pseudo differential operator of order \(0\), and \(T_{\Omega,\alpha}(\cdot)\) denotes the Stokes-Coriolis semigroup corresponding to the following linear Stokes problem with Coriolis force \begin{equation*} \left\{\textstyle \begin{array}{l} u_{t}+\mu(-\Delta)^{\alpha} u+\Omega e_{3}\times u +\nabla \pi=0\;\;\;\;(t,x)\in \mathbb{R}^{+}\times \mathbb{R}^{3},\\ \nabla.u = 0,\\ u(0,x) = u_{0}(x)\;\;x\in \mathbb{R}^{3}\,.\\ \end{array} \displaystyle \right. \end{equation*} Besides, \(T_{\Omega,1}(\cdot)\) is given explicitly by \begin{align*}T_{\Omega,1}(t)f= \mathcal{F}^{-1}[\cos(\Omega\frac{\xi_{3}}{|\xi|}t)I +\sin(\Omega\frac{\xi_{3}}{|\xi|}t)R(\xi)]*(e^{\mu\Delta t}f) \end{align*} for \(t\geq 0\) and divergence-free vector fields \(f\). Here, \(I\) is the identity matrix in \(\mathbb{R}^{3}\) and \(R(\xi)\) is the skew-symmetric matrix symbol related to the Riesz transform, which is defined by \[R(\xi):= \frac{1}{|\xi|}\left(\begin{array}{lcr} 0&\xi_{3}&-\xi_{2}\\ -\xi_{3}&0&\xi_{1}\\ \xi_{2}&-\xi_{1}&0 \end{array}\right).\] We refer to Babin-Mahalov-Nikolaenko [25, 26, 27], Giga-Inui-Mahalov-Saal [8] and Hieber- Shibata [3] for the derivation of the explicit form of \(T_{\Omega,1}(\cdot)\). For \(\alpha\in\big(\frac{2}{3},\frac{2}{3}+\frac{1}{p'}+\frac{\lambda}{3p'}\big]\), the same argument given in [3, 10] gives \begin{align*}T_{\Omega,\alpha}(t)f= \mathcal{F}^{-1}[\cos(\Omega\frac{\xi_{3}}{|\xi|}t)I +\sin(\Omega\frac{\xi_{3}}{|\xi|}t)R(\xi)]*(e^{-\mu(-\Delta)^{\alpha} t}f). \end{align*} The first estimate corresponds to the Stokes-Coriolis semigroup \(T_{\Omega,\alpha}\).

Lemma 9. Let \(0< T\leq \infty,\,s\in\mathbb{R},\,0\leq\lambda< 3, 1\leq p< \infty,\, 1\leq q, \rho, r\leq\infty\) and \(f\in\mathcal{L}^{r}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\). There exists a constant \(C>0\) such that \(\|\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)f(\tau)d\tau\|_{\mathcal{L}^{\rho}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s })}\leq C\|f\|_{\mathcal{L}^{r}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s-2\alpha-\frac{2\alpha}{\rho}+\frac{2\alpha}{r} })}.\)

Proof. Set \(1+\frac{1}{\rho}=\frac{1}{\tilde{\rho}}+\frac{1}{r}.\) The definition of the space-time norm of \(\mathcal{L}^{\rho}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\) and Young's inequality give \begin{eqnarray*} \Big\|\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)f(\tau)d\tau\Big\|_{\mathcal{L}^{\rho}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s })}&=&\Big\{\sum_{j\in\mathbb{Z}}2^{jqs}\Big(\int_{0}^{T}\|\varphi_{j}\int_{0}^{t}\mathcal{F}(T_{\Omega,\alpha}(t-\tau)f)(\tau)d\tau\| _{\mathrm{M}_{p}^{\lambda}}^\rho dt\Big)^{\frac{q}{\rho}}\Big\}^{1/q}\\ &\leq& \Big\{\sum_{j\in\mathbb{Z}}2^{jqs}\Big(\int_{0}^{T}\|\varphi_{j} \int_{0}^{t}e^{-\mu|\xi|^{2\alpha}(t-\tau)}\hat{f}(\tau)d\tau\| _{\mathrm{M}_{p}^{\lambda}}^\rho dt\Big)^{\frac{q}{\rho}} \Big\}^{1/q}\\ &\leq& \Big\{\sum_{j\in\mathbb{Z}}2^{jqs}\Big(\int_{0}^{T} \|\varphi_{j}\int_{0}^{t}e^{-\mu2^{2\alpha j}(t-\tau)}\hat{f}(\tau)d\tau\| _{\mathrm{M}_{p}^{\lambda}}^\rho dt\Big)^{\frac{q}{\rho}} \Big\}^{1/q}\\ &\leq& \Big\{\sum_{j\in\mathbb{Z}}2^{jqs}\Big(\int_{0}^{T}e^{-t\mu\tilde{\rho}2^{2\alpha j}}dt\Big)^{\frac{q}{\tilde{\rho}}} \|\varphi_{j}\hat{f}(\tau)\| _{L^{r}([0,T),\mathrm{M}_{p}^{\lambda})}^{q} \Big\}^{1/q}\\ &\leq& C \Big\{\sum_{j\in\mathbb{Z}}2^{jq(s-2\alpha-\frac{2\alpha}{\rho}+\frac{2\alpha}{r})}\|\varphi_{j}\hat{f}(\tau)\| _{L^{r}([0,T),\mathrm{M}_{p}^{\lambda})}^{q} \Big\}^{1/q}\\ &\leq&C\|f\|_{\mathcal{L}^{r}([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{s-2\alpha-\frac{2\alpha}{\rho}+\frac{2\alpha}{r}}}. \end{eqnarray*}

Lemma 10. Let \(I=[0,T),\, 0< T\leq \infty,\,0\leq\lambda< 3,\,1\leq p< \infty,\,1\leq q\leq\infty\) and \(u_{0}\in\mathcal{FN}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}(\mathbb{R}^{3})\). Then there exists a constant \(C>0\) such that

Proof. To prove the first inequality (9), it suffices to write that $$ \|T_{\Omega,\alpha}(t)u_{0}\|_{\mathcal{L}^{\infty}\Big([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \leq \Big(\sum_{j\in\mathbb{Z}}2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})q}\|\varphi_{j}\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}^{q}\Big)^{\frac{1}{q}} \leq C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. $$ In order to prove the second estimate (10), we write $$ \|T_{\Omega,\alpha}(t)u_{0}\|_{\mathcal{L}^{1}\Big([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big) } \leq \Big(\sum_{j\in\mathbb{Z}}2^{j(1+\frac{3}{p'}+\frac{\lambda}{p})q}\Big(\int_{0}^{T}e^{-t\mu2^{2\alpha j}} \|\varphi_{j}\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}dt\Big)^{q}\Big)^{\frac{1}{q}} \leq C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. $$ To estimate the inequality (11), it suffices to write that \begin{eqnarray*} \|T_{\Omega,\alpha}(t)u_{0}\|_{\mathcal{L}^{4}\Big([0,T),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big) }&\leq& \Big(\sum_{j\in\mathbb{Z}}2^{j(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})q}\Big(\int_{0}^{T}e^{-t\mu2^{2\alpha j+2}} \|\varphi_{j}\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}^{4}dt\Big)^{\frac{q}{4}}\Big)^{\frac{1}{q}}\\ &\leq&C \Big(\sum_{j\in\mathbb{Z}}2^{j(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})q} 2^{-\frac{1}{2}\alpha jq}\|\varphi_{j}\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}^{q}\Big)^{\frac{1}{q}} \leq C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. \end{eqnarray*}

Proposition 11. Let \(0\leq\lambda< 3,\,\max\{1,\frac{3-\lambda}{2}\}\leq p< \infty,\,1\leq q \leq 2,\,I=[0,T),\,0< T\leq+\infty\) and \(\frac{2}{3}< \alpha\leq\frac{2}{3}+\frac{1}{p'}+\frac{\lambda}{3p}\), and set \begin{equation*} Y=\mathcal{L}^4\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big), \end{equation*} there exists a constant \(C=C(p,q)>0\) depending on \(p,q\) such that

Proof. We need to introduce some notations about the standard localization operators. We set \begin{align*} u_{j}=\dot{\Delta}_{j}u,\;\;\;\dot{S}_{j}u=\sum_{k\leq j-1}\dot{\Delta}_{k}u,\;\;\; \widetilde{\dot{\Delta}}_{j}u=\sum_{|k-j|\leq 1}\dot{\Delta}_{k}u,\;\;\; j\in \mathbb{Z}\,. \end{align*} Applying Bony paraproduct decomposition and quasi-orthogonality property for Littlewood-Paley decomposition, for fixed \(j\), we obtain \begin{align*} \dot{\Delta}_{j}(uv) &=\sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}u \dot{\Delta}_{k}v)+ \sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}v \dot{\Delta}_{k}u)+\sum_{k\geq j-3}\dot{\Delta}_{j}(\dot{\Delta}_{k}u \widetilde{\dot{\Delta}}_{k}v)\\ &=I_{j}+II_{j}+III_{j}\,. \end{align*} The triangular inequality gives

We evaluate the above three terms individually. First, using Young's inequality (3) and Lemma 8 with \(|\gamma|=0\), we get \begin{eqnarray} 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\|\widehat{I_{j}}\|_{L^{2}(I ,\mathrm{M}_{p}^{\lambda})} &\leq & 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{|k-j|\leq 4}\|\widehat{\dot{S}_{k-1}u\dot{\Delta}_{k}v}\|_{L^{2}(I ,\mathrm{M}_{p}^{\lambda})}\\ &\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})} \sum_{|k-j|\leq 4}\|\widehat{v}_{k}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}\| \widehat{u}_{l}\|_{L^{4}(I ,L^{1})}\\ &\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{|k-j|\leq 4}\|\widehat{v}_{k}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\| \widehat{u}_{l}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda})}\\ &\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{|k-j|\leq 4}\|\widehat{v}_{k}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}2^{(1-\frac{3}{2}\alpha)l}2^{(\frac{3}{2}\alpha-1)l}\| \widehat{u}_{l}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda})}\\ &\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{|k-j|\leq 4}\|\widehat{v}_{k}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda} )}\Big(\sum_{l\leq k-2}2^{(\frac{3}{2}\alpha-1)lq'}\Big)^{\frac{1}{q'}} \|u\|_{ \mathcal{L}^4\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\ &\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{|k-j|\leq 4}2^{(\frac{3}{2}\alpha-1)k}\|\widehat{v}_{k}\|_{L^{4}(I ,\mathrm{M}_{p}^{\lambda} )} \|u\|_{ \mathcal{L}^4\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}. \end{eqnarray} Applying \(l^{q}-\)norm on both sides in the above estimate we obtain Likewise, we prove that To estimate \(III_{j}\), the so-called "remainder term", we use a different approach. Let \begin{eqnarray*} III_{jk}:=\dot{\Delta}_{j}\Big(\sum_{|i-k|\leq 1}\dot{\Delta}_{i}v\dot{\Delta}_{k}u\Big)=\sum_{i=-1}^{1}\dot{\Delta}_{j}(\dot{\Delta}_{k}u\dot{\Delta}_{i+k}v). \end{eqnarray*} First, we use Young's inequality (3) and Lemma 8 with \(|\gamma|=0\) to obtain \begin{eqnarray*} \label{SIII2} 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\|\widehat{III_{j}}\|_{L^{2}(I ,\mathrm{M}_{p}^{\lambda})}&\leq& 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{k\geq j-3}\|\widehat{III_{jk}}\|_{L^{2}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ & \leq & 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{k\geq j-3}\sum_{|k-i|\leq 1} \big\|\widehat{\dot{\Delta}_{k}u\dot{\Delta}_{i}v}\big\|_{L^{2}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ & \leq & 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{k\geq j-3}\sum_{|k-i|\leq 1} \big\|\widehat{u}_{k}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\big\|\widehat{v}_{i}\big\|_{L^{4}(I,L^{1})}\nonumber\\ & \leq & 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{k\geq j-3}\sum_{|k-i|\leq 1} 2^{i(\frac{3}{p'}+\frac{\lambda}{p})}\big\|\widehat{u}_{k}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})} \big\|\widehat{v}_{i}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ & \leq & C 2^{j(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\sum_{k\geq j-3}\Big(\sum_{|k-i|\leq 1}2^{iq(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})} \big\|\widehat{v}_{i}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}^{q}\Big)^{\frac{1}{q}}\\ && \times2^{(\frac{3}{2}\alpha-1)k}\big\|\widehat{u}_{k}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ & \leq & C \|v\|_{\mathcal{L}^4\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \sum_{k\geq j-3}2^{(j-k)(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p})}2^{k(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})} \big\|\widehat{u}_{k}\big\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\,. \end{eqnarray*} When \(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p}>0\), we take the \(l^{q}-\)norm on both sides in the above estimate, and then we apply Young's inequality for series to get For the case \(2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p}=0\), we write \begin{eqnarray*}\label{IIII2} \Big(\sum_{j\in\mathbb{Z}} \|\widehat{III_{j}}\|_{L^{2}(I,\mathrm{M}_{p}^{\lambda})}^{q}\Big)^{\frac{1}{q}}&\leq&\sum_{j\in\mathbb{Z}}\Big(\sum_{k\geq j-3}\Big(\int_{I} \|\varphi_{j}(\xi)\times\sum_{i=-1}^{1} \hat{u}_{k}*\hat{v}_{k+i}\|_{\mathrm{M}_{p}^{\lambda}}^{2}\Big)^{\frac{1}{2}}\Big)\nonumber\\ &\leq&\sup_{\xi}\Big(\sum_{j\in\mathbb{Z}}\varphi_{j}(\xi)\Big)\sum_{k\in\mathbb{Z}}\Big(\int_{I} \|\sum_{i=-1}^{1} \hat{u}_{k}*\hat{v}_{k+i}\|_{\mathrm{M}_{p}^{\lambda}}^{2}\Big)^{\frac{1}{2}}\Big)\nonumber\\ &\leq&\sum_{i=-1}^{1}\sum_{k\in\mathbb{Z}}\Big( \| \hat{u}_{k}\|_{L^{4}(I,L^{1})}\|\hat{v}_{k+i}\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\Big)\nonumber\\ &\leq&\sum_{i=-1}^{1}\sum_{k\in\mathbb{Z}}\Big( 2^{k(\frac{3}{p'}+\frac{\lambda}{p})} \|\hat{u}_{k}\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})} \|\hat{v}_{k+i}\|_{L^{4}(I,\mathrm{M}_{p}^{\lambda})}\Big)\nonumber \end{eqnarray*} where we have used the fact \(1\leq q\leq2\) implies \(\mathcal{F\dot{N}}_{p,\lambda,q}^{\frac{3}{2}\alpha-1}\hookrightarrow \mathcal{F\dot{N}}_{p,\lambda,q'}^{\frac{3}{2}\alpha-1}\) with \(q'\) is the conjugate of \(q\). Estimates (13), (14), (15), (16) and (17) yield (12).

Proposition 12. Let \(1\leq p< \infty,\,1\leq \rho\leq \infty,\,1\leq q \leq 2,\,0\leq\lambda< 3\) and \(\frac{1}{2}< \alpha\leq\frac{2+\frac{3}{p'}+\frac{\lambda}{p}}{4-\frac{2}{\rho}}\), and set \begin{equation*} X=\mathcal{L}^{\infty}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{\rho}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{2\alpha}{\rho}+\frac{\lambda}{p}}\Big), \end{equation*} with the norm \begin{equation*} \|u\|_{X}=\|u\|_{\mathcal{L}^{\infty}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} +\mu\|u\|_{\mathcal{L}^{\rho}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{2\alpha}{\rho}+\frac{\lambda}{p}}\Big) }\,. \end{equation*} There exists a constant \(C=C(\alpha,p,q)>0\) depending on \(\alpha,p,q\) such that

Proof. This proposition is given in [28] for \(\frac{1}{2}< \alpha< \frac{2+\frac{3}{p'}+\frac{\lambda}{p}}{4-\frac{2}{\rho}}\). For the case \(\alpha=\frac{2+\frac{3}{p'}+\frac{\lambda}{p}}{4-\frac{2}{\rho}}\) the argument is similar to the method described for (17).

Now, we give an abstract lemma on the existence of fixed point solutions.

Lemma 13. Let \(X\) be a Banach space with norm \(\|.\|_{X}\) and \(B:X\times X\longmapsto X\) be a bounded bilinear operator satisfying \begin{equation*} \|B(u,v)\|_{X}\leq \eta \|u\|_{X}\|v\|_{X} \end{equation*} for all \(u,v\in X \) and a constant \(\eta >0\). Then, if \(0< \varepsilon< \frac{1}{4\eta}\) and if \(y\in X\) such that \(\|y\|_{X}\leq\varepsilon\), the equation \(x:=y+B(x,x)\) has a solution \(\overline{x}\) in \(X\) such that \(\|\overline{x}\|_{X}\leq 2 \varepsilon\). This solution is the only one in the ball \(\overline{B}(0,2\varepsilon)\). Moreover, the solution depends continuously on \(y\) in the sense: if \(\|y'\|_{X}< \varepsilon ,\;x'=y'+B(x',x')\), and \(\|x'\|_{X}\leq2\varepsilon\), then \begin{equation*} \|\overline{x}-x'\|_{X}\leq \frac{1}{1-4\varepsilon \eta}\|y-y'\|_{X}\,. \end{equation*}

Proof. [Proof of Theorem 5] We will use Lemma 13 to sketch the proof of the existence results. The functions here are vector fields, whose norm is the sum of the norms of the three components.\\ For the local existence, we set \begin{equation*} Y=\mathcal{L}^{4}\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big),\,I=[0,T)\,. \end{equation*} Here, as usual, we begin with the mild integral equation

and we consider the bilinear operator \(B\) given by \begin{eqnarray*} B(u,v)=\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)\mathbb{P}\nabla.(u\otimes v)(\tau)d\tau\,. \end{eqnarray*} According to Lemma 9 and Proposition 11 we obtain \begin{eqnarray*} \|B(u,v)\|_{\mathcal{L}^{4}\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}&=& \Big\|\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)\mathbb{P}\nabla.(u\otimes v)(\tau)d\tau\Big\|_{\mathcal{L}^{4}\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\ &\leq& C\|\nabla.(u\otimes v)\|_{\mathcal{L}^{2}\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-3\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\leq C\|uv\|_{\mathcal{L}^{2}\Big(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{2-3\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\leq C\|u\|_{Y}\|v\|_{Y}\,. \end{eqnarray*} Lemma 10 yields Now, we shall decompose the initial data \(u_{0}\) into two terms \begin{align*}u_{0}=\mathcal{F}^{-1}(\chi_{B(0,\delta)}\hat{u_{0}})+\mathcal{F}^{-1}(\chi_{B^{C}(0,\delta)}\hat{u_{0}}) :=u_{0,1}+u_{0,2}, \end{align*} where \(\delta=\delta(u_{0})>0\) is a real number. Since \(u_{0,2}\) converge to \(0\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) as \(\delta\rightarrow+\infty\), by (21) there exists \(\delta\) large enough such that \begin{eqnarray*} \big\|T_{\Omega,\alpha}(t)u_{0,2}\big\|_{Y}\leq\frac{1}{8C}\,. \end{eqnarray*} For the first term \(u_{0,1}\), \begin{align*} \Big\|T_{\Omega,\alpha}(t)u_{0,1}\Big\|_{Y} &\leq\Big\|2^{j(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})} \big\|\varphi_{j}e^{-\mu t|\xi|^{2\alpha}}\chi_{B(0,\delta)}\hat{u_{0}}\big\|_{{L}^{4} (I,\mathrm{M}_{p}^{\lambda})}\Big\|_{\ell^{q}}\\ &\leq\Big\|2^{j(1-\frac{3}{2}\alpha+\frac{3}{p'}+\frac{\lambda}{p})} \big\|\sup_{\xi\in B(0,\delta)}e^{-\mu t|\xi|^{2\alpha}}|\xi|^{\frac{\alpha}{2}}\big\|_{L^{4}([0,T))} \|\varphi_{j}|\xi|^{-\frac{\alpha}{2}}\hat{u_{0}}\|_{\mathrm{M}_{p}^{\lambda}}\Big\|_{\ell^{q}}\\ &\leq C \delta^{\frac{\alpha}{2}}T^{\frac{1}{4}} \Big\|u_{0}\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. \end{align*} Thus for arbitrary \(u_{0}\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\), (20) has a unique local solution in \(Y\) on \([0,T)\) where \begin{align*}T\leq \Big(\frac{1}{8C^{2}\delta^{\frac{\alpha}{2}}\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha +\frac{3}{p'}+\frac{\lambda}{p}}}}\Big)^{4}\,. \end{align*} For the global existence, we will again use Lemma 13 to ensure the existence of global mild solution with small initial data in the Banach space \(X\) given by \begin{equation*} X=\mathcal{L}^{\infty}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\,. \end{equation*} According to Lemma 9 and Proposition 12, we obtain \begin{eqnarray*} \|B(u,v)\|_{\mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} &=& \Big\|\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)\mathbb{P}\nabla.(u\otimes v)(\tau)d\tau\Big\|_{\mathcal{L}^{1}([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1 +\frac{3}{p'}+\frac{\lambda}{p}})}\\ &\leq& C\|\nabla.(u\otimes v)\|_{\mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\leq C\mu^{-1}\|u\|_{X}\|v\|_{X}\,. \end{eqnarray*} Similarly, \begin{eqnarray*} \|B(u,v)\|_{\mathcal{L}^{\infty}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} &=& \Big\|\int_{0}^{t}T_{\Omega,\alpha}(t-\tau)\mathbb{P}\nabla.(u\otimes v)(\tau)d\tau\Big\|_{\mathcal{L}^{\infty}([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}})}\\ &\leq& C\|\nabla.(u\otimes v)\|_{\mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\ &\leq& C\mu^{-1}\|u\|_{X}\|v\|_{X}\,. \end{eqnarray*} Finally, \begin{eqnarray*} \|B(u,v)\|_{X}\leq C\mu^{-1}\|u\|_{X}\|v\|_{X}\,. \end{eqnarray*} Lemma 10 yields \begin{eqnarray*} \|T_{\Omega,\alpha}(t)u_{0}\|_{X}\leq C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. \end{eqnarray*} If \(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}< C_{0}\mu\) with \(C_{0}=\frac{1}{4C^{2}}\), then (20) has a unique global solution \(u\in X\) satisfying \begin{eqnarray*} \|u\|_{X} \leq 2 C \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. \end{eqnarray*} This completes the proof of Theorem 5.

4. The decay property

In this section, we first present the following interpolation inequalities which have their specific utility in the sequel.

Lemma 14.[18] Let \(\alpha< \frac{5}{4},\,s>\frac{5}{2}-2\alpha,\,1\leq p,q\leq2\) and \(0\leq\lambda\leq3-\frac{3}{2}p\). Then we have \begin{align*} \|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'} +\frac{\lambda}{p}}} \lesssim\|u\|_{L^2}^{1-\frac{5/2-2\alpha}{s}} \|u\|_{\dot{H}^{s}}^{\frac{5/2-2\alpha}{s}}\,. \end{align*}

Lemma 15[18] Let \(1\leq p,q\leq2\) and \(\frac{1}{2}< \alpha\leq1\). Then we have

Proof. [Proof of Theorem 6] In this part, we will focus on the asymptotic behavior of global solutions when \(t \rightarrow \infty\), which was developed in different papers such as [11, 29, 30] and [31, chap.11]. For \(k\in \mathbb{N}\), define \begin{eqnarray*} \mathcal{A}_{k}=\{\xi\in\mathbb{R}^{3};|\xi|\leq k\; and\; |\hat{u}_{0}(\xi)|\leq k\}\,. \end{eqnarray*} Obviously \(\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_{0})\) converge to \(u_{0}\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) as \(k\rightarrow +\infty\).\\ Then, there exists \(k\in \mathbb{N}\) such that \begin{eqnarray*} \|u_{0}-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_{0})\|_{\mathcal{F\dot{N}}_{p, \lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \leq\frac{\varepsilon}{2}\,. \end{eqnarray*} Put \begin{eqnarray*} u_{0,k}=\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_{0}),\,\, w_{0,k}=u_{0}-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_{0})\,. \end{eqnarray*} Then \(u_{0,k}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\cap \mathrm{L}^{2}\) and

Now, we insert the following system For \(\frac{\varepsilon}{2}\leq C_{0}\mu\), we infer from Theorem 5 that the system (24) has a unique global solution such that \begin{equation*} w_{k}\in \mathcal{C}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\,. \end{equation*} Moreover, for any \(t\geq 0\) we have So let's consider the difference \(u_{k}=u-w_{k}\), which satisfies \begin{equation*} \left \{ \textstyle\begin{array}{l} \partial_{t}u_{k}+\mu(-\Delta)^{\alpha}u_{k}+\Omega e_{3}\times u_{k}+(u\cdot\nabla) u_{k} +(u_{k}\cdot\nabla) w_{k}+\nabla (\pi-\pi_{k})=0, \\ \nabla\cdot u_{k}=0,\\ \end{array}\displaystyle \right . \end{equation*} where \(\pi\) and \(\pi_{k}\) are the associated pressures to the solutions \(u\) and \(w_{k}\), respectively. By taking the inner products with \(u_{k}\) and integrating by parts, we can show that where we have used \(\Omega (e_{3}\times u_{k}).u_{k}=0.\)
Integrating by parts, Hölder's inequality and Lemma 15 lead to \begin{eqnarray*} \big|\big< \nabla.(u_{k}\otimes w_{k}),u_{k}\big>\big| &\leq&\big\|(-\Delta)^{\frac{1}{2}-\frac{\alpha}{2}}(u_{k}\otimes w_{k})\big\|_{L^{2}} \big\|(-\Delta)^{\frac{\alpha}{2}}u_{k}\big\|_{L^{2}}\\ &\leq& C\big\|u_{k}\otimes w_{k}\big\|_{\dot{H}^{1-\alpha}}\big\|u_{k}\big\|_{\dot{H}^{\alpha}} \leq C\big\|u_{k}\big\|_{L^{2}} \big\|w_{k}\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \big\|u_{k}\big\|_{\dot{H}^{\alpha}}\\ &&+C\big\|u_{k}\big\|_{\dot{H}^{\alpha}}^{2} \big\|w_{k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\ &\leq& \frac{2C^{2}}{\mu}\big\|u_{k}\big\|_{L^{2}}^{2} \big\|w_{k}\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{2} +\frac{\mu}{8}\big\|u_{k}\big\|_{\dot{H}^{\alpha}}^{2}+C\big\|u_{k}\big\|_{\dot{H}^{\alpha}}^{2} \big\|w_{k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,. \end{eqnarray*} By (23) and (25) we have \(\Big\|w_{k}\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\leq C\frac{\varepsilon}{2}\). In addition, we assume \(\varepsilon\) small enough such that \(C^{2}\varepsilon\leq\frac{\mu}{4}\), thus We conclude that \begin{equation*}\label{eh6} \begin{aligned} \frac{d}{dt}\| u_{k}\|_{L^2}^2 +\mu\| u_{k}\|_{\dot{H}^{\alpha}}^2 \leq \frac{8C^{2}}{\mu}\big\|u_{k}\big\|_{L^{2}}^{2} \big\|w_{k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{2}\,. \end{aligned} \end{equation*} Gronwall's inequality leads to Since \(q\leq2\), by Hölder's inequality, we get \begin{eqnarray*} \int_{0}^{t}\big\|w_{k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{2} &\leq& \Big\{\sum_{j\in\mathbb{Z}}2^{j(1-\alpha+\frac{3}{p'}+\frac{\lambda}{p})q}\Big(\int_{0}^{t}\|\varphi_{j}\hat{w}_{k}\|_{\mathrm{M}_{p}^{\lambda} }^{2}\Big)^{\frac{q}{2}}\Big\}^{2/q}\\ &\leq&\Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})\frac{q}{2}} 2^{j(1+\frac{3}{p'}+\frac{\lambda}{p})\frac{q}{2}} \|\varphi_{j}\hat{w}_{k}\|_{L^{\infty}([0,t),\mathrm{M}_{p}^{\lambda}) }^{\frac{q}{2}}\|\varphi_{j}\hat{w}_{k}\|_{L^{1}([0,t),\mathrm{M}_{p}^{\lambda}) }^{\frac{q}{2}}\Big\}^{2/q}\\ &\leq&\big\|w_{k}\big\|_{\mathcal{L}^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}})} \big\|w_{k}\big\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\ &\leq&\frac{1}{2\mu}\left(\big\|w_{k}\big\|_{\mathcal{L}^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+ \frac{\lambda}{p}})} +\mu\big\|w_{k}\big\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\right)^{2}\\ &\leq& \frac{C^{2}}{2\mu} \big\|w_{0,k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{2}\,, \end{eqnarray*} where we have used (25). Consequently Using Lemma 14 with \(s=\alpha\) we obtain \begin{equation*}\label{eh14} \int_{0}^{\infty}\big\|u_{k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{\frac{4\alpha}{5-4\alpha}} \lesssim C^{\frac{4\alpha}{5-4\alpha}}\mu^{-1}\big\|u_{0,k}\big\|_{L^{2}} ^{\frac{4\alpha}{5-4\alpha}}\\ \times \exp\Big\{\frac{8C^{4}\alpha}{\mu^{2}(5-4\alpha)} \big\|w_{0,k}\big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^{2}\Big\}\,. \end{equation*} The continuity of \(u_{k}\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\) ensure that there exists a time \(t_{0}\) such that \begin{equation*} \|u_{k}(t_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \leq\frac{\varepsilon}{2}\,. \end{equation*} Then we have \begin{equation*} \begin{aligned} \|u(t_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} &\leq \|u_{k}(t_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\|w_{k}(t_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\ &\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}\,. \end{aligned} \end{equation*} Now, we consider the fractional Navier-Stokes equations with Coriolis forces starting at \(t=t_{0}\) \begin{equation*} \left \{ \textstyle\begin{array}{l} u_{t}+u\cdot\nabla u+\mu(-\Delta)^{\alpha} u+\Omega e_{3}\times u+\nabla \pi=0, \\ \nabla\cdot u=0, \\ u(t_{0},x)=u(t_{0}). \end{array}\displaystyle \right . \end{equation*} By Theorem 5 and using the method mentioned in the proof of (25), we directly obtain \begin{eqnarray*} \|u(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\mu\|u\|_{\mathcal{L}^{1}\Big([t_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \leq C\|u(t_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \leq C\varepsilon\,, \end{eqnarray*} for all \(t\geq t_{0}\). We have completed the proof of Theorem 6.

5. Stability of global solutions

In this section we prove Theorem 7. Let \(T^*\) be the maximal existence time of a solution \(u\) of (1) in
\(\mathcal{L}^{\infty}\Big([0,T^*);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\,.\) In order to prove a blow-up criterion of the solution given by Theorem 5, assume that \(T^*< \infty\) and \(\|u\|_{\mathcal{L}^{1}([0,T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}})} < \infty\), then we can find \(0< T_{0}< T^*\) satisfying \begin{align*} \|u\|_{\mathcal{L}^{1}\Big([T_{0},T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} < \frac{1}{2}\,. \end{align*} For \(t\in[T_{0},T^*)\), we explicitly consider the integral equation \begin{eqnarray*} u(t)=T_{\Omega,\alpha}(t)u(T_{0})-\int_{T_{0}}^{t}T_{\Omega,\alpha}(t-s)\mathbb{P}\nabla.(u\otimes u)(s)ds\,, \end{eqnarray*} we obtain \begin{eqnarray*} |\widehat{u}(t,\xi)|\leq e^{-\mu|\xi|^{2\alpha}t}|\widehat{u}(T_{0},\xi)|+ \int_{T_{0}}^{t} e^{-\mu(t-s)|\xi|^{2\alpha}}|\mathbb{P}\nabla.(u\otimes u)(s,\xi)| \, \mathrm{d}s \,. \end{eqnarray*} The same reasoning as in the proof of Proposition 12 gives \begin{equation*} \|u\|_{\mathcal{L}^{\infty}\Big([T_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \lesssim\|u(T_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\|u\|_{\mathcal{L}^{\infty}\Big([T_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\Big)}\|u\|_{\mathcal{L}^{1}\Big([T_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p} }\Big)}\,. \end{equation*} It follows that \begin{align*} \|u\|_{\mathcal{L}^{\infty}\Big([T_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} \lesssim\|u(T_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\frac{1}{2}\|u\|_{\mathcal{L}^{\infty}\Big([T_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\Big)}\,. \end{align*} We can deduce that \begin{align*} \sup_{T_{0}\leq s\leq t}\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \lesssim2\|u(T_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\,,\forall t\in[T_{0},T^*)\,. \end{align*} Setting \begin{align*} M=\max\Big(2\|u(T_{0})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }},\max_{t\in[0,T_{0}]}\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\Big)\,, \end{align*} we have \begin{align*} \|u(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\lesssim M,\,\,\forall t\in[0,T^*)\,. \end{align*} On the other side \begin{align*} u(t)=e^{-t\mu (-\Delta)^{\alpha}}u_{0}-\Omega\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}\mathbb{P}(e_{3}\times u)(\tau)d\tau-\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}\mathbb{P}\nabla.(u\otimes u)(\tau)d\tau\,. \end{align*} Then \begin{align*} u(t')-u(t)&=(e^{-\mu t'(-\Delta)^{\alpha}}u_{0}-e^{-\mu t(-\Delta)^{\alpha}}u_{0})\\ &\quad-\Big(\int_0^{t'}e^{-\mu(t'-\tau)(-\Delta)^{\alpha}}\mathbb{P}\nabla.(u\otimes u)(\tau)d\tau -\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}\mathbb{P}\nabla.(u\otimes u)(\tau)d\tau\Big)\\ &\quad-\Omega\Big(\int_0^{t'}e^{-\mu(t'-\tau)(-\Delta)^{\alpha}}\mathbb{P}(e_{3}\times u)(\tau)d\tau-\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}\mathbb{P}(e_{3}\times u))(\tau)d\tau\Big)\\ &=[e^{-\mu t'(-\Delta)^{\alpha}}u_{0}-e^{-\mu t(-\Delta)^{\alpha}}u_{0}]-\Big[\int_{t}^{t'}e^{-\mu(t'-\tau)(-\Delta)^{\alpha}}\mathbb{P}\nabla.(u\otimes u)(\tau)d\tau\Big]\\ &\quad - \Big[\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}(e^{-\mu(t'-t)(-\Delta)^{\alpha}}-1)\mathbb{P}\nabla.(u\otimes u)(\tau)d\tau\Big]\\ &\quad-\Omega\Big[\int_{t}^{t'}e^{-\mu(t'-\tau)(-\Delta)^{\alpha}}\mathbb{P}(e_{3}\times u)(\tau)d\tau\Big]\\ &\quad - \Omega\Big[\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}(e^{-\mu(t'-t)(-\Delta)^{\alpha}}-1)\mathbb{P}(e_{3}\times u)(\tau)d\tau\Big]\\ &:=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}\,. \end{align*} We will estimate \(J_{1}, \,J_{2},\, J_{3},\,J_{4}\) and \(J_{5}\); \begin{align*} \|J_{1}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}&=\Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\|\varphi_{j}(e^{- \mu t'|\xi|^{2\alpha}}-e^{-\mu t|\xi|^{2\alpha}})\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}\Big\|_{\ell^{q}}\\ &\leq \Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\|\varphi_{j}(e^{-\mu(t'-t)|\xi|^{2\alpha}}-1)\hat{u}_{0}\|_{\mathrm{M}_{p}^{\lambda}}\Big\|_{\ell^{q}}\,, \end{align*} \begin{align*} \|J_{2}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}&\leq \Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_t^{t'}\|\varphi_{j}e^{-\mu(t'-\tau)|\xi|^{2\alpha}}\mathcal{F}(\nabla.u\otimes u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\\ &\leq \Big\| 2^{j(2-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_t^{t'}\|\varphi_{j}\mathcal{F}(u\otimes u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\,, \end{align*} \begin{align*} \|J_{3}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}&\leq\Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\times\int_0^{t}\|\varphi_{j}e^{-\mu(t'-\tau)|\xi|^{2\alpha}}(1-e^{-\mu(t'-t)|\xi|^{2\alpha}})\mathcal{F}(\nabla.(u\otimes u)(\tau))\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\\ &\leq\Big\| 2^{j(2-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_0^{t}\|\varphi_{j}(e^{-\mu(t'-t)|\xi|^{2\alpha}}-1)\mathcal{F}(u\otimes u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\,, \end{align*} \begin{align*} \|J_{4}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}&\lesssim \Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_t^{t'}\|\varphi_{j}e^{-\mu(t'-\tau)|\xi|^{2\alpha}}\mathcal{F}(e_{3}\times u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\\ &\lesssim \Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_t^{t'}\|\varphi_{j}\mathcal{F}(e_{3}\times u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\,, \end{align*} and \begin{align*} \|J_{5}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}&\lesssim\Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\times\int_0^{t}\|\varphi_{j}e^{-\mu(t'-\tau)|\xi|^{2\alpha}}(1-e^{-\mu(t'-t)|\xi|^{2\alpha}}) \mathcal{F}(e_{3}\times u(\tau))\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\\ &\lesssim\Big\| 2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} )}\int_0^{t}\|\varphi_{j}(e^{-\mu(t'-t)|\xi|^{2\alpha}}-1)\mathcal{F}(e_{3}\times u)(\tau)\|_{\mathrm{M}_{p}^{\lambda}}d\tau\Big\|_{\ell^{q}}\,. \end{align*} The dominated convergence theorem gives \begin{align*} \limsup_{t,t'\nearrow T^*,t\leq t'} \|u(t)-u(t')\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}=0\,. \end{align*} This means that \(u(t)\) satisfies the Cauchy criterion at \(T^*\). As \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) is a Banach space, then there exists an element \(u^*\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) such that \(u(t)\to u^*\) in \(\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) as \(t\to T^*\). Set \(u(T^*)=u^*\) and consider the fractional Navier-Stokes equations with Coriolis force starting by \(u^*\). By the well-posedness we obtain a solution existing on a larger time interval than \([0,T^*)\), which is a contradiction. Now, let \( v\in \mathcal{C}\Big([0,T^*);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap \mathcal{L}^{1}\Big([0,T^*),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big) \) be the maximal solution of (1) corresponding to the initial condition \(v_{0}\). We want to prove \(T^*=\infty\). Put \(w=v-u\) and \(w_{0}=v_{0}-u_{0}\). We have \begin{align*} w_{t}+\mu(-\Delta)^{\alpha} w+\Omega e_{3}\times w+w\cdot\nabla w+u\cdot\nabla w+w\cdot\nabla u=-\nabla \pi\,. \end{align*} We first apply \(\mathbb{P}\) to the above equation, then we have \begin{align*} w_{t}+\mu(-\Delta)^{\alpha} w =-\Omega \mathbb{P}e_{3}\times w-\mathbb{P}\nabla.(w\otimes w)-\mathbb{P}\nabla.(u\otimes w)-\mathbb{P}\nabla.(w\otimes u)\,. \end{align*} Due to Duhamel's formula, we write \begin{align*} |\widehat{w}(t,\xi)|&\leq e^{-\mu|\xi|^{2\alpha}t}|\widehat{w}(0,\xi)|+ \int_{0}^{t} e^{-\mu(t-s)|\xi|^{2\alpha}}|\mathcal{F}(\mathbb{P}\nabla.(w\otimes w))(s,\xi)| \, \mathrm{d}s \\ &\quad+\int_{0}^{t} e^{-\mu(t-s)|\xi|^{2\alpha}}|\mathcal{F}(\mathbb{P}\nabla.(u\otimes w))(s,\xi)| \, \mathrm{d}s\\ &\quad +\int_{0}^{t} e^{-\mu(t-s)|\xi|^{2\alpha}}|\mathcal{F}(\mathbb{P}\nabla.(w\otimes u))(s,\xi)| \, \mathrm{d}s\\ &\quad+|\Omega|\int_{0}^{t} e^{-\mu(t-s)|\xi|^{2\alpha}}|\mathcal{F}(\mathbb{P}e_{3}\times w)(s,\xi)| \, \mathrm{d}s\,. \end{align*} Then, for \(t\in[0,T^*)\) we get \begin{equation*} \begin{aligned} \mu \|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} &\leq C\Big\{\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\|\nabla.(w\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\quad+\|\nabla.(u\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)} \\ &\quad+\|\nabla.(w\otimes u)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\quad+|\Omega|\|e_{3}\times w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\Big\}\,. \end{aligned} \end{equation*} Similarly, \begin{equation*} \begin{aligned} \|w\|_{\mathcal{L}^{\infty}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} &\leq\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\|\nabla.(w\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\quad+\|\nabla.(u\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)} \\ &\quad+\|\nabla.(w\otimes u)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\quad+|\Omega|\|e_{3}\times w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\,. \end{aligned} \end{equation*} Consequently, for \(t\in[0,T^*)\) we get \begin{eqnarray*} \|w(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\mu \|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} &\leq& C\Big\{\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\|\nabla.(w\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &&+\|\nabla.(u\otimes w)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)} \\ &&+\|\nabla.(w\otimes u)\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &&+|\Omega|\|e_{3}\times w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\Big\}\\ &\lesssim&\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+L_{1}+L_{2}+L_{3}\,. \end{eqnarray*} where \begin{align*} L_{1}&=\|\nabla.(w\otimes w)\|_{\mathcal{L}^{1}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} )},\\ L_{2}&=\|\nabla.(u\otimes w)\|_{\mathcal{L}^{1}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} )} +\|\nabla.(w\otimes u)\|_{\mathcal{L}^{1}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} )} \end{align*} and \(L_{3}=|\Omega|\|e_{3}\times w\|_{\mathcal{L}^{1}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} ) }\,.\)
The same calculus as in the proof of Proposition 12 gives \begin{align*} L_{1}&\lesssim \|w\|_{\mathcal{L}^{\infty}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)} \|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\,,\\ L_{2}&\lesssim \int_0^{t}\|w\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} } \|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} }\,,\\ L_{3}&\lesssim |\Omega|\|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\,. \end{align*} Then \begin{equation*} \begin{aligned} &\|w(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\mu\|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\leq C\Big\{\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\|w\|_{\mathcal{L}^{\infty}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)} \|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\,\,\,+\int_0^{t}\|w\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} } \|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} }+|\Omega|\|w\|_{\mathcal{L}^{1}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} )}\Big\}\,. \end{aligned} \end{equation*} Put For \(t\in[0,T)\), we have \begin{equation*} \begin{aligned} &\|w(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\frac{\mu}{2}\|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}\\ &\leq C\big\{\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}+\int_{0}^{t}\|w\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} (|\Omega|+\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}})\big\}\,. \end{aligned} \end{equation*} Gronwall's Lemma yields \begin{equation*} \begin{aligned} \|w(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\frac{\mu}{2} \int_{0}^{t}\|w\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}} &\leq C\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\exp\Big\{\int_{0}^{t}C\Big(|\Omega|+\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}}\Big)\Big\}\\ &\leq C\|w_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }}\exp\Big\{\int_{0}^{\infty}C\Big(|\Omega|+\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}}\Big)\Big\}\,. \end{aligned} \end{equation*} Thus if we take \(C_{0}\) sufficiently small in (7), we have \begin{equation*} \begin{aligned} \|w(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\frac{\mu}{2}\|w\|_{\mathcal{L}^{1}\Big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}} \Big)}< \frac{\mu}{8C}\,, \end{aligned} \end{equation*} which contradicts the definition (30). Then \(T=T^*\) and \(\|w\|_{\mathcal{L}^{1}([0,T^{*}),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}} )}< \infty\), therefore \(T^*=\infty\). This completes the proof of Theorem 7.

Conclusion

Using harmonic analysis tools such as decomposition of Littlewood-Paley and the fixed point argument in Banach space, we obtain some results of existence, uniqueness, stability and asymptotic behaviour of solutions of the fractional Navier-Stokes equations with Coriolis force for small initial data. Moreover, local well-posedness results of these equations for large initial data are also discussed. The adopted functional framework is the critical Fourier-Besov-Morrey space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) which covers many classical spaces, e.g. the Fourier-Herz space \(\mathcal{\dot{B}}_{q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\), the Fourier-Besov-Lebesgue space \(\mathrm{F\dot{B}}_{p,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) and the Lei-Lin's space \(\chi^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\). These spaces \(\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p} }\) are some refined functional spaces, more suitable and more adapted for studying these equations.

Acknowledgments

The authors are grateful to the referee and the editorial board for some useful comments that improved the presentation of the paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15] and the references therein.

Zhang in [16] investigated the existence and uniqueness of positive solutions for the nonlinear fractional differential equation \begin{equation*} \left\{ \begin{array}{l} D^{\alpha }x\left( t\right) =f(t,x(t)), 0< t\leq 1, \\ x\left( 0\right) =0, \end{array} \right. \end{equation*}

where \(D^{\alpha }\) is the standard Riemann Liouville fractional derivative of order \(0< \alpha < 1\) and \(f:\left[ 0,1\right] \times \left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) is a given continuous function. By using the method of the upper and lower solution and cone fixed-point theorem, the author obtained the existence and uniqueness of a positive solution.

The nonlinear fractional differential equation \begin{equation*} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right) =f(t,x(t))+^{C}D^{\alpha -1}g\left( t,x\left( t\right) \right) ,\ 0< t\leq T, \\ x\left( 0\right) =\theta _{1}>0,\ x^{\prime }\left( 0\right) =\theta _{2}>0, \end{array} \right. \end{equation*} has been investigated in [5], where \(^{C}D^{\alpha }\) is the standard Caputo's fractional derivative of order \(1< \alpha \leq 2\), \(g,f:\left[ 0,T \right] \times \left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) are given continuous functions, \(g\) is non-decreasing on \(x\) and \(\theta _{2}\geq g\left( 0,\theta _{1}\right) \). By employing the method of the upper and lower solutions and Schauder and Banach fixed point theorems, the authors obtained positivity results.

In [2], Abdo, Wahash and Panchat discussed the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary conditions \begin{equation*} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right) =f\left( t,x\left( t\right) \right) , 0< t\leq 1, \\ x\left( 0\right) =\lambda \int_{0}^{1}x\left( s\right) ds+d, \end{array} \right. \end{equation*} where \(0< \alpha \leq 1\), \(\lambda \geq 0\), \(d>0\) and \(f:\left[ 0,1\right] \times \left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) is a given continuous function. By using the method of the upper and lower solutions and Schauder and Banach fixed point theorems, the existence and uniqueness of solutions has been established.

Ahmad and Ntouyas in [4] studied the existence and uniqueness of solutions to the following boundary value problem \begin{equation*} \begin{cases} \mathfrak{D}_{1}^{\alpha }\left( \mathfrak{D}_{1}^{\beta }u(t)-g\left( t,u_{t}\right) \right) =f(t,u_{t}),\ t\in \lbrack 1,b], \\[4pt] u(t)=\phi (t), t\in \lbrack 1-r,1], \\ \mathfrak{D}_{1}^{\beta }u(1)=\eta \in \mathbb{R}, \end{cases} \end{equation*} where \(\mathfrak{D}_{1}^{\alpha }\) and \(\mathfrak{D}_{1}^{\beta }\) are the Caputo-Hadamard fractional derivatives, \(0< \alpha ,\beta < 1\). By employing the fixed point theorems, the authors obtained existence and uniqueness results.

In this paper, we are interested in the analysis of qualitative theory of the problems of the positive solutions to fractional differential equations with integral boundary conditions. Inspired and motivated by the works mentioned above and the papers [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15] and the references therein, we concentrate on the positivity of the solutions for the nonlinear fractional differential equation with integral boundary conditions

1. Introduction

he theory of discrete dynamical systems of difference equations has been utilized to study natural phenomena which change over discrete time. A massive number of researchers investigated various real life problems that occur in population dynamics, genetics in biology, engineering, queuing problems, electrical networks, physics, economics, etc [1]. The long-term behaviours of such problems have been recently discussed by some scholars. For example, in [2] Asiri et al. explained the periodic solutions of the following system of difference equations: \begin{equation*} x_{n+1}=\frac{y_{n-2}}{1-y_{n-2}x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{x_{n-2}}{ \pm 1\pm x_{n-2}y_{n-1}x_{n}}. \end{equation*} Cinar [3] explored the periodicity of non-negative solutions of a system of fractional difference equations given by the form: \begin{equation*} x_{n+1}=\frac{1}{y_{n}},\ \ \ y_{n+1}=\frac{y_{n}}{x_{n-1}y_{n-1}}. \end{equation*} Din [4] analyzed and obtained the equilibrium points, local asymptotic stability and global behaviour of the fixed points of Lotka-Volterra model which is illustrated by the system: \begin{equation*} x_{n+1}=\frac{\alpha x_{n}-\beta x_{n}y_{n}}{1+\gamma x_{n}},\ \ \ y_{n+1}= \frac{\delta y_{n}+\epsilon x_{n}y_{n}}{1+\eta y_{n}}. \end{equation*} El-Metwally et al. [5] found the solutions and the periodicity of the following third order system of rational recursive equations: \begin{equation*} x_{n+1}=\frac{y_{n-2}}{-1-y_{n-2}x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{x_{n-2}}{ \pm 1\pm x_{n-2}y_{n-1}x_{n}}. \end{equation*} In [6], Elsayed presented the solutions of the following system of second order difference equations: \begin{equation*} x_{n+1}=\frac{x_{n-1}}{\pm 1\pm x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{y_{n-1}}{ \mp 1+y_{n-1}x_{n}}. \end{equation*} Elsayed and Alzubaidi [7] found the solutions of following systems of rational difference equations: \[ x_{n+1}=\frac{y_{n-8}}{1+y_{n-2}x_{n-5}y_{n-8}},\ \ \ \ \ y_{n+1} =\frac{x_{n-8}}{\pm1\pm x_{n-2}y_{n-5}x_{n-8}}. \] G\H{u}m\H{u}\c{s} and \H{O}calan [8] explored the positive solutions of the systems: \[ u_{n+1}=\frac{\alpha u_{n-1}}{\beta+\gamma v_{n}^{p}v_{n-2}^{q}} ,\ \ \ v_{n+1}=\frac{\alpha_{1}v_{n-1}}{\beta_{1}+\gamma_{1}u_{n}^{p_{1} }u_{n-2}^{q_{1}}}. \] Kurbanli et al. [9] obtained the solutions of the following system of difference equations: \begin{equation*} x_{n+1}=\frac{x_{n-1}}{y_{n}x_{n-1}-1},\ \ y_{n+1}=\frac{y_{n-1}}{ x_{n}y_{n-1}-1},\ \ z_{n+1}=\frac{x_{n}}{y_{n}z_{n-1}}. \end{equation*} In [10], Touafek et al. discovered the periodicity and solution of the system: \begin{equation*} x_{n+1}=\frac{x_{n-3}}{\pm 1\pm x_{n-3}y_{n-1}},\ \ \ y_{n+1}=\frac{y_{n-3}}{ \pm 1\pm y_{n-3}x_{n-1}}. \end{equation*} The author in [11] examined the dynamics of the following system of recursive equations: \begin{equation*} x_{n+1}=\frac{x_{n-2}}{B+y_{n}y_{n-1}y_{n-2}},\ \ \ y_{n+1}=\frac{y_{n-2}}{ A+x_{n}x_{n-1}x_{n-2}}. \end{equation*} For more information about basic theory and qualitative behaviour of dynamical systems of difference equations, one can see references [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 35, 26, 27].
This work aims to present the solutions of discrete dynamical systems of difference equations which are given by \begin{equation*} x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) } ,\ \ \ n=0,1,\ldots, \end{equation*} where the initial values \(x_{-3},\ x_{-2},\ x_{-1},\ \)\ \(x_{0},\ y_{-3},\ y_{-2},\ y_{-1}\) and\(\ \ y_{0}\) are required to be real numbers.

2. Main Results

2.1. First system \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( 1+y_{n-1}x_{n-3}\right) }\)

This subsections is devoted for the solutions of the following system of recursive equations: The initial values of this system are required to be arbitrary real numbers.

Theorem Let \(\left\{ x_{n},y_{n}\right\} \) be a solution to system (1) and let \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ \ldots\), we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}\underset{i=0}{ \overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) b\omega +1\right] }{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{c^{n+1}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{ d^{n+1}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }{b^{n}\omega ^{n}}, \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{i=0}{\overset {n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] },\ \ \ y_{4n-2}= \frac{b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-1}{\Pi }} \left[ \left( 2i+1\right) b\omega +1\right] }, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] },\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}\underset{ i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) b\omega +1\right] }. \end{eqnarray*}

Proof. It is easy to see that the results hold for \(n=0\). Next, we assume that \(n>1\) and suppose that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{ a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{ \left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i\right) b\omega +1\right] }{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1}}, \\ x_{4n-5} &=&\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}= \frac{d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] },\ \ \ y_{4n-6}=\frac{b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma +1\right] },\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) b\omega +1\right] }. \end{eqnarray*} Following this, system (1) gives us \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\frac{ a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+1\right) a\gamma +1\right] }}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1 \right] }\left[ -1-\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi } }\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\frac{ a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+1\right) a\gamma +1\right] }\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}c^{n}\alpha ^{n}\underset{i=0}{\overset{n-2 }{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Moreover, one can observe from system (1) that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left[ 1+y_{4n-5}x_{4n-7}\right] } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma +1\right] }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1+\frac{ \left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1 \right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \end{eqnarray*} \begin{eqnarray*} &=&\frac{\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }}{\frac{c^{n}\alpha ^{n-1}\underset{i=0 }{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{ a^{n-1}\gamma ^{n-1}}\left[ 1+\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{\underset{i=0}{\overset{n-2}{ \Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }\right] } \\ &=&\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] a^{n-1}\gamma ^{n-1}}{c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] \left[ \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1 \right] +a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] \right] } \\ &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-1}{ \Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }. \end{eqnarray*} Similarly, other results can be proved. The proof has been completed.

Second System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( -1-y_{n-1}x_{n-3}\right) }\)

In this section, we obtained the solutions of the following system of differential equations: The initial values of system (2) are required to be arbitrary real numbers.

Theorem 2. Let \(\left\{ x_{n},y_{n}\right\} \) be a solution to system (2) and assume that \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ...\) we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}}{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{\left( -1\right) ^{n}c^{n+1}\alpha ^{n}\left( a\gamma +1\right) ^{n}}{a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{\left( -1\right) ^{n}d^{n+1}\beta ^{n}\left( b\omega +1\right) ^{n}}{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n}},\ \ \ y_{4n-2}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\left( b\omega +1\right) ^{n}}, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}},\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}}. \end{eqnarray*}

Proof. The solutions are verified for \(n=0\). Next, we let \(n>1\) and assume that the results hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{\left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}}{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1} }, \\ x_{4n-5} &=&\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{\left( -1\right) ^{n-1}d^{n}\beta ^{n-1}\left( b\omega +1\right) ^{n-1}}{ b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}},\ \ \ y_{4n-6}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\left( b\omega +1\right) ^{n-1}}, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}},\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}}. \end{eqnarray*} Now, the first relation is given by \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\left[ -1-\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1} }\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}}\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}c\alpha \ c^{n-1}\alpha ^{n-1}}{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }= \frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Similarly, system (2) leads \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( -1-y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}}\left[ -1-\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}a\gamma \ a^{n-1}\gamma ^{n-1}}{ c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}\left[ 1+a\gamma \right] }= \frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n}}. \end{eqnarray*} Accordingly, the remaining relations of system (2) can be verified. Hence, this achieves the proof.

2.3. Third System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( 1-y_{n-1}x_{n-3}\right) }\)

In this subsection, the solutions of the following dynamic discrete system will be formulated: where the initial values are required to be arbitrary real numbers.

Theorem 3. Assume that \(\left\{ x_{n},y_{n}\right\} \) is a solution to system (3) and suppose that \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ...\) we have \begin{eqnarray*} x_{4n-3} &=&\frac{c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{d^{n}\beta ^{n}\underset{i=0}{\overset{ n-1}{\Pi }}\left[ \left( 2i\right) b\omega -1\right] }{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{\left( -1\right) ^{n}c^{n+1}\alpha ^{n}\underset{i=0}{ \overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{\left( -1\right) ^{n}d^{n+1}\beta ^{n} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega -1 \right] }{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] },\ \ \ y_{4n-2}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n}}{ d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega -1\right] }, \\ y_{4n-1} &=&\frac{a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{ c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] },\ \ \ y_{4n}=\frac{b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) b\omega -1\right] }. \end{eqnarray*}

Proof. The solutions hold for \(n=0\). Now, we suppose that \(n>1\) and assume that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i\right) a\gamma -1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{d^{n-1}\beta ^{n-1}\underset{ i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) b\omega -1\right] }{ b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1}}, \end{eqnarray*} \begin{eqnarray*} x_{4n-5} &=&\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{\left( -1\right) ^{n-1}d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega -1 \right] }{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] },\ \ \ y_{4n-6}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n-1}}{ d^{n-1}\beta ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega -1\right] }, \\ y_{4n-5} &=&\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] },\ \ \ y_{4n-4}=\frac{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{d^{n-1}\beta ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+2\right) b\omega -1\right] }. \end{eqnarray*} Next, it can be simply seen from system (3) that \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{ c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }}{\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma -1\right] }\left[ -1-\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }\right] } \\ &=&-\frac{c\alpha \ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{c^{n}\alpha ^{n} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Furthermore, system (3) gives that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( 1-y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] }\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{ a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1-\frac{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1 \right] }\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \\ &=&\frac{\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] }}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1-\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1 \right] }\right] } \\ &=&\frac{\left( -1\right) ^{-n+1}a^{n}\gamma ^{n}\underset{i=0}{\overset{n-2} {\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{c^{n}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] \left[ \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] -a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] \right] } \\ &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{ i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }. \end{eqnarray*} The remaining formulas can be shown in a similar way.

2.4 Fourth System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( -1+y_{n-1}x_{n-3}\right) }\)

In the next theorem, we will discover the solutions of the following nonlinear system of difference equations: where the initial conditions are required to be arbitrary real numbers.

Theorem 4. Assume that \(\left\{ x_{n},y_{n}\right\} \) is a solution to system (4) and let \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ... \) we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}}{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{c^{n+1}\alpha ^{n}\left( a\gamma -1\right) ^{n}}{ a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{d^{n+1}\beta ^{n}\left( b\omega -1\right) ^{n}}{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n}},\ \ \ y_{4n-2}=\frac{b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\left( b\omega -1\right) ^{n}}, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}},\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}}. \end{eqnarray*}

Proof. It is clear that the relations hold for \(n=0.\) Now, we suppose that \(n>1\) and assume that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{\left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}}{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1} }, \\ x_{4n-5} &=&\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{d^{n}\beta ^{n-1}\left( b\omega -1\right) ^{n-1}}{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}},\ \ \ y_{4n-6}=\frac{b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\left( b\omega -1\right) ^{n-1}}, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}},\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}}. \end{eqnarray*} Now, it can be obviously observed from system (4) that \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}}\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\left[ -1-\frac{ c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}} \frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}}\right] } \\ &=&\frac{\left( -1\right) ^{-n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Similarly, one can obtain from system (4) that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( -1+y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}}\left[ -1+\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \end{eqnarray*} \begin{eqnarray*} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1} \left[ -1+a\gamma \right] }=\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n}}. \end{eqnarray*} Other relations can be likewise proved. Therefore, this completes our proof.

2.5. Numerical Examples

This subsection is allocated to confirm our theoretical discussion by illustrating some numerical examples. These examples show the behaviour of the solutions of each system.

Example 1. In this example, we describe the behaviour of the solution of system (1). Our initial data has been taken as follows: \(x_{-3}=0.21,\ x_{-2}=-0.25,\ x_{-1}=-0.032,\ x_{0}=2,\ y_{-3}=1.06,\ y_{-2}=-0.4,\ y_{-1}=-1.55\) and \(\ y_{0}=0.082.\) See Figure 1.

Example 2. This example demonstrates the plot of system (2). The initial values are considered as follows: \( x_{-3}=0.22,\ x_{-2}=-0.25,\ x_{-1}=0.2,\ x_{0}=0.6,\ y_{-3}=1.03,\ y_{-2}=-0.43,\ y_{-1}=1.5\) and \(\ y_{0}=0.8,\) as depicted in Figure 2.

Example 3. Here, we plot the curves of solutions of system (3). Figure 3 presents this behaviour under the following initial values: \(x_{-3}=0.03,\ x_{-2}=-0.2,\ x_{-1}=10,\ x_{0}=-0.05,\ y_{-3}=2,\ y_{-2}=-4.1,\ y_{-1}=1.4\) and \(\ y_{0}=1.03.\)

Example 4. Figure \ref{fig4} illustrates the behaviour of the solution of system (4). The considered initial conditions in this example are given as follows: \(x_{-3}=0.02,\ x_{-2}=-0.2,\ x_{-1}=0.21,\ x_{0}=-0.03,\ y_{-3}=0.1,\ y_{-2}=-4.9,\ y_{-1}=1,\ y_{0}=-1.\)

3. Conclusion

This paper has been written to highlight the analytical and numerical solutions of four different systems of difference equations. In subsection 1, we have provided the solution of System 1 and illustrated its behaviour under specific conditions in Figure 1. Theorem 2 and Theorem 3 presented the exact solutions of System 2 and System 3, respectively. Finally, Figure 4 demonstrated the curve of the solution of System 4 which is given in Subsection 4.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

This work is concerned with the global existence, uniqueness, and asymptotic behavior of solution for the Euler-Bernoulli viscoelastic equation where \(\Omega\) is a bounded domain of \(\mathbb{R}^{n}\) with smooth boundary \(\partial\Omega\), and \(\nu\) is the unit outer normal on \(\partial\Omega\). Here \(g_{1}\) and \(g_{2}\) are positive functions satisfying some conditions to be specified later, and \begin{equation*} g_{i}\ast\chi(t)=\int_{0}^{t}g_{i}(t-\tau)\chi(\tau)d\tau,\quad i=1,2. \end{equation*} The Euler-Bernoulli equation describes the deflection \(u(x,t)\) of a beam (when \(n=1)\) or a plate (when \(n=2\)), where \( \Delta^{2}u:=\Delta(\Delta u)=\sum_{j=1}^{n}\left(\sum_{i=1}^{n}u_{x_{i}}u_{x_{i}}\right)_{x_{j}x_{i}}, \) and \(h\) and \(f\) represent the friction damping and the source respectively.
Lange and Menzala [1] considered where \(x\in\mathbb{R}^{n}\), \(t\geq0\), \(a(t)=m(\|\nabla v(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{n}})\) and the real-valued function \(m:[0,+\infty)\rightarrow [1,+\infty)\) will be assumed to be of class \(C^{1}\) satisfying the condition \(m(s)\geq 1+s\) for all \(s\geq0\). They remarked that the imaginary part of the solutions of Schr\"{o}dinger's equation \begin{equation*} iw_{t}=\Delta w+im\left(\|\nabla(Imw)\|_{L^{2}(\mathbb{R}^{n}}^{2}\right)Rew=0, \end{equation*} are precisely the solutions for (3). Then, using Fourier transform, the existence of global classical solutions and algebraic decay rate were proved for initial data whose regularity depends on the spacial dimension \(n\). Messaoudi [2] studied the equation where \(a,b>0\), \(p,m>2\). He established an existence result for (4) and showed that the solution continued to exist globally if \(m\geq p\). If we take the viscoelastic materials into consideration, the model (2) becomes where \(g\) is so-called viscoelastic kernel. The term \(\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds\) describes the hereditary properties of the viscoelastic materials [3]. It expresses the fact that the stress at any instant \(t\) depends on the past history of strains which the material has undergone from time \(0\) up to \(t\). Tatar [4] obtained the property of the energy decay of the model (5) for \(h=f=0\) and from this, we know that the term \(\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds\), similar to the friction damping, can cause the inhibition of the energy. Messaoudi and Mukiawa [5] studied the fourth-order viscoelastic plate equation \begin{equation*} u_{tt}(x,t)+\Delta^{2}u(x,t)-\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds=0, \end{equation*} in the bounded domain \(\Omega=(0,\pi )\times (-l,l)\subset\mathbb{R}^{2}\) with nontraditional boundary conditions. The authors established the well-posedness of the solution and a decay result.
Rivera et al. [6] investigated the plate model: \begin{equation*} u_{tt}+\Delta^{2}u-\sigma\Delta u_{tt}+\int_{0}^{t}g(t-s)\Delta^{2} u(s)ds=0, \end{equation*} in the bounded domain \(\Omega\subset \mathbb{R}^{2}\) with mixed boundary condition and suitable geometrical hypotheses on \(\partial\Omega\). They established that the energy decays to zero with the same rate of the kernel \(g\) such as exponential and polynomial decay. To do so in the second case they made assumptions on \(g, g'\) and \(g''\) which means that \(g\simeq(1+t)^{-p}\) for \(p>2\). Then they obtained the same decay rate for the energy. However, their approach can not be applied to prove similar results for \(1< p\leq2\).
Cavalcanti et al. [7] investigated the global existence, uniqueness and stabilization of energy of \begin{equation*} u_{tt}+\Delta^{2}u-\int_{0}^{t}g(t-s)\Delta^{2}u(s)ds+a(t)u_{t}=0 \end{equation*} where \begin{equation*} a(t)=M\left(\int_{\Omega}|\nabla u(x,t)|^{2}dx\right) \ \ \mbox {with} \ \ \ M\in C^{1}([0,+\infty)). \end{equation*}

By taking a bounded or unbounded open set \(\Omega\) where \(M(s)>m_{0}>0\) for all \(s\geq0\), the authors showed in [7] that the energy goes to zero exponentially, provided that \(g\) goes to zero at the same form.

The aim of this work is to study the global existence of regular and weak solutions of problem (1) for the bounded domain, then for \(\xi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) a increasing \(C^{2}\) function such that the solution features the asymptotic behavior \begin{equation*} E(t)\leq E(0)e^{-\kappa \xi(t)},\quad \forall t\geq0, \end{equation*} where \(E(t)\) is defined in (38) and \(\kappa\) is a positive constant independent of the initial energy \(E(0)\).

2. Preliminaries and main results

We begin by introducing some notation that will be used throughout this work. For functions \(u(x,t)\), \(v(x,t)\) defined on \(\Omega\), we introduce \[(u,v)=\int_{\Omega}u(x)v(x)dx \ \ \mbox {and} \ \ \|u\|_{2}=\left(\int_{\Omega}|u(x)|^{2}dx\right)^{\frac{1}{2}}.\] Define \[X=\left\{u\in H^{2}_{0}(\Omega);\Delta^{2}u\in L^{2}(\Omega)\right\}\] Then, \(X\) is a Hilbert space endowed with the natural inner product \[(u,v)_{X}=(u,v)_{H^{2}_{0}}+(\Delta^{2}u,\Delta^{2}v).\] Now let us precise the hypotheses on \(g_{1}\) and \(g_{2}\).
(H1) \(g_{1}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a bounded function satisfying \begin{equation*} g_{1}(t)\in C^{2}(\mathbb{R}^{+})\cap L^{1}(\mathbb{R}^{+}),\quad g_{1}(0)>0. \end{equation*} (H2) There exist positive constants \(\alpha_{1},\alpha_{2}\) and \(\alpha_{3}\) such that \begin{equation*} -\alpha_{1}g_{1}(t)\leq g_{1}'(t)\leq-\alpha_{2}g_{1}(t),\quad \forall t\geq0, \end{equation*} (H3) \begin{equation*} 0\leq g_{1}''(t)\leq\alpha_{3}g_{1}(t),\quad \forall t\geq0, \end{equation*} (H4) \(g_{2}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a bounded function satisfying \begin{equation*} g_{2}(t)\in C^{1}(\mathbb{R}^{+})\cap L^{1}(\mathbb{R}^{+}),\quad g_{2}(0)>0. \end{equation*} (H5) There exist positive constants \(\eta_{1}\) and \(\eta_{2}\) such that \begin{equation*} -\eta_{1}g_{2}(t)\leq g_{2}'(t)\leq-\eta_{2}g_{2}(t),\quad \forall t\geq0, \end{equation*} (H6) \begin{equation*} 1-\int_{0}^{t}\left(g_{1}(s)+\lambda_{1}^{-1} g_{2}(s)\right)ds=l>0, \end{equation*} where \(\lambda_{1}>0\) is the first eigenvalue of the spectral Dirichlet problem \begin{equation*} \Delta^{2}u=\lambda_{1}u\quad \ \mbox {in} \ \ \Omega,\quad u=\frac{\partial u}{\partial \nu}=0 \ \ \mbox {in} \ \ \partial\Omega, \end{equation*} \begin{equation*} \|\nabla u\|_{2}\leq \frac{1}{\sqrt{\lambda_{1}}}\|\Delta u\|_{2}. \end{equation*}

Lemma 1. For \(\phi,\psi\in C^{1}([0,+\infty[,\mathbb{R})\) we have \begin{equation*} 2\int_{0}^{t}\int_{\Omega}\phi(t-s)\psi\psi'dxds=-\frac{d}{dt}\left((\phi\square\psi)(t) -\int_{0}^{t}\phi(s)ds\|\psi\|^{2}_{2}\right)+(\phi'\square\psi)(t) -\phi(t)\|\phi\|^{2}_{2}, \end{equation*} where \begin{equation*} (\phi\square\psi)(t)=\int_{0}^{t}\phi(t-s)\|\psi(t)-\psi(s)\|_{2}^{2}ds. \end{equation*}

Theorem 2. Assume that \((H1)-(H6)\) hold, and that \(\left\{u_{0},u_{1}\right\}\) belong to \(H^{2}_{0}(\Omega)\times L^{2}(\Omega)\). Then, Problem (1) admits a unique weak solution \(u\) in the class $$u\in C^{0}([0,\infty);H^{2}_{0}(\Omega))\cap C^{1}([0,\infty);L^{2}(\Omega)).$$ Moreover, for \(\xi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) a increasing \(C^{2}\) function satisfying (\ref{eq6}) and, if \(\|g_{1}\|_{L^{1}(0,\infty)}\) is sufficiently small, we have for \(\kappa>0\) \begin{equation*} E(t)\leq E(0)e^{-\kappa \xi(t)},\quad \forall t\geq0. \end{equation*}

3. Existence of Solutions

In this section we first prove the existence and uniqueness of regular solutions to Problem (1). Then, we extend the same result to weak solutions using density arguments.

3.1. Regular solutions

Let \((w_{j})\) be a Galerkin basis in \(X\), and let \(V_{m}\) be the subspace generated by the first \(m\) vectors \(w_{1},...,w_{m}\). We search for a function \begin{equation*} u_{m}(t)=\sum_{i=1}^{m}k_{im}(t)w_{i}(x),\ m=1,2,.... \end{equation*} satisfying the approximate Cauchy problem By standard methods in differential equations, we can prove the existence of solutions to the problem \((5)-(6)\) on \([0,t_{m})\) with \(0< t_{m}< T\). In order to extend the solution of \((7)-(8)\) to the whole \([0,T]\), we need the following priori estimate.
Estimate 1. Taking \(v=2u'_{m}(t)\) in \((7)\), we have Using Lemma 1, we obtain and Inserting Equations (10) and (11) into Equation (9) and integrating over \([0,t]\subset[0, T]\), we obtain By using the fact that \begin{eqnarray*} &&(g_{1}\square\Delta u_{m})(t)+(g_{2}\square\nabla u_{m})(t)-\int_{0}^{t}(g'_{1}\square\Delta u_{m})(s)ds-\int_{0}^{t}(g'_{2}\square\nabla u_{m})(s)ds\nonumber\\ &&+\int_{0}^{t}\int_{\Omega}g_{1}(s)|\Delta u_{m}(s)|^{2}dxds+\int_{0}^{t}\int_{\Omega}g_{2}(s)|\nabla u_{m}(s)|^{2}dxds\geq0, \end{eqnarray*} and \begin{eqnarray*} \left(1-\int_{0}^{t}g_{1}(s)ds\right)\|\Delta u_{m}(t)\|^{2}_{2}-\left(\int_{0}^{t}g_{2}(s)ds\right)\|\nabla u_{m}(t)\|^{2}_{2}&\geq&\left(1-\int_{0}^{t}\left[g_{1}(s)+\lambda_{1}^{-1} g_{2}(s)\right]ds\right)\|\Delta u_{m}(t)\|^{2}_{2}\nonumber\\ &\geq& l\|\Delta u_{m}(t)\|^{2}_{2}, \end{eqnarray*} Equation (12) yields Taking the convergence of Equation (8) into consideration, we arrive at where \(L_{1}=\|u_{1}\|^{2}_{2}+\|\Delta u_{0}\|^{2}_{2}\).
Estimate 2. Firstly, we obtain an estimate for \(u''_{m}(0)\) in the \(L^{2}\) norm. indeed, setting \(v=u''_{m}(0)\) and \(t=0\) in Equation (7), we obtain From Equations (8), (14) and (15), it follows that where \(L_{2}\) is a positive constant independent of \(m\in\mathbb{N}\). Differentiating Equation (7) with respect to \(t\), and setting \(v=u''_{m}(t)\), we obtain By (H5), H\"{o}lder's inequality and Young's inequality give From Equation (14) we obtain and From (H3), we deduce Inserting Equations (18)-(21) in Equation (17), we get where $$ C_{3}=\left[\frac{\eta_{1}^{2}\|g_{2}\|_{L^{1}}}{2}+\frac{[g_{2}(0)]^{2}L_{1}}{2l} +\frac{|g'_{1}(0)|L_{1}}{2l}\right]T+\left[\frac{\eta_{1}^{2}\|g_{2}\|_{L^{1}(0,\infty)}\|g_{2}\|_{L^{\infty} (0,\infty)}}{2}+\frac{\alpha_{1}^{2}\|g_{1}\|_{L^{1}(0,\infty)}\|g_{1}\|_{L^{\infty} (0,\infty)}}{2}\right]\frac{L_{1}T}{l}$$ and $$C_{4}=\frac{|g'_{1}(0)|}{2}+\frac{1}{2}.$$ Using H\"{o}lder's inequality, we know that, for any \(\delta>0\), where \[C_{5}=\left[\frac{[g_{1}(0)]^{2}}{4\delta}+\frac{\alpha_{1}^{2}}{4\delta}\|g_{1}\|_{L^{1}(0,\infty)}\|g_{1}\|_{L^{\infty} (0,\infty)}T\right]\frac{L_{1}}{l}.\] Combining Equation (22) and Equation (23), we get Fixing \(\delta>0\), sufficiently small, so that \(\frac{1}{2}-2\delta>0\) in Equation (24), and taking into account Equations (8) and (16), we get from Gronwall's Lemma the second estimate, where \(L_{3}\) is a positive constant independent of \(m\in\mathbb{N}\) and \(t\in[0,T]\).
Estimate 3. Let \(m_{1}\geq m_{2}\) be two natural numbers, and consider \(z_{m}=u_{m_{1}}-u_{m_{2}}\). Then, applying the same way as in the estimate 1 and observing that \(\left\{u_{0m}\right\}\) and \(\left\{u_{1m}\right\}\) are Cauchy sequence in \(X\) and \(H^{2}_{0}(\Omega)\), respectively, we deduce for all \(t\in[0,T]\).
Therefore, from Equations (24), (25) and (26), we deduce that there exist a subsequence \(\left\{u_{\mu}\right\}\) of \(\left\{u_{m}\right\}\) and \(u\) such that The above convergences (27)-(29) are enough to pass to the limit in Equation (7), to obtain \begin{equation*}\label{W} \begin{array}{ll} u''+\Delta^{2}u -\int_{0}^{t}g_{1}(t-s)\Delta^{2}u(s)ds+\int_{0}^{t}g_{2}(t-s)\Delta u(s)ds+u'=0\ \ \mbox {in} \ \ \ L^{\infty}(0,\infty;L^{2}(\Omega)),\\ u(0)=u_{0},\quad u'(0)=u_{1}. \end{array}. \end{equation*} Next, we want to show the uniqueness of solution of (7)-(8). Let \(u^{(1)}\), \(u^{(2)}\) be two solutions of (7)-(8). Then \(z=u^{(1)}-u^{(2)}\) satisfies \begin{equation*} z(x,0)=z'(x,0)=0,\quad x\in\Omega, \end{equation*} \begin{equation*} z=0,\quad \frac{\partial z}{\partial\nu}=0,\quad x\in\partial\Omega,\ t>0. \end{equation*} Setting \(v=2z'(t)\) in (30), then as in deriving (14), we see that Therefore, we have the uniqueness.

3.2. Weak solutions

Let \((u_{0},u_{1})\in H^{2}_{0}(\Omega)\times L^{2}(\Omega)\). Then, since \(X\times H^{2}_{0}(\Omega)\) is dense in \(H^{2}_{0}(\Omega)\times L^{2}(\Omega)\) there exists \((u_{0\mu},u_{1\mu})\subset X\times H^{2}_{0}(\Omega)\) such that Then, for each \(\mu\in\mathbb{N}\), there exists a unique regular solution \(u_{\mu}\) of Problem (1) in the class In view of Equation (33) and using an analogous argument to that in Estimate 1 and Estimate 3, we find a sequence \(\left\{u_{\mu}\right\}\) of solutions to Problem (1) such that The convergences (33)-(36) are sufficient to pass to the limit in order to obtain a weak solution of Problem (1), which satisfies \begin{array}{ll} u''+\Delta^{2}u -\int_{0}^{t}g_{1}(t-s)\Delta^{2}u(s)ds+\int_{0}^{t}g_{2}(t-s)\Delta u(s)ds+u'=0\ \ \mbox {in} \ \ \ L^{2}(0,\infty;H^{-2}(\Omega)),\\ u(0)=u_{0},\quad u'(0)=u_{1}. \end{array}. The uniqueness of weak solutions requires a regularization procedure and can be obtained using the standard method of Visik-Ladyzhenskaya, c.f. Lions and Magenes [8, Chap. 3, Sec. 8.2.2].

4. Asymptotic Behaviour

In this section, we discuss the asymptotic behavior of the above-mentioned weak solutions. Let us define the energy associated to Problem (1) as To demonstrate our decay result, the lemmas below are essential.

Lemma 3. For any \(t>0\) \begin{equation*} 0\leq E(t)\leq \frac{1}{2}\Big[\|u_{t}(t)\|_{2}^{2}+\|\Delta u(t)\|^{2}_{2}+(g_{1}\square\Delta u)(t)+(g_{2}\square\nabla u)(t)\Big]. \end{equation*}

Proof. Using the fact that \(\|\nabla u(t)\|^{2}_{2}\leq\lambda_{1}^{-1}\|\Delta u(t)\|^{2}_{2}\), we have \begin{eqnarray*} &&\left(1-\int_{0}^{t}g_{1}(\tau)d\tau\right)\|\Delta u(t)\|^{2}_{2}-\left(\int_{0}^{t}g_{2}(\tau)d\tau\right)\|\nabla u(t)\|^{2}_{2}\nonumber\\ &&\quad \geq\left(1-\int_{0}^{t}\left[g_{1}(s)+\lambda_{1}^{-1}g_{2}(s)\right]ds\right)\|\Delta u(t)\|^{2}_{2} \end{eqnarray*} and according to \((H6)\) we have \(E(t)\geq0\),
and \begin{eqnarray*} E(t)&=&\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}\|\Delta u(t)\|^{2}_{2}+\frac{1}{2}(g_{1}\square\Delta u)(t)+\frac{1}{2}(g_{2}\square\nabla u)(t)\nonumber\\ &&-\frac{1}{2}\left\{\left(\int_{0}^{t}g_{1}(s)ds\right)\|\Delta u(t)\|^{2}_{2}+\left(\int_{0}^{t}g_{2}(s)ds\right)\|\nabla u(t)\|^{2}_{2}\right\}\nonumber\\ &\leq&\frac{1}{2}\Big[\|u_{t}(t)\|_{2}^{2}+\|\Delta u(t)\|^{2}_{2}+(g_{1}\square\Delta u)(t)+(g_{2}\square\nabla u)(t)\Big]. \end{eqnarray*}

Lemma 4. The energy \(E(t)\) satisfies

Proof. Multiplying the first equation in (1) by \(u_{t}\) and integrating over \(\Omega\), we obtain \begin{equation*} \frac{d}{dt}\left[\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}\|\Delta u(t)\|^{2}_{2}\right]+\|u_{t}(t)\|^{2}_{2} = \int_{0}^{t}g_{1}(t-\tau)\Delta u(\tau)\cdot\Delta u_{t}(t)dxd\tau +\int_{0}^{t}g_{2}(t-\tau)\nabla u(\tau)\cdot\nabla u_{t}(t)dxd\tau. \end{equation*} Exploiting (10)-(11) and by \((H1)-(H5)\), we deduce

From assumptions \((H2)\) and since \(\int_{0}^{t}g_{1}'(\tau)d\tau=g_{1}(t)-g_{1}(0)\), we obtain Combining Equation (40) and Equation (41), we conclude that \begin{eqnarray*} E'(t)&\leq&-\|u_{t}(t)\|^{2}_{2}-\frac{1}{2}\alpha_{2}(g_{1}\square\Delta u)(t)-\frac{1}{2}\eta_{2}(g_{2}\square\nabla u)(t)\nonumber\\ &&-\frac{1}{2}\left[g_{1}(0)-\alpha_{1}\|g_{1}\|_{L^{1}(0,\infty)}\right]\|\Delta u(t)\|^{2}_{2}\leq0. \end{eqnarray*} Multiplying Equation (39) by \(e^{\kappa\xi(t)}\) \((\kappa>0)\) and utilizing Lemma 3, we have Using the fact that \(\xi'\) is decreasing we arrive at Choosing \(\|g_{1}\|_{L^{1}(0,\infty)}\) sufficiently small so that \[g_{1}(0)-\alpha_{1}\|g_{1}\|_{L^{1}(0,\infty)}=L>0,\] and choosing \(\kappa\) sufficiently small in order to have \[2-\kappa\xi'(0)>0,\quad \alpha_{2}-\kappa\xi'(0)>0,\quad \eta_{2}-\kappa\xi'(0)>,\quad L-\kappa\xi'(0)>0.\] from Equation (43) we arrive at Integrating the above inequality over \((0,t)\), it follows that

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The problem of the small motions of a layer of viscous fluid heated from below was the subject of numerous works that can be found in the book (chapter II,[1]). Some cases of a heated viscous fluid partially or completely filling a fixed container has been studied and discussed in the book [2].

In this paper, we consider a mass of visco-elastic fluid heated from below and filling completely a container, restricting to the more simple Oldroyd model [2]. We obtain the operatorial equations for the small motions of the fluid and for the heat transfert by using for the classical Boussinesq hypothesis [1, 2].

At first, we prove the existence and the symmetry of the spectrum and the stability of the system, if the coefficients of kinematic viscosity and temperature conductivity are sufficiently large. We show that there exists a set of positive real eigenvalues having a point on the real axis as point of accumulation and that the possible non real eigenvalues are in a suitable disk. Then, we prove that the problem can be reduced to the study of a Krein- Langer pencil [3], so that we obtain new results. In particular, there is only a finite number of non real eigenvalues. Finally, we obtain an existence and unicity theorem of the solution of the associated evolution problem by means of the semigroups theory.

2. The more simple Oldroyd model for a viscoelastic fluid

It is a matter of a viscoelastic fluid for which the tensor of viscous stresses \(\sigma'\) and the double tensor of deformation velocities \(\tau\) verify a differential relation [2] where \(\eta\), \(\kappa_{0}\), \(\kappa_{1}\) are some positive physical constants.
We set with \[ \gamma=\eta^{-1};\qquad \mu=\kappa_{1} \gamma;\qquad \alpha=\frac{\kappa_{0}}{\kappa_{1}}-\gamma \] which are positive constants.
In the following, we set We suppose that, if the tensor of deformation velocities is equal to zero at \(t=0\), the same condition is true for the tensor of viscous stresses. Under this hypothesis, we can replace the differential relation (1) by an integral relation Indeed, if we denote the Laplace transformed of \(\sigma'\) and \(\tau\) by \(\Sigma'(p)\) and \(\mathcal{T}(p)\), we have From (5), noting that \(\frac{1}{\gamma+p}\mathcal{T}(p)\) is a product of transformed, we deduce that

3. The Boussinesq approximation

When a viscous or viscoelastic fluid is nonuniformly heated, its density \(\tilde{\rho}(t,x)\) depends on modifications of the temperature field, that is, the deviation \(T(t,x)\) of the temperature from some mean value \(T_{\text{m}}=\text{constant}\). For small \(T(t,x)\), we admit that \[ \tilde{\rho}(t,x)=\rho \left[ 1-\delta T(t,x) \right] \] where \(\rho=\text{constant}\) is the density corresponding to \(T_{\text{m}}\) and \(\delta>0\) is the coefficient of thermal extension.
In accordance with the Boussinesq approximation [1, 2], we admit that, in the equation of motion of the fluid, the density can be considered as constant, except in the external forces.

4. Position and equations of the problem

We introduce orthogonal axes of coordinates \(Ox_{1}x_{2}x_{3}\), \(Ox_{3}\) directed vertically upwards. The heavy viscoelastic fluid occupies a bounded domain \(\Omega\) with regular boundary \(S\) and \(g\) is the constant acceleration of the gravity. The fluid is nonuniformly heated from below.

The equations of the problem are: taking into account the Boussinesq approximation, the hydrodynamic equations take the form, the continuity equation the equation of heat conduction where \(\vec{u}(t,x)\) is the velocity of the fluid particle from its position at the state of mechanical equilibrium, \(P(t,x)\) is the deviation of pressure from the hydrostatic pressure corresponding to the constant temperature \(T_{\text{m}}\), \(X>0\) is the coefficient of temperature conductivity.
If we take \[ \tau_{ij}=\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}} \qquad (i,j=1,2,3) \] the components of the tensor \(\tau\), we have the components of the tensor \(\sigma'\) \[ \sigma'_{ij}=\mu \hat{I}_{0}\tau_{ij} \] and then \[ \frac{\partial \sigma'_{ij}}{\partial x_{j}}= \mu \hat{I}_{0} \frac{\partial \tau_{ij}}{\partial x_{j}} =\mu \hat{I}_{0} \left( \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}}+\frac{\partial}{\partial x_{i}} \left( \frac{\partial u_{j}}{\partial x_{j}} \right) \right)=\mu \hat{I}_{0} \Delta u_{i} \] since \(\frac{\partial u_{j}}{\partial x_{j}}=\text{div} \vec{u}=0\). Therefore, the vectorial equation of the motion of the fluid can be written We add the stickiness condition Now we will give boundary condition for the temperature.

5. Conditions of mechanical equilibrium

There exist conditions of heating such that the system is in the state of mechanical equilibrium: \(\vec{u}=0\). (But, then, there is no thermodynamic equilibrium, the temperature varying with the coordinates).
Setting \[ \vec{u}(t,x)=0;\qquad P=p_{0}(x);\qquad T=T_{0}(x), \] we obtain \[ \overrightarrow{\text{grad}}p_{0}=\rho g \delta T_{0} \vec{x}_{3};\qquad \Delta T_{0}=0 \] \(p_{0}\) and \(T_{0}\) are functions of \(x_{3}\) only and we have \[ \frac{\text{d}^2 T_{0}}{\text{d}x^{2}_{3}}=0 \] i.e \[ T_{0}=-\tilde{\alpha}x_{3}+\tilde{\alpha}_{0} \qquad (\tilde{\alpha}\;\;, \tilde{\alpha}_{0}\; \text{constants}). \] In the following, we consider the case \(\tilde{\alpha}>0\), corresponding to the heating from below.
The pressure in the state of mechanical equilibrium is given by \[ \frac{\text{d}p_{0}}{\text{d}x_{3}}=\rho g \delta(-\tilde{\alpha}x_{3}+\tilde{\alpha}_{0}) \] i.e \[ p_{0}(x_{3})=\rho g \delta \left(-\tilde{\alpha}\frac{x^{2}_{3}}{2}+\tilde{\alpha}_{0}x_{3} \right)+\text{constant} \] Concerning the boundary condition for the temperature, we suppose [2] that the temperature remain equal to \(T_{0}\) on the wall \(S\), so that, if we set \[ \theta(t,x)=T(t,x)-T_{0}(x_{3}) \] we have

6. Final formulation of the problem

Supposing the existence of the previous state of mechanical equilibrium, we seek the solutions of the equations in the form \[ \vec{u}=\vec{u}(t,x);\qquad P=p_{0}(x_{3})+p(t,x);\qquad T=T_{0}(x_{3})+ \theta(t,x) \] where \(\vec{u} \;, p\;, \theta\) are of the first order infinitesimal.
We get where \(\nu=\frac{\mu}{\rho}\) being the kinematic coefficient of viscosity. The equations (8), (11), (12) remains unchanged.

7. Transition to a system of operatorial equations

(1) We are going to seek \(\vec{u}(t,x),\; p(t,x)\), \(\theta (t,x)\) functions of \(t\) with values in \(J^{1}_{0}(\Omega)\), \(H^{1}(\Omega)\), \(H^{1}_{0}(\Omega)\) respectively, where \[ J^{1}_{0}(\Omega)=\left\{ \vec{u} \in \left[ H^1(\Omega)\right]^3;\;\; \text{div}\vec{u}=0 \;\; ; \;\; \vec{u}_{|S}=0 \right\} \] We introduce the space \[ J_{0}(\Omega)=\left\{ \vec{u} \in \mathcal{L}^2(\Omega) \overset{\text{def}}{=} \left[ L^2(\Omega)\right]^3;\;\; \text{div}\vec{u}=0\;\; ;\; u_{n|S}=0 \right\}, \] where \(u_{n|S}\) is the external normal component of \(\vec{u}\) on \(S\).\\ It is well-known [4] that the embedding from \(J^{1}_{0}(\Omega)\) into \(J_{0}(\Omega)\) is continuous, dense and compact and that we have the orthogonal decomposition where \(\mathcal{G} \left(\Omega \right)\) is the space of potential fields.
We have, for each \(\vec{\tilde{u}} \in J^{1}_{0}(\Omega) \): The first integral of the right-hand side is equal to zero, because \( \overrightarrow{\text{grad}}p \in \mathcal{G} \left(\Omega \right)\) and \(\vec{\tilde{u}} \in J^{1}_{0}(\Omega) \subset J_{0}(\Omega) \).
The vectorial Laplacian formula [4] can be written by setting \( \epsilon_{ij}=\frac{1}{2}\tau_{ij}\) as \[ \int_\Omega \, \Delta {\vec{u}} \cdot \bar{\vec{\tilde{u}}} \, \mathrm d\Omega= -2\int_\Omega \,\epsilon_{ij}(\vec{u})\epsilon_{ij}(\bar{\vec{\tilde{u}}})+ 2\int_S \,\epsilon_{ij}(\vec{u})n_{j}\bar{\tilde{u}}_{i} \, \mathrm dS \] and the last integral is equal to zero by virtue of (11).
It is well-known that \[ \left( 2\int_\Omega \,\epsilon_{ij}(\vec{u})\epsilon_{ij}(\bar{\vec{u}}) \, \mathrm d\Omega \right)^{1/2} \] defines a norm on \(J^{1}_{0}(\Omega)\), that is equivalent to the classical norm of \( \left[ H^{1}(\Omega) \right]^{3}\).
We denote by \(A_{0}\) the unbounded operator of \(J_{0}(\Omega)\) associated to the pair \((J^{1}_{0}(\Omega),J_{0}(\Omega)) \) and to this norm.
Finally, denoting by \(P_{0}\), the orthogonal projector from \(\mathcal{L}^2(\Omega)\) into \(J_{0}(\Omega)\), we can write
\[ \left( \delta g \theta \vec{x}_{3},\vec{\tilde{u}} \right)_{\mathcal{L}^2(\Omega)}= \left(P_{0} ( \delta g \theta \vec{x}_{3}) ,\vec{\tilde{u}}\right)_{J_{0}(\Omega)} \] So, the variational equation (16) is equivalent to the operatorial equation [5] (2) In the same manner, for every \(\tilde{\theta} \in H^{1}_{0}(\Omega)\), we have Since \(\tilde{\theta}|_{S}=0\), the Green formula gives \[ \int_\Omega \,\Delta \theta \cdot \bar{\tilde{\theta}} \, \mathrm d\Omega =-\int_\Omega \,\overrightarrow{\text{grad}} \theta \cdot \overrightarrow{\text{grad}}\bar{\tilde{\theta}} \, \mathrm d\Omega \] It is well-known that \((\int_\Omega \,|\overrightarrow{\text{grad}} \theta |^{2}\, \mathrm d\Omega)^{1/2} \) defines norm a on \( H^{1}_{0}(\Omega)\), that is equivalent to the classical norm of \(H^{1}(\Omega)\). We denote by \(A_{1}\) the unbounded operator of \(L^{2}(\Omega)\) associated to the pair \((H^{1}_{0}(\Omega), L^{2}(\Omega))\) and to this norm. So the variational equation (18) is equivalent to the operatorial equation [5] (3) By setting [2] \[ \theta (t,x)=\left( \frac{\tilde{\alpha}}{\delta g}\right)^{1/2} w(t,x) ; \qquad \epsilon=(\tilde{\alpha} \delta g)^{1/2} \] The equations (17) and (19) take the form (4) Now, we aim to replace the equation (20) by two equations whose coefficients do not depend on \(t \) [2].
As \[ \hat{I}_{0} A_{0}\vec{u}=A_{0}\vec{u}(t,x)+\alpha \int_{0}^{t} e^{-\gamma(t-s)}A_{0}\vec{u}(s,x) \, \mathrm ds \] By setting \[ \vec{u}_{0}(t,x)=\vec{u}(t,x); \qquad \vec{u}_{1}(t,x)= (\nu \alpha)^{1/2} \int_{0}^{t} e^{-\gamma(t-s)}A^{1/2}_{0}\vec{u}(s,x) \, \mathrm ds \] We have \[ (\nu \alpha)^{1/2} A^{1/2}_{0}\vec{u}(t,x)= \nu \alpha \int_{0}^{t} e^{-\gamma(t-s)}A_{0}\vec{u}(s,x) \, \mathrm ds \] and the equation (20) becomes Hence, we obtain The equation (21) remains unchanged.

8. The normal oscillations

(1) We seek the solutions of the equations (22), (23), (21) in the form \[ \vec{u}_{0}=e^{-\lambda t}\vec{u}_{0}(x) \, ;\qquad \vec{u}_{1}=e^{-\lambda t}\vec{u}_{1}(x)\, ;\qquad w=e^{-\lambda t}w(x) \, ;\qquad \lambda \in \mathbb{C} \] We obtain It is easy to verify that \(\lambda =\gamma\) is not an eigenvalue.
Indeed, if \(\lambda =\gamma\), (25) gives \(\vec{u}_{0}=0\), then (26) gives $$A_{1}w=\frac{\gamma }{X}w $$and consequently \(w=0\), if dismiss the exceptional case where \(\frac{\gamma }{X}\) is an eigenvalue of \(A_{1}\); finally, (24) gives \(\vec{u}_{1}=0\). Therefore, we have \[ \vec{u}_{1}=-\frac{(\nu \alpha)^{1/2} A_{0}^{1/2}\vec{u}_{0}}{\lambda - \gamma} \] so that, taking (3) into account, the equation (24) takes the form \[ \nu I_{0}(\lambda)A_{0}\vec{u}_{0}-\epsilon P_{0}(w\vec{x}_{3})=\lambda \vec{u}_{0} \] Let us introduce the operators \(C\) from \(L^2(\Omega)\) into \(J_{0}(\Omega)\) and \(C^{\ast}\) from \(J_{0}(\Omega)\) into \(L^2(\Omega)\) defined by \[ Cw=P_{0}(w\vec{x}_{3}) \qquad;\qquad C^{\ast}\vec{u}=\vec{u}\cdot \vec{x}_{3} \] It is easy to verify that these operators are bounded, with a norm smaller that one, and mutually adjoints. Since \(\vec{u}_{0}=\vec{u}\), we obtain the equations of the normal oscillations in the form (2) We can deduce from Equations (27), (28) with bounded coefficients by setting \[ A_{0}^{1/2}\vec{u}=\vec{U}\in J_{0}(\Omega);\qquad A_{1}^{1/2}w=V\in L^{2}(\Omega) \] and by applying the operators \(A_{0}^{-1/2} \) and \(A_{1}^{-1/2} \) to the equations (27), (28) respectively.
Setting \[ D=A_{0}^{-1/2} C A_{1}^{-1/2};\qquad D^{\ast}=A_{1}^{-1/2} C^{\ast} A_{0}^{-1/2} \] we obtain It is easy to see that \(D\) and \(D^{\ast}\) are mutually adjoint and compact.

9. Existence and symmetry of the spectrum

We write the equations (29) and (30) in the form \[ \mathbf{J} \begin{pmatrix} \vec{U} \\ V \end{pmatrix} + \begin{pmatrix} 0 &-\epsilon D \\ -\epsilon D^{\ast} & 0 \end{pmatrix} \begin{pmatrix} \vec{U} \\ V \end{pmatrix} -\lambda \begin{pmatrix} A_{0}^{-1} &0 \\ 0 & A_{1}^{-1} \end{pmatrix} \begin{pmatrix} \vec{U} \\ V \end{pmatrix} =0 \] where \[ \mathbf{J}= \begin{pmatrix} \nu I_{0}(\lambda)I_{J_{0}(\Omega) } &0 \\ 0 & XI_{L^{2}(\Omega)} \end{pmatrix} \] has an inverse and \[ \begin{pmatrix} \vec{U} \\ V \end{pmatrix} \in \chi \overset{\text{def}}{=} J_{0}(\Omega)\oplus L^{2}(\Omega) \] Applying \(\mathbf{J}^{-1}\), we obtain \begin{equation*} \left\{ \begin{array}{l} I_{\chi} \begin{pmatrix} \vec{U} \\ V \end{pmatrix} + \begin{pmatrix} 0 &\nu^{-1} I_{0}(\lambda)^{-1}\epsilon D \\ -X^{-1}\epsilon D^{\ast} & 0 \end{pmatrix} \begin{pmatrix} \vec{U} \\ V \end{pmatrix}\\[.2cm] \\[.2cm] -\lambda \begin{pmatrix} \nu^{-1} I_{0}(\lambda)^{-1}A_{0}^{-1} &0 \\ 0 & X^{-1} A_{1}^{-1} \end{pmatrix} \begin{pmatrix} \vec{U} \\ V \end{pmatrix} =0 \\[.2cm] \end{array} \right. \end{equation*} that has the form \[ \left( I_{\chi} +\Phi (\lambda) \right)\begin{pmatrix} \vec{U} \\ V \end{pmatrix}=0 \] where \(\Phi(\lambda)\) is a operatorial function with compact values. Consequently, we obtain a Fredholm pencil in \(\mathbb{C}-\left\lbrace\beta \right\rbrace\), since \(\beta\) is a pole of \(I_{0}(\lambda)^{-1}\). This pencil is regular, then since \(\lambda=\gamma\) is not an eigenvalue, \(I+\Phi(\lambda)\) has a bounded inverse [4]. Then, the spectrum of the problem exists and consists of isolated points and its accumulation points may be located in \(\lambda=\beta\) and \(\lambda=\infty\). All points of the spectrum are eigenvalues and the corresponding eigenelements have finite multiplicities. On the other hand, it is obvious that the initial pencil is self adjoint, so that the spectrum is symmetrical with respect to the real axis.

10. Location of the spectrum in the complex right half-plane

From the equations (29) and (30), we deduce We have \[ \left\{ \begin{array}{l} \Re \left(DV, \vec{U} \right)_{J_{0}(\Omega)}\leq \left|\left(DV, \vec{U} \right)_{J_{0}(\Omega)}\right| = \\[.2cm] \\[.2cm] \left|\left(A_{0}^{-1/2} C A_{1}^{-1/2}V, \vec{U} \right)_{J_{0}(\Omega)}\right|= \left|\left( C A_{1}^{-1}V, A_{0}^{-1/2}\vec{U} \right)_{J_{0}(\Omega)}\right| \\[.2cm] \end{array} \right. \] and consequently Noting that \[ \Re \left(\frac{1}{\gamma-\lambda}\right)=\frac{\gamma- \Re \lambda}{\left|\lambda - \gamma \right|^2 } \] and taking the real parts in (31), we obtain \[ \left\{ \begin{array}{l} \Re \lambda\left[ \left(A_{0}^{-1} \vec{U}, \vec{U} \right)_{J_{0}(\Omega)}+ \left(A_{1}^{-1} V, V \right)_{L^{2}(\Omega)} +\frac{\nu \alpha}{\left|\lambda - \gamma \right|^2 } \left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 \right] \\[.2cm] \\[.2cm] = \nu \left( 1+\frac{ \alpha \gamma}{\left|\lambda - \gamma \right|^2 }\right) \left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 +X\left\| V \right\|_{L^{2}(\Omega)}^2 -2\epsilon \Re \left(DV, \vec{U} \right)_{J_{0}(\Omega)} \\[.2cm] \end{array} \right. \] so that, using (32) \[ \left\{ \begin{array}{l} \Re \lambda\left[ \left(A_{0}^{-1} \vec{U}, \vec{U} \right)_{J_{0}(\Omega)}+ \left(A_{1}^{-1} V, V \right)_{L^{2}(\Omega)} +\frac{\nu \alpha}{\left|\lambda - \gamma \right|^2 } \left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 \right] \\[.2cm] \\[.2cm] \geq \left( \nu -\epsilon \left\| A_{0}^{-1} \right\| \right)\left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 +\left( X -\epsilon \left\| A_{1}^{-1} \right\| \right) \left\| V \right\|_{L^{2}(\Omega)}^2 \\[.2cm] \end{array} \right. \] Finally, under the sufficient conditions i.e. if \(\nu \) and \(X\) are sufficiently large, we have \[ \Re \lambda >0 \] so that the system is stable.

11. Existence of a set of positive real eigenvalues having \(\lambda =\beta\) as point of accumulation

The equation (30) can be written as \[ \left( X I_{L^{2}(\Omega)}-\lambda A_{1}^{-1} \right)V+\epsilon D^{\ast} \vec{U}=0 \] \(A_{1}^{-1}\) has a denumerable infinity of positive real eigenvalues, the largest being \( \left\| A_{1}^{-1} \right\|\). Consequently, \( XI_{L^{2}(\Omega)}-\lambda A_{1}^{-1}\) has a bounded inverse if \(\lambda\) is not real and if \(\lambda\) is real with \(\left|\lambda\right|< \frac{X}{ \left\| A_{1}^{-1} \right\|}\). Under this condition, we have \[ V=-\epsilon \left(XI_{L^{2}(\Omega)}-\lambda A_{1}^{-1} \right)^{-1}D^{\ast}\vec{U} \] Carrying out in the equation (29), we obtain \[ \nu I_{0}(\lambda) \vec{U}+\epsilon^{2}D\left(XI_{L^{2}(\Omega)}-\lambda A_{1}^{-1} \right)^{-1}D^{\ast}\vec{U}-\lambda A_{0}^{-1}\vec{U}=0 \] If \(X\) is sufficiently large, the inequality \(\left|\lambda\right|< \frac{X}{ \left\| A_{1}^{-1} \right\|}\) is satisfied for \(\lambda=\beta\) and for \(\lambda\) sufficiently close to \(\beta\).
Setting \[ \lambda=\lambda'+\beta,\qquad \left|\lambda'\right|\text{ sufficiently small} \] we obtain the equation \(\mathcal{L}(\lambda')\) is an operatorial function holomorphic in the vicinity of \(\lambda'=0\) and self adjoint. We have \[ \mathcal{L}(0)= \epsilon^{2}D\left(XI_{L^{2}(\Omega)}-\beta A_{1}^{-1} \right)^{-1}D^{\ast}-\beta A_{0}^{-1} \] that is compact.\\ Calculating the derivative, we obtain \[ \mathcal{L}'(0)= \frac{\nu}{\beta-\gamma}I_{J_{0}(\Omega)} +\epsilon^{2}D \left[ XI_{L^{2}(\Omega)}-\beta A_{1}^{-1} \right]^{-1}A_{1}^{-1}\left[ XI_{L^{2}(\Omega)}-\beta A_{1}^{-1} \right]^{-1} D^{\ast} - A_{0}^{-1} \] that is strongly positive if \(\nu\) is sufficiently large. So, [4], if \(\nu\) and \(X\) are sufficiently large, for each \(\eta>0\) sufficiently small, there exists in \(]\beta-\eta,\beta+\eta [\) a set of real eigenvalues of the spectrum of the problem having \(\beta\) as point of accumulation.

12. Location of the non real eigenvalues

We consider a non real eigenvalue \(\lambda\) and \( \begin{pmatrix} \vec{U} \\ V \end{pmatrix} \) an associated eigenelement.
The equations (29) and (30) can be written \[ \left\{ \begin{array}{l} \nu \frac{I_{0}(\lambda)}{\lambda} \vec{U} - \frac{\epsilon}{\lambda} DV=A_{0}^{-1} \vec{U} \\[.2cm] \\[.2cm] -\frac{\epsilon}{\lambda}D^{\ast}\vec{U}+\frac{X}{\lambda}VI_{L^{2}(\Omega)}=A_{1}^{-1} V \\[.2cm] \end{array} \right. \] From these equations, we deduce \[ \left\{ \begin{array}{l} \nu \frac{I_{0}(\lambda)}{\lambda} \left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 - \frac{\epsilon}{\lambda} \left(DV, \vec{U} \right)_{J_{0}(\Omega)}=\left(A_{0}^{-1} \vec{U}, \vec{U} \right)_{J_{0}(\Omega)} \\[.2cm] \\[.2cm] -\frac{\epsilon}{\bar{\lambda}}\left(DV, \vec{U} \right)_{J_{0}(\Omega)}+\frac{X}{\bar{\lambda}}\left\| V \right\|_{L^{2}(\Omega)}^2=\left(A_{1}^{-1} V, V \right)_{L^{2}(\Omega)} \\[.2cm] \end{array} \right. \] and then \[ \frac{\nu}{\epsilon} \Im \frac{I_{0}(\lambda)}{\lambda} \left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 -\frac{X}{\epsilon}\Im \frac{1}{\bar{\lambda}} \left\| V \right\|_{L^{2}(\Omega)}^2 =-\frac{2 \Im \lambda}{\left|\lambda\right|^{2}} \Re \left(DV, \vec{U} \right)_{J_{0}(\Omega)} \] Calculating \( \Im \frac{I_{0}(\lambda)}{\lambda}\) and dividing by \(\Im \lambda \neq 0\), we obtain \[ \frac{\nu}{2}\left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 \left[ \frac{\frac{\beta-\gamma}{\gamma}}{\left|\lambda-\gamma\right|^{2}}-\frac{\frac{\beta}{\gamma}}{\left|\lambda\right|^{2}} \right] -\frac{X}{\epsilon} \frac{1}{\left|\lambda\right|^{2}} \left\| V \right\|_{L^{2}(\Omega)}^2 = -\frac{2}{\left|\lambda\right|^{2}} \Re \left(DV, \vec{U} \right)_{J_{0}(\Omega)} \] or, using (32) \[ \left\{\frac{\nu}{2}\left[ \frac{\frac{\beta-\gamma}{\gamma}}{\left|\lambda-\gamma\right|^{2}}-\frac{\frac{\beta}{\gamma}}{\left|\lambda\right|^{2}} \right]+\frac{\left\| A_{0}^{-1} \right\|}{\left|\lambda\right|^{2}} \right\}\left\|\vec{U} \right\|_{J_{0}(\Omega)}^2 \geq \frac{1}{\left|\lambda\right|^{2}} \left[ \frac{X}{\epsilon}-\left\| A_{1}^{-1} \right\| \right]\left\| V \right\|_{L^{2}(\Omega)}^2 \] The right-hand side of above equation is positive according to (33), so that the eventual nonreal eigenvalues must verify with \[ k=\frac{\frac{\nu}{\epsilon} \cdot \frac{\beta}{\gamma}-\left\| A_{0}^{-1} \right\|}{\frac{\nu}{2} \cdot \frac{\beta-\gamma}{\gamma}} \] which is greater than \(1\) according to (33).
Setting \(\lambda=x+iy\), we see that the domain of the complex plane \((\lambda)\) defined by (35) is the disk \(\mathcal{D}\) defined by \[ \left(x-\frac{k\gamma}{k-1} \right)^2+y^{2}< \frac{k\gamma^{2}}{(k-1)^2} \] \(\lambda=\gamma\) belongs to \(\mathcal{D}\) and it is easy to verify that \(\lambda =\beta\) belongs to \(\mathcal{D}\).

13. Reduction to a Krein - Langer pencil

(1) Multiplying the equations 29) and (30) by \(\lambda - \gamma \neq 0\), we obtain We set \[ \mathbf{Q}= \begin{pmatrix} \nu \beta I_{J_{0}(\Omega)} & -\epsilon \gamma D \\ -\epsilon \gamma D^{\ast} & X \gamma I_{L^{2}(\Omega)} \end{pmatrix} \;\;; \;\; \mathbf{\tilde{B}}= \begin{pmatrix} \nu I_{J_{0}(\Omega)}+\gamma A_{0}^{-1} & -\epsilon D \\ -\epsilon D^{\ast} & XI_{L^{2}(\Omega)}+\gamma A_{1}^{-1} \end{pmatrix} \] \[ \mathcal{\tilde{C}}= \begin{pmatrix} A_{0}^{-1} &0 \\ 0 & A_{1}^{-1} \end{pmatrix}\;;\;\; \tilde{\lambda}=-\lambda\;\;;\;\; z=\begin{pmatrix} \vec{U} \\ V \end{pmatrix} \] We can rewrite the equations (36), (38) in the form (2) Let us study the properties of the operators of the equation (38). (a) \(\mathcal{\tilde{C}}\) is obviously self adjoint, positive definite, compact. (b) \(\mathbf{\tilde{B}}\) is bounded and self adjoint.
We have \[ \begin{aligned} &\left(\mathbf{\tilde{B}} z, z \right)_{\chi} = \nu \left\| \vec{U}\right\|^{2}_{J_{0}(\Omega)} +X\left\| V\right\|^{2}_{L^{2}(\Omega)} +\gamma \left[ \left(A_{0}^{-1} \vec{U}, \vec{U} \right)_{J_{0}(\Omega)} +\left(A_{1}^{-1} V, V \right)_{L^{2}(\Omega)} \right] \\ & \qquad \qquad \qquad -2\epsilon \Re \left(DV, \vec{U} \right)_{J_{0}(\Omega)} \end{aligned} \] Using the inequality (32), we obtain \[ \begin{aligned} &\left(\mathbf{\tilde{B}} z, z \right)_{\chi} \geq \left( \nu -\epsilon \left\| A_{0}^{-1} \right\| \right) \left\| \vec{U}\right\|^{2}_{J_{0}(\Omega)} + \left( X -\epsilon \left\| A_{1}^{-1} \right\| \right)\left\| V\right\|^{2}_{L^{2}(\Omega)} \\ & \qquad \qquad \qquad + \gamma \left[ \left(A_{0}^{-1} \vec{U}, \vec{U} \right)_{J_{0}(\Omega)} +\left(A_{1}^{-1} V, V \right)_{L^{2}(\Omega)} \right] \end{aligned} \] By virtue of the inequalities (33), \(\mathbf{\tilde{B}}\) is strongly positive. (c) \( \mathbf{Q}\) is bounded and self adjoint.
On the other hand, we have \[ \begin{aligned} &\left(\mathbf{Q} z, z \right)_{\chi} \geq \nu \left( \beta-\gamma \right)\left\| \vec{U}\right\|^{2}_{J_{0}(\Omega)} \\ & \qquad \qquad \qquad + \gamma \left[ \left( \nu -\epsilon \left\| A_{0}^{-1} \right\| \right) \left\| \vec{U}\right\|^{2}_{J_{0}(\Omega)} + \left( X -\epsilon \left\| A_{1}^{-1} \right\| \right)\left\| V\right\|^{2}_{L^{2}(\Omega)} \right] \end{aligned} \] so that \(\mathbf{Q}\) is strongly positive and then admits an inverse that has the same properties. (3) Setting \[ \mathbf{Q}^{1/2}z=W\in \chi \] carrying out in (38) and applying \(\mathbf{Q}^{-1/2}\), we obtain \[ W+\tilde{\lambda}\mathbf{Q}^{-1/2} \mathbf{\tilde{B}} \mathbf{Q}^{-1/2}W+ \tilde{\lambda}^{2}\mathbf{Q}^{-1/2} \mathcal{\tilde{C}} \mathbf{Q}^{-1/2}W=0 \] It is easy to see that \(\mathbf{B} =\mathbf{Q}^{-1/2} \mathbf{\tilde{B}} \mathbf{Q}^{-1/2}\) and \(\mathcal{C}=\mathbf{Q}^{-1/2} \mathcal{\tilde{C}} \mathbf{Q}^{-1/2}\) are bounded, self adjoint and positive definite and \(\mathcal{C}\) is compact.
So we obtain a Krein-Langer pencil [2; pp 295-309] (4) The theory of this pencil is treated in [3]. We restrict ourselves to an additional result.
Since \(\mathbf{B}\) is self adjoint, the set of the nonreal eigenvalues may have only the infinity as point of accumulation.
Consequently, since these eigenvalues are in the disk \(\mathcal{D}\), there are no more than a finite number of such eigenvalues.

14. Existence and unicity of the solution of the associated evolution problem

We write the equations (22), (23), (21) in the form \[ \dot{\mathbf{W}}+\mathcal{A} \mathbf{W}=0 \] with \[ \mathbf{W}= \begin{pmatrix} \vec{u}_{0}\\ \vec{u}_{1} \\ w \end{pmatrix} \qquad , \quad \mathcal{A}= \begin{pmatrix} \nu A_{0} & (\nu \alpha)^{1/2}A^{1/2}_{0} & -\epsilon C \\ -(\nu \alpha)^{1/2}A^{1/2}_{0} & \gamma I_{J_{0}(\Omega)} & 0 \\ -\epsilon C^{\ast} & 0 & XA_{1} \end{pmatrix} \] We are going to consider \(\mathcal{A}\) as unbounded operator of \[ H=J_{0}(\Omega)\oplus J_{0}(\Omega) \oplus L^{2}(\Omega) \] with domain \[ D(\mathcal{A})=D(A_{0})\oplus D(A^{1/2}_{0}) \oplus D(A_{1}) \] We are going to prove that \(-\mathcal{A}\) is maximal dissipative. (1) \(\mathcal{A}\) is obviously closed.
By direct calculations, we have \[ \begin{aligned} & \left(\mathcal{A} \mathbf{W}, \mathbf{W} \right)_{H}= \nu \left(A_{0} \vec{u}_{0}, \vec{u}_{0}\right)_{J_{0}(\Omega)} +\gamma \left\| \vec{u}_{1}\right\|^{2}_{J_{0}(\Omega)} +X \left(A_{1} w, w \right)_{L^{2}(\Omega)} -2\epsilon \Re \left(Cw, \vec{u}_{0}\right)_{J_{0}(\Omega)}\\ & \qquad \qquad \qquad +2i\Im \left(A^{1/2}_{0} \vec{u}_{1}, \vec{u}_{0}\right)_{J_{0}(\Omega)} \end{aligned} \] so that \[ \Re \left(\mathcal{A} \mathbf{W}, \mathbf{W} \right)_{H} \geq 0 \qquad \forall \mathbf{W}\in D(\mathcal{A}) \] under the condition (33). (2) We have \[ \mathcal{A}^{\ast}= \begin{pmatrix} \nu A_{0} & -(\nu \alpha)^{1/2}A^{1/2}_{0} & -\epsilon C \\ (\nu \alpha)^{1/2}A^{1/2}_{0} & \gamma I_{J_{0}(\Omega)} & 0 \\ -\epsilon C^{\ast} & 0 & XA_{1} \end{pmatrix} \] so that, in the same manner, we have \[ \Re \left(\mathcal{A}^{\ast} \mathbf{W}, \mathbf{W} \right)_{H} \geq 0 \qquad \forall \mathbf{W}\in D(\mathcal{A}^{\ast})=D(\mathcal{A}) \] Consequently, [3;p 54,55], if the initial value \(\mathbf{W}(0)\in D(\mathcal{A})\), i.e \[ \vec{u}_{0}(0)\in D(A_{0}) \qquad,\qquad \vec{u}_{1}(0)\in D(A^{1/2}_{0}) \qquad,\qquad w(0)\in D(A_{1}) \] the evolution problem has a solution and only one belonging to \(D(\mathcal{A})\) for \(t\geq 0\).

Acknowledgment

The authors are grateful to the referee and the editorial board for some useful comments that improved the presentation of the paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Muckenhoupt and Wheeden [1] have proved the boundedness of the fractional integral operators \(T_{\Omega,\mu}\) with power weights from \(L^{p}\) to \(L^{q}\). The boundedness of the fractional integral operators were studied by Calderón and Zygmund [2]. Ding, Chen and Fan [3] introduced the properties of \(T_{\Omega,\mu}\) on Hardy spaces. The theory of the variable exponent function spaces has been rapidly developed after the work [4] where Kováčik and Rákosník proved fundamental properties of Lebesgue spaces with variable exponent. After that, many researchers work in this direction, see [5, 6, 7, 8, 9, 10, 11]. Izuki [7] defined the class of Herz-Morrey spaces with variable exponent and considered the boundedness of the fractional integral on these spaces. In [11] the author's studied the boundedness of the fractional integral with variable kernel on \(M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})\) spaces.

Let \( S^{n-1} (n\geq 2) \) be the unit sphere in \(\mathbb{R}^{n}\) with normalized Lebesgue measure \( d\sigma(x')\). A function \(\Omega(x,z)\) defined on \( \mathbb{R}^{n} \times \mathbb{R}^{n}\) is said to be in \({L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq{1})\), if \(\Omega\) satisfies the following two conditions:
(i) For any \( x,z \in{\mathbb{R}^{n}}\) and any \(\lambda > 0\), one has \(\Omega{(x,\lambda{z})}=\Omega(x,z)\);
(ii) \(\|\Omega\|_{L^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}}:=\sup\limits_{x\in\mathbb{R}^{n}}\left(\int_{s^{n-1}}|{\Omega(x ,z')}|^{r}\mathrm{d}{\sigma}(z')\right)^{\frac{1}{r}}{< \infty}.\)
For \( 0\leq \mu < n \) the fractional integral operator with variable kernel \(T_{\Omega,\mu}\) is defined by $$T_{\Omega,\mu}f (x)= \int_{\mathbb{R}^{n}}\frac{\Omega(x, x-y )}{|x -y|^{n-\mu}}f(y)\mathrm{d}y.$$ The commutators of the fractional integral is defined by $$ [b^{m},T_{\Omega,\mu}]f (x)= \int_{\mathbb{R}^{n}}\frac{\Omega(x, x-y )}{|x -y|^{n-\mu}}(b(x)- b(y))^{m}f(y)\mathrm{d}y. $$ Throughout this paper, Let \(E\) be a Lebesgue measurable set in \(\mathbb{R}^{n}\) with measure \(|E|>0\), \(\chi_{E}\) means its characteristic function. Now, introduce the definition of the variable exponent Lebesgue spaces.

Definition [5] Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function, the Lebesgue space with variable exponent \(L^{p(\cdot)}(E)\) is defined by $$ L^{p(\cdot)}(E)= \{{ f~ \mbox{is measurable}: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} \mathrm{d}x <\infty}~ \mbox{for some constant } \eta > 0\}. $$

The space \(L _{loc}^{p(\cdot)} {(E)}\) is defined by $$ L_{loc}^{p(\cdot)} {(E)}= \{ \mbox {f is measurable}: f\in {L^{p(\cdot)} {(K)}}~\mbox{for all compact}~K\subset E\}. $$ The Lebesgue spaces \(L^{p(\cdot)} {(E)}\) is a Banach spaces with the norm defined by $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\left\{\eta> 0 : \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}\mathrm{d}x \leq 1\right\}. $$

We denote \(p_{-}=\) ess inf \(\{p(x): x \in E\} , \) \( p_{+}=\) ess sup\( \{p(x): x \in E\} \). Then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\). Let \(B_{k}=\{ x\in\mathbb{R}^{n} : |x|\leq 2^{k}\}, C_{k}= B_{k}\backslash B_{k-1}, \chi_{k}= \chi_{C_{k}}, \) \( k \in{\mathbb{Z}}\).

Definition [7] Let \(\alpha \in\mathbb{R}\), \(0< q < \infty\), \(p(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) and \(0 \leq\lambda< \infty\). The homogeneous Herz\(-\) Morrey spaces with variable exponent \(M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})\) is defined by $$ M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})= \left\{f\in {L_{Loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|f\|_{M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})}< \infty \right\}, $$ where $$ \|f\|_{M\dot{K}_{q,p{(\cdot)}}^{\alpha,\lambda}(\mathbb{R}^{n})}:= \sup_{L\in \mathbb{Z}} 2^{-L\lambda} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q}\|f\chi_{k}\|_{L^{p(\cdot)}}^{q}\right\}^{1/q}. $$

2. Properties of variable Lebesgue spaces

In this section, we state some properties of variable exponent Lebesgue spaces.

Proposition 3. Assume that \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies the follows inequalities: \begin{align*} | p(x) - p(y)| &\leq \frac{ -C}{Log( |x - y|)},\qquad | x - y| \leq 1/ 2;\\ | p(x) - p(y)| &\leq \frac{ C}{Log( e +|x|)},\qquad |y|\geq|x|; \end{align*} then, we have \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\).

Lemma 4 [4] Let \(p(\cdot): \mathbb{R}^{n} \rightarrow [1 , \infty)\), for all function \(f\) and \(g\), there exists the fact $$ \int_{\mathbb{R}^{n}}|f(x) g(x)| dx \leq C\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 5. [5] Assume that \(E\) is a Lebesgue measurable subset of \(\mathbb{R}^{n}\) with positive measure and \( p(\cdot)\in \mathcal{P}\), if \(f: E\times E \longrightarrow \mathbb{R}\) be a measurable function (with respect to product measure) such that for almost every \(y\in E\), \(f(\cdot,y)\in L^{p(\cdot)}(E)\). Then we have $$ \left\|\int_{E}f(\cdot,y)dy\right\|_{L^{p(\cdot)}(E)}\leq C \int_{E}\left\|f(\cdot,y)\right\|_{L^{p(\cdot)}(E)}\mathrm{d}y. $$

Lemma 6. 12 If \(0 < \mu < n \), and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq1)\) satisfies the \(L^{r}-\) Dini condition. If there exists an \( 0< \alpha < 1/2\) such that \( |y|< \alpha_{0}R ,\) we have $$ \left(\int_{R< |x|< 2 R} \left|\frac{\Omega(x, x- y)}{|x - y|^{n - \mu})}- \frac{\Omega(x,x)}{|x|^{n-\mu}}\right|^{r} \mathrm{d}x \right)^{{\frac{1}{r}}}\leq CR^{(\frac{n}{r}- n +\mu)} \left(\frac{|y|}{R} + \int_{|y|/2R}^{|y|/R} \frac{\omega_{r}(\delta)}{\delta} \mathrm{d}\delta \right). $$

Lemma 7. [13] If \(x \in \mathbb{R}^{n}\) and defined \(\widetilde{q}(x)\) by \(\frac{1}{p(x)}= \frac{1}{q} + \frac{1}{\widetilde{q}(x)}\), for all measurable function \(f\) and \(g\), we have $$ \|f(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{r}(\mathbb{R}^{n})}\|f(x)\|_{L^{\widetilde{q}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 8. [13] Suppose that \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) and \( 0 < p^{-}\leq p^{+} < \infty,\) (1)\ For any cube (or ball) and \(|Q|\leq 2^{n}\), all the \(\chi \in Q\), then: \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p(x)}\). (2)\ For any cube (or ball) and \(|Q|\geq 1\), then \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p_{\infty}}\), where \( p_{\infty} = \lim_{ x \rightarrow\infty} p(x)\).

Lemma 9. [8] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), then there exist constants \(C, \delta,\delta_{1} > 0 \) such that for all balls \(B\) in \(\mathbb{R}^{n}\) and all measurable subset \(S\subset B\) $$ \frac{\|\chi_{S}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} \leq C \left( \frac{|S|}{|B|}\right)^{\delta}, \frac{\|\chi_{S}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left( \frac{|S|}{|B|}\right)^{\delta_{1}}. $$

Lemma 10. [8] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), there exists constant \(C > 0\) such that for any balls B in \(\mathbb{R}^{n}\), we have $$ \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$

Lemma 11. [14] Assume that \(b^{m} \in BMO(\mathbb{R}^{n})\), \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}} \), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$ \|T^{b^{m}}_{\Omega , \gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C\|b\|^{m}_{BMO(\mathbb{R}^{n})} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 12. [15] Assume that \(b^{m}\in Lip_{\beta}(\mathbb{R}^{n})\), \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}},\) \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma+m\beta}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\) we have $$ \|T^{b^{m}}_{\Omega , \gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|^{m}_{Lip_{\beta}} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 13. [10] Let \(b^{m}\in Lip_{\beta}(\mathbb{R}^{n})\); \(m\) is a positive integer, and there exist constants \(C> 0\), such that for any \( k,j \in\mathbb{Z}\) with \(k>j\), we have (1)\ \(C^{-1}\|b\|^{m}_{Lip_{\beta}}\leq |B|^{-m\beta/n}\|\chi_{B}\|^{-1}_{L^{p(\cdot)}(\mathbb{R}^{n})}\|( b- b_{B})^{m}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C \|b\|^{m}_{Lip_{\beta}}\); (2)\ \(\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C |B_{k}|^{m\beta/n} \|b\|^{m}_{Lip_{\beta}}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

Lemma 14. [16] Let \(b^{m} \in BMO(\mathbb{R}^{n})\); \(m\) is a positive integer, and there exist constants \(C> 0\), such that for any \( k,j \in\mathbb{Z}\) with \(k>j\), we have (1)\ \( C^{-1}\|b\|^{m}_{*}\leq \sup\limits_{B} \frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|( b- b_{B})^{m}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C\|b\|_{*}^{m};\) (2)\ \(\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C (k - j)^{m}\|b\|^{m}_{*}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

3. Main theorems and their proofs

In this section, we will prove the main results of this paper.

Theorem 15. Assume that \(b^{m} \in BMO(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2}< \infty.\) Let \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r > p_{2}^{+})\), and the integral modulus of continuity \(\omega_{r}(\delta)\) satisfying

If \(p_{1}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) such that \(0< \mu\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\mu}{n}\). Then for all \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\), we have $$\|T^{b^{m}}_{\Omega , \mu} f\|_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} \leq C\|b\|^{m}_{*} \| f\|_{M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}.$$

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\) arbitrarily, we use the following inequality $$\left(\sum\limits_{k=1}^{\infty}a_{k}\right)^{q}\leq \sum\limits_{k=1}^{\infty}a_{k}^{q} ~~~~~~~~~~ \mbox{such that}~~~~ (a_{1}, a_{2}.....) \geq 0,$$ we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}&= \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{2}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{1}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{1}}\right\}. \end{align*} Let \( f(x) = \sum\limits_{k=-\infty}^{\infty}f(x)\chi_{k}= {\sum\limits_{k=- \infty}^{ \infty}} f_{j}(x)\). Then we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{k-2}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &~~~~+ \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &= U_{1}+ U_{2}. \end{align*} First, we consider \( U_{1}\). By the vanishing condition of \(f_{j}\), applying Lemma 5 and Minkowski inequality when \(j\leq k-2\) we have \begin{eqnarray*} \quad\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b(y)|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y \\ &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b_{B}|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &&+ \int_{B_{j}} |b_{B}-b(y)|^{m}|f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &=& U_{11} + U_{12}. \end{eqnarray*} For \(U_{11}\), we define \( \frac{1}{p_{2}(x)}= \frac{1}{r} + \frac{1}{\widetilde{p}_{2}(x)},\) such that \({\widetilde{p_{2}}(x)}> 1\), by Lemma 7 and Lemma 14, we have \begin{align*} &~~\left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} |b(x)-b_{B}|^{m} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B}|^{m}\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} (k-j)^{m}\|b\|^{m}_{*} \|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}. \end{align*} According to Lemma 8 and the formula \( \frac{1}{\tilde{p}_{2}(x)} = \frac{1}{p_{2}(x)} - \frac{1}{r}\), we have $$\|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})} \approx \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{\mu}{n}}.$$ Applying Lemma 6, noting that \(2^{j -k }\leq 2 ^{(j - k)\beta}\) we get \begin{align*} \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} &\leq CR^{(\frac{n}{r}- n +\mu)} \left(\frac{|y|}{2^{k-1}} + \int\limits_{|y|/2^{k}}^{|y|/2^{k-1}} \frac{\omega_{r}(\delta)}{\delta} \mathrm{d}\delta \right) \\ &\leq CR^{(\frac{n}{r}- n +\mu)} \left( 2^{j-k} + 2^{(j-k)\beta} \int\limits_{0}^{1} \frac{\omega_{r}(\delta)}{\delta^{1+\beta}} \mathrm{d}\delta\right) \\ &\leq CR^{(\frac{n}{r}- n +\mu)} 2^{(j-k)\beta} \left( 1+ \int\limits_{0}^{1} \frac{\omega_{r}(\delta)}{\delta^{1+\beta}} \mathrm{d}\delta \right)\\ &=C2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta}. \end{align*} Thus, we have \begin{align*} U_{11}&\leq C(k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta}\|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} \int_{B_{j}}|f_{j}(y)| \mathrm{d}y\\ &\leq C(k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} For \(U_{12}\), similar to \(U_{11}\), we have \begin{eqnarray*} \left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq& C 2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} By Lemma 14, we obtain \begin{align*} U_{12} &\leq C 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\||b_{B_{j}}-b(y)|^{m}\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C \|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} So, $$\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C (k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Using Lemma 4, Lemma 9 and Lemma 10, we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& C (k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(k-j)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-kn} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C (k-j)^{m}\|b\|^{m}_{*}2^{(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\frac{\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B_{k}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)\beta}2^{(j- k)n\delta_{1}}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)(\beta+n\delta_{1})} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} Then we have \begin{align*} U_{1}&\leq C \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=-\infty}^{k-2} (k-j)^{m}2^{\alpha{k}}2^{(j-k)(\beta+n\delta_{1})}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=-\infty}^{k-2} (k-j)^{m}2^{\alpha{j}}2^{(j-k)(\beta+n\delta_{1}-\alpha)} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}} \end{align*} When \(1< q_{1}< \infty\), take \(1/q_{1}+1/q'_{1}=1\). Noting that \( \alpha < n\delta_{1} + \beta \), by the Hölder's inequality we obtain

When \(0< q_{1}\leq1\), we have Finally, we estimate \(U_{2}\). By the boundedness of the \(T^{b^{m}}_{\Omega,\mu}\) on \(L^{p(\cdot)}(\mathbb{R}^{n})\)(Lemma 11 ), we have \begin{align*} U_{2}&\leq C \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left(\sum\limits_{j=k-1}^{\infty}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=k-1}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &~~~~~+ \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k+2}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &= U_{21} + U_{22}. \end{align*} First, we consider \(U_{21}\), then we have \begin{align*} U_{21} &\leq C\|b\|^{m q_{1}}_{*}\sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\end{align*} For \(U_{22}\), when \(\lambda < \alpha \) we have Thus, by (2)-(5), we finishes the proof of Theorem 15.

Theorem 16. Assume that \(b^{m} \in Lip_{\beta}(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2}< \infty.\) Let \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r > p_{2}^{+})\), and the integral modulus of continuity \(\omega_{r}(\delta)\) satisfying 1. If \(p_{1}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) such that \(0< \mu+m\beta\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\mu+m\beta}{n}\). Then for all \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\), we have $$\|T^{b^{m}}_{\Omega , \mu} f\|_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} \leq C\|b\|^{m}_{Lip_{\beta}} \| f\|_{M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}.$$

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\) arbitrarily, we use the following inequality $$\left(\sum\limits_{k=1}^{\infty}a_{k}\right)^{q}\leq \sum\limits_{k=1}^{\infty}a_{k}^{q} ~~~~~~~~~~ \mbox{such that}~~~~ (a_{1}, a_{2}.....) \geq 0,$$ we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}&= \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{2}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{1}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{1}}\right\}. \end{align*} Let \( f(x) = \sum_{k=-\infty}^{\infty}f(x)\chi_{k}= {\sum_{k=- \infty}^{ \infty}} f_{j}(x)\). Then we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{k-2}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &~~~~+ \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &= A_{1}+ A_{2}. \end{align*} First, we consider \( A_{1}\). By the vanishing condition of \(f_{j}\) and Lemma 5, the Minkowski inequality when \(j\leq k-2\) we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq&\int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b(y)|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y \\ &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b_{B}|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &&+ \int_{B_{j}} |b_{B}-b(y)|^{m}|f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &=& A_{11} + A_{12}. \end{eqnarray*} For \(A_{11}\), we define \( \frac{1}{p_{2}(x)}= \frac{1}{r} + \frac{1}{\widetilde{p}_{2}(x)}\) such that \({\widetilde{p_{2}}(x)}> 1\), by Lemma 7 and Lemma 13, we have \begin{align*} &~~\left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} |b(x)-b_{B}|^{m} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B}|^{m}\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} |B_{k}|^{m\beta/n}\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}. \end{align*} According to Lemma 8 and the formula \( \frac{1}{\tilde{p}_{2}(x)} = \frac{1}{p_{2}(x)} - \frac{1}{r}\), we have $$\|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})} \approx \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{(\mu+m\beta)}{n}}.$$ By Lemma 7 know that $$\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\leq C2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta}.$$ Thus, we have \begin{align*} A_{11}&\leq C |B_{k}|^{m\beta/n}\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{(\mu+m\beta)}{n}} \int_{B_{j}} |f_{j}(y)|\mathrm{d}y\\ &\leq C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(j-k)\beta}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \int_{B_{j}}|f_{j}(y)| \mathrm{d}y.\\ &\leq C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} For \(A_{12}\), we obtain that \begin{eqnarray*} \left\| { \left|\frac{\Omega(x, x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C |B_{k}|^{-m\beta/n}2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \end{eqnarray*} Then, by Lemma 13, we get \begin{align*} A_{12}&\leq C |B_{k}|^{-m\beta/n}2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\||b_{B_{j}}-b(y)|^{m}\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\end{align*}\begin{align*} &\leq C\|b\|^{m}_{Lip(\mathbb{R}^{n})} \frac{|B_{j}|^{m\beta/n}}{|B_{k}|^{m\beta/n}} 2^{-kn+(j-k)\beta}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn+(j-k)\beta}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} So, we have that $$\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Using Lemma 4, Lemma 9 and Lemma 10, we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(k-j)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})}\\ &\leq& C\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{(j-k)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-kn} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\frac{\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B_{k}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}}\\ &\leq& \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{(j-k)\beta}2^{(j- k)n\delta_{1}}\| f_{j}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})}\\ &\leq& \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{(j-k)(\beta+n\delta_{1})} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} When \(1< q_{1}< \infty\), take \(1/q_{1}+1/q'_{1}=1\). Noting that \( \alpha < n\delta_{1} + \beta \), by the Hölder's inequality we have

When \(0< q_{1}\leq1\), we have Finally, we estimate \(A_{2}\). By the boundedness of the \(T^{b^{m}}_{\Omega,\mu}\) on \(L^{p(\cdot)}(\mathbb{R}^{n})\)(Lemma 12 ), we have \begin{align*} A_{2}&\leq C \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left(\sum\limits_{j=k-1}^{\infty}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=k-1}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &+ \|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k+2}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}.\\ \end{align*} The rest of the proof is the same as the proof of \(U_{2}\) in Theorem 15, we omit the details there. Then, we can easily see that By (6)-(8) the proof of Theorem 16 is complete.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The main purpose of this paper (see Theorem 7) is to establish the existence and uniqueness of solutions for the Navier problem \[ (P)\left\{ \begin{array}{lll} & Lu(x) = f(x) - {\textrm{div}}(G(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u(x) = {\Delta}u(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] where $$ Lu(x) = {\Delta}{\big[}{\omega}_1(x)\,{\vert{\Delta}u\vert}^{p-2}{\Delta}u + {\nu}_1(x)\,{\vert{\Delta}u\vert}^{q-2}{\Delta}u{\big]}-{\textrm{div}}{\big[}{\omega}_2(x){\vert {\nabla}u\vert}^{p-2}{\nabla}u + {\nu}_2(x)\, {\vert{\nabla}u\vert}^{s-2}{\nabla}u){\big]},$$ \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) is a bounded open set, \(\displaystyle{\dfrac{f}{{\omega}_2}}\,{\in}L^{p\,'}(\Omega,{\omega}_2)\), \(\displaystyle{\dfrac{G}{{\nu}_2}}\, {\in}\,[L^{s\,'}(\Omega , {\nu}_2)]^N\), \({\omega}_1\), \({\omega}_2\), \({\nu}_1\) and \({\nu}_2\) are four weight functions (i.e., \({\omega}_i\) and \({\nu}_i\), \(i=1,2\) are locally integrable functions on \({\mathbb{R}}^N\) such that \(0< {\omega}_i(x), {\nu}_i(x)< {\infty}\) a.e. \(x{\in}{\mathbb{R}}^N\)), \({\Delta}\) is the Laplacian operator, \(1< q,s< p< {\infty}\), \(1/p + 1/p\,' =1\) and \(1/s+1/s\,'=1\).

For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1, 2, 3, 4, 5, 6, 7, 8]). The type of a weight depends on the equation type.

A class of weights, which is particularly well understood, is the class of \(A_p\) weights that was introduced by B.Muckenhoupt in the early 1970's (see [7]). These classes have found many useful applications in harmonic analysis (see [9] and [10]). Another reason for studying \(A_p\)-weights is the fact that powers of the distance to submanifolds of \({\mathbb{R}}^N\) often belong to \(A_p\) (see [8] and [11]). There are, in fact, many interesting examples of weights (see [7] for p-admissible weights).

In the non-degenerate case (i.e. with \({\omega}(x) \equiv 1\)), for all \(f\, {\in}\,L^p(\Omega)\) the Poisson equation associated with the Dirichlet problem \[ \left\{ \begin{array}{ll} &\, - \, {\Delta}u = f(x), \ {\textrm{in}} \ {\Omega} \\ & u(x) = 0, \ {\textrm{in}} \ {\partial\Omega} \end{array} \right. \] is uniquely solvable in \(W^{2,p}(\Omega)\,{\cap}\, W_0^{1,p}(\Omega)\) (see [12]), and the nonlinear Dirichlet problem \[ \left\{ \begin{array}{ll} &\, - \, {\Delta}_p u = f(x), \ {\textrm{in}} \ {\Omega} \\ & u(x) = 0, \ {\textrm{in}} \ {\partial\Omega} \end{array} \right. \] is uniquely solvable in \(W_0^{1,p}(\Omega)\) (see [13]), where \({\Delta}_p u = {\textrm{div}}({\vert {\nabla}u\vert}^{p-2}{\nabla}u)\) is the p-Laplacian operator. In the degenerate case, the degenerated p-Laplacian has been studied in [11].

The paper is organized as follow. In Section 2 we present the definitions and basic results. In Section 3 we prove our main result about existence and uniqueness of solutions for problem \((P)\).

2. Definitions and basic results

Let \(\Omega\) be an open set in \({\mathbb{R}}^n\). By the symbol \({\mathcal{W}}(\Omega)\) we denote the set of all measurable, a.e. in \(\Omega\) positive and finite functions \({\omega}={\omega}(x)\), \(x\, {\in}\, {\Omega}\). Elements of \({\mathcal{W}}(\Omega)\) will be called weight functions. Every weight \(\omega\) gives rise to a measure on the measurable subsets of \({\mathbb{R}}^N\) through integration. This measure will be denoted by \({\mu}_{\omega}\). Thus, \(\displaystyle {\mu}_{\omega}(E) = \int_E{\omega}(x)\, dx\) for measurable sets \(E\,{\subset}\,{\mathbb{R}}^N\).

Definition 1. Let \(1\,{\leq}\,p< {\infty}\). A weight \(\omega\) is said to be an \(A_p\)-weight, if there is a positive constant \(C = C({p , \omega})\) such that, for every ball \(B\,{\subset}\,{\mathbb{R}}^N\) \begin{eqnarray*} & & {\biggr(}{\frac{1}{{\vert B \vert}}} \int_B {\omega}(x)\,dx{\biggr)}{\biggl(}{\frac{1}{{\vert B \vert}}} \int_B{\omega}^{1/(1-p)}(x)\,dx{\biggr)}^{p-1}\,{\leq}\, C, \ \ {\textrm{if}} \ \ p>1,\\ & & {\biggr(}{\frac{1}{{\vert B \vert}}} \int_B{\omega}(x)\,dx{\biggr)}{\biggl(} { \textrm{ess}}\sup_{x\,{\in}\,B}{\frac{1}{{\omega}(x)}}{\biggr)}\, {\leq}C, \ \ {\textrm{if}} \ \ p=1, \end{eqnarray*} where \(\vert . \vert\) denotes the \(N\)-dimensional Lebesgue measure in \({\mathbb{R}}^N\).

If \(1< q\,{\leq}\,p\), then \(A_q\,{\subset}\,A_p\) (see [5, 6, 8] for more information about \(A_p\)-weights). As an example of an \(A_p\)-weight, the function \({\omega}(x) = {\vert x \vert}^{\alpha}\), \(x\,{\in}\,{\mathbb{R}}^N\), is in \(A_p\) if and only if \(-N< {\alpha}< N(p-1)\) (see [8], Chapter IX, Corollary 4.4). If \({\varphi}\, {\in}\, BMO({\mathbb{R}}^N)\), then \({\omega}(x) = {\textrm{e}}^{{\alpha}\, {\varphi}(x)}\,{\in}\,A_2\) for some \({\alpha}>0\) (see [9]).

Remark 1. If \({\omega}\, {\in}\, A_p\), \(1< p< {\infty}\), then $${\biggl(} {\frac{\vert E \vert}{\vert B \vert}}{\biggr)}^p\, {\leq} \, C \, {\frac{{\mu}_{\omega}(E)}{{\mu}_{\omega}(B)}}$$ for all measurable subsets \(E\) of \(B\) (see 15.5 strong doubling property in [6]). Therefore, \({\mu}_{\omega}(E)=0\) if and only if \({\vert E \vert}=0\); so there is no need to specify the measure when using the ubiquitous expression almost everywhere and almost every, both abbreviated a.e..

Definition 2. Let \(\omega\) be a weight. We shall denote by \(L^p(\Omega ,\omega)\) (\(1\,{\leq}\,p< {\infty}\)) the Banach space of all measurable functions \(f\) defined in \(\Omega\) for which $${\Vert f \Vert}_{L^p(\Omega ,\omega)} = {\bigg(}\int_{\Omega} {\vert f(x) \vert}^p{\omega}(x)\,dx{\bigg)}^{1/p}< {\infty}.$$ We denote \(\displaystyle [L^{p}(\Omega , \omega)]^N = L^{p}(\Omega , \omega)\,{\times}...{\times}\, L^{p}(\Omega , \omega)\).

Remark 2. If \({\omega}\,{\in}\,A_p\), \(1< p< \infty\), then since \({\omega}^{-1/(p-1)}\) is locally integrable, we have \(L^p(\Omega , \omega)\,{\subset}\,L^1_{\textrm{loc}}(\Omega)\) (see [8], Remark 1.2.4). It thus makes sense to talk about weak derivatives of functions in \(L^p(\Omega , \omega)\).

Definition 3. Let \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) be a bounded open set, \(1< p< {\infty}\), \(k\) be a nonnegative integer and \({\omega}\,{\in}\,A_p\). We shall denote by \(W^{k,p}(\Omega , \omega)\), the weighted Sobolev spaces, the set of all functions \(u\,{\in}\,L^p(\Omega , \omega)\) with weak derivatives \(D^{\alpha}u\,{\in}\,L^p(\Omega , \omega)\), \(1\,{\leq}\,{\vert \alpha\vert}\,{\leq}\,k\). The norm in the space \(W^{k,p}(\Omega , \omega)\) is defined by

We also define the space \(W_0^{k,p}(\Omega , \omega)\) as the closure of \(C_0^{\infty}(\Omega)\) with respect to the norm (1). We have that the spaces \(W^{k,p}(\Omega , \omega)\) and \(W_0^{k,p}(\Omega , \omega)\) are Banach spaces (see Proposition 2.1.2 in [8]).

The dual space of \(W_0^{1,p}(\Omega , \omega)\) is the space \([W_0^{1,p}(\Omega , \omega)]^* = W^{-1,p\,'}(\Omega , \omega)\), $$W^{-1,p\,'}(\Omega , \omega) = \{T=f-{\textrm{div}}(G): G=(g_1,...,g_N), {\dfrac{f}{\omega}}, {\dfrac{g_j}{\omega}}\, {\in}\, L^{p\,'}(\Omega , \omega)\}.$$ It is evident that a weight function \(\omega\) which satisfies \(0< C_1\,{\leq}\,{\omega}(x)\,{\leq}\,C_2\), for a.e. \(x\,{\in}\,{\Omega}\), gives nothing new (the space \({\textrm{W}}^{k,p}(\Omega , \omega)\) is then identical with the classical Sobolev space \({\textrm{W}}^{k,p}(\Omega)\)). Consequently, we shall be interested in all above such weight functions \(\omega\) which either vanish somewhere in \({\Omega}\,{\cup}\,{\partial\Omega}\) or increase to infinity (or both). We need the following basics results.

Theorem 4. (The weighted Sobolev inequality) Let \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) be a bounded open set and let \({\omega}\) be an \(A_p\)-weight, \(1< p< {\infty}\). Then there exists positive constants \(C_{\Omega}\) and \(\delta\) such that for all \(\,u{\in}\,W_0^{1,p}(\Omega, \omega)\) and \(\displaystyle 1\,{\leq}\,{\eta}\,{\leq}\, N/(N-1) + {\delta}\)

Proof. Its suffices to prove the inequality for functions \(u\, {\in}\, C_0^{\infty}(\Omega)\) (see Theorem 1.3 in [4]). To extend the estimates (2) to arbitrary \(u\, {\in}\, W_0^{1,p}(\Omega , \omega)\), we let \(\{u_m\}\) be a sequence of \(C_0^{\infty}(\Omega)\) functions tending to \(u\) in \(W_0^{1,p}(\Omega , \omega)\). Applying the estimates (2) to differences \(u_{m_1}-u_{m_2}\), we see that \(\{u_m\}\) will be a Cauchy sequence in \(L^p(\Omega , \omega)\). Consequently the limit function \(u\) will lie in the desired spaces and satisfy (2).

Lemma 5. (a) Let \(1\,< p< {\infty}\), then exists a constant \(C_p>0\) such that for all \({\xi}, {\eta}\, {\in}\, {\mathbb{R}}^N\), $${\big\vert}{\vert{\xi}\vert}^{p-2}\,{\xi} - {\vert{\eta}\vert}^{p-2}{\eta} {\big\vert}\, {\leq}\, C_p\, {\vert \xi - \eta \vert}(\,{\vert \xi \vert} + {\vert \eta \vert})^{p-2}.$$ (b) Let \(1< p< {\infty}\). There exist two positive constants \({\alpha}_p\) and \({\beta}_p\) such that for every \({\xi},{\eta}\, {\in}\,{\mathbb{R}}^N\) (\(N\,{\geq}\,1)\) $${\alpha}_p(\,{\vert \xi \vert}+{\vert \eta \vert})^{p-2}{\vert \xi - \eta \vert}^2\, {\leq}\,{\langle}\,{\vert \xi \vert}^{p-2}{\xi} - {\vert \eta \vert}^{p-2}{\eta} , {\xi - \eta}{\rangle}\,{\leq}\, {\beta}_p(\,{\vert \xi\vert} + {\vert {\eta}\vert})^{p-2}{\vert \xi - \eta\vert},$$ where \({\langle}. , . {\rangle}\) denotes here the Euclidian scalar product in \({\mathbb{R}}^N\).

Proof. See Proposition 17.2 and Proposition 17.3 in [13].

3. Weak Solutions

Let \({\omega}_1, {\omega}_2\, {\in}\, A_p\) and \({\nu}_1,{\nu}_2\,{\in}\, {\cal W}(\Omega)\), \(1< q,s< p< {\infty}\). We denote by \(X\) the space \linebreak \(\displaystyle X = W^{2,p}(\Omega , {\omega}_1)\, {\cap}\, W_0^{1,p}(\Omega , {\omega}_2)\) with the norm $${\Vert u \Vert}_X = {\bigg(} \int_{\Omega}{\vert {\nabla}u\vert}^p\, {\omega}_2\, dx + \int_{\Omega}{\vert {\Delta}u\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p}.$$ In this section we prove the existence and uniqueness of weak solutions \(u\, {\in}\,X\) to the Navier problem \[ (P)\left\{ \begin{array}{lll} & Lu(x) = f(x) - {\textrm{div}}(G(x)),\ \ {\textrm{in}} \ \ {\Omega}, \\ & u(x) = {\Delta}u= 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] where \(\Omega\) is a bounded open set of \({\mathbb{R}}^N\) (\(N\,{\geq}\,2\)), \(\displaystyle{\dfrac{f}{{\omega}_2}}\,{\in}\,L^{p\,'}(\Omega, {\omega}_2)\) and \(\displaystyle{\dfrac{G}{{\nu}_2}}\,{\in}\, [L^{s\,'}(\Omega , {\nu}_2)]^N\), \(G=(g_1,...,g_N)\).

Definition 6. We say that \(u\, {\in}\,X\) is a weak solution for problem \((P)\) if

for all \({\varphi}\,{\in}\,X\), with \(f/{\omega}_2\, {\in}\, L^{p\,'}(\Omega , {\omega}_2)\) and \(G/{\nu}_2\, {\in}\, [L^{s\,'}(\Omega , {\nu}_2)]^N\), where \({\langle}.,.{\rangle}\) denotes here the Euclidean scalar product in \({\mathbb{R}}^N\).

Remark 3 (a) Since \(1< q,s< p < {\infty}\) and if \(\displaystyle {\dfrac{{\nu}_1}{{\omega}_1}}\, {\in}\, L^{p/(p-q)}(\Omega , {\omega}_1)\) and \(\displaystyle {\dfrac{{\nu}_2}{{\omega}_2}}\, {\in}\, L^{p/(p-s)}(\Omega , {\omega}_2)\), there exist two constants \(M_1, M_2>0\) such that $${\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}\, {\leq}\, M_1{\Vert u \Vert}_{L^p(\Omega, {\omega}_1)} \ {\textrm{and}} \ {\Vert u \Vert}_{L^s(\Omega , {\nu}_2)}\, {\leq}\, M_2{\Vert u \Vert}_{L^p(\Omega, {\omega}_2)}$$ where \(\displaystyle M_1= {\bigg[}\int_{\Omega} {\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}{\omega}_1\,dx{\bigg]}^{(p-q)/p\,q}\) and \(\displaystyle M_2= {\bigg[}\int_{\Omega} {\bigg(}{\dfrac{{\nu}_2}{{\omega}_2}}{\bigg)}{\omega}_2\,dx{\bigg]}^{(p-s)/p\,s}\). In fact, since \(1< q,s< p< \infty\), we have \(r =p/q>1\) and \(r' = p/(p-q)\), \begin{eqnarray*} {\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}^q & = & \int_{\Omega}{\vert u \vert}^q\,{\nu}_1\, dx = \int_{\Omega}{\vert u \vert}^q {\dfrac{{\nu}_1}{{\omega}_1}}\, {\omega}_1\, dx\\ &{\leq}& {\bigg(}\int_{\Omega}{\vert u \vert}^{q\, r}\, {\omega}_1\, dx{\bigg)}^{1/r}{\bigg(}\int_{\Omega}{\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}^{r\,'}\,{\omega}_1\, dx{\bigg)}^{1/r\,'}\\ & = & {\bigg(}\int_{\Omega}{\vert u \vert}^p\, {\omega}_1\, dx{\bigg)}^{q/p}{\bigg(}\int_{\Omega}{\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}^{p/(p-q)}\,{\omega}_1\, dx{\bigg)}^{(p-q)/p}. \end{eqnarray*} Hence, \(\displaystyle {\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}\, {\leq}\, M_1\, {\Vert u \Vert}_{L^p(\Omega , {\omega}_1)}\). Analogously, we obtain \(\displaystyle {\Vert u \Vert}_{L^s(\Omega , {\nu}_2)}\, {\leq}\, M_2\, {\Vert u \Vert}_{L^p(\Omega , {\omega}_2)}\).
(b) Using the estimate in (a) we have \begin{eqnarray*} {\bigg\vert}\int_{\Omega}{\vert {\Delta}u\vert}^{q-2}{\Delta}u\, {\Delta}{\varphi}\, {\nu}_1\, dx{\bigg\vert} & {\leq} & \int_{\Omega}{\vert {\Delta}u\vert}^{q-1}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx\\ & {\leq}&{\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^{(q-1)\,q\,'}{\nu}_1\, dx{\bigg)}^{1/q\,'}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q}\\ & = & {\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^q\, {\nu}_1\, dx{\bigg)}^{(q-1)/q}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\,{\nu}_1\, dx{\bigg)}^{1/q}\\ & = & {\Vert {\Delta} u \Vert}_{L^q(\Omega , {\nu}_1)}^{q-1}{\Vert {\Delta}{\varphi}\Vert}_{L^q(\Omega , {\nu}_1)}\\ & {\leq} & M_1^{q-1}\, {\Vert {\Delta}u \Vert}_{L^p(\Omega , {\omega}_1)}^{q-1} \, M_1\, {\Vert {\Delta}{\varphi}\Vert}_{L^p(\Omega , {\omega}_1)}\\ & {\leq}& M_1^q\, {\Vert u \Vert}_X\, {\Vert \varphi \Vert}_X, \end{eqnarray*} and, analogously, we also have $${\bigg\vert}\int_{\Omega}{\vert{\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx{\bigg\vert}\, {\leq}\, M_2^s\,{\Vert u \Vert}_X\, {\Vert \varphi \Vert}_X.$$

Theorem 7. Let \({\omega}_i\, {\in}\, A_p\), \({\nu}_i\, {\in}\,{\cal W}(\Omega)\) (\(i=1,2\)), \(1< q, s < p< {\infty}\). Suppose that
(a) \(\displaystyle {\dfrac{{\nu}_1}{{\omega}_1}}\, {\in}\,L^{p/(p-q)}(\Omega , {\omega}_1)\) and \(\displaystyle {\dfrac{{\nu}_2}{{\omega}_2}}\, {\in}\,L^{p/(p-s)}(\Omega , {\omega}_2)\);
(b) \(f/{\omega}_2\, {\in}\,L^{p\,'}(\Omega , {\omega}_2)\) and \(G/{\nu}_2\, {\in}\, [L^{s\,'}(\Omega , {\nu}_2)]^N\).
Then the problem \((P)\) has a unique solution \(u\, {\in}\,X\) and $${\Vert u \Vert}_X\, {\leq}\,{\bigg[}C_{\Omega} {\bigg\Vert {\dfrac{f}{{\omega}_2}}\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert {\dfrac{{\vert G \vert}}{{\nu}_2}}\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg]}^{1/(p-1)},$$ where \(C_{\Omega}\) is the constant in Theorem 3 and \(M_2\) is the constant in 3 (a).

Proof. (I) Existence. By Theorem 4 (with \({\eta}=1\)), we have that \begin{eqnarray*} {\bigg\vert}\int_{\Omega}f\, {\varphi}\,dx{\bigg\vert} & {\leq} & {\bigg(}\int_{\Omega} {\bigg\vert}{\dfrac{f} {{\omega}_2}}{\bigg\vert}^{p\,'}\, {\omega}_2\, dx{\bigg)}^{1/p\,'}{\bigg(}\int_{\Omega}{\vert\,\varphi\vert}^p\, {\omega}_2\, dx{\bigg)}^{1/p} \end{eqnarray*}

and by Remark 3(a) Define the functional \(J:X\, {\rightarrow}\, {\mathbb{R}}\) by \begin{eqnarray*} J(\varphi) & = & {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\Delta}{\varphi}{\vert}^p\, {\omega}_1\, dx + {\dfrac{1}{q}}\, \int_{\Omega}{\vert}{\Delta}{\varphi}{\vert}^q\, {\nu}_1\, dx\\ && + {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\, {\omega}_2\, dx + {\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\, {\nu}_2\, dx - \int_{\Omega}\, f\, {\varphi}\, dx - \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx. \end{eqnarray*} Using (4), (5), Remark 3(a) and Young's inequality (\(a\,b\, {\leq}\, {\dfrac{a^p}{p}} + {\dfrac{b^{p\,'}}{p\,'}}\)), we have that \begin{eqnarray*} J(\varphi) & {\geq}& {\dfrac{1}{p}}\int_{\Omega}{\vert {\Delta}{\varphi} \vert}^p\,{\omega}_1\, dx +{\dfrac{1}{q}}\int_{\Omega}{\vert {\Delta}{\varphi}\vert}^q\,{\nu}_1\,dx + {\dfrac{1}{p}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx +{\dfrac{1}{s}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx\\ && -{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega ,{\omega}_2)}{\Vert \varphi \Vert}_{L^p(\Omega , {\omega}_2)} -{\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega ,{\nu}_2)}{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\geq} & {\dfrac{1}{p}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx+{\dfrac{1}{s}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx - C_{\Omega}{\bigg\Vert}{\dfrac{ f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}{\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}\\ && - {\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)} {\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\geq} & {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx +{\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx - {\dfrac{C_{\Omega}^{p'}}{p'}}{\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}^{p'} \\&&- {\dfrac{1}{p}}\,{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}^p - {\dfrac{1}{s'}}\,{\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}^{s'} - {\dfrac{1}{s}}\,{\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}^s\\ & = & - {\dfrac{C_{\Omega}^{p'}}{p\,'}}{\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}^{p'} - {\dfrac{1}{s\,'}}{\bigg\Vert} {\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}^{s'} \end{eqnarray*} that is, \(J\) is bounded from below. Let \(\{u_n\}\) be a minimizing sequence, that is, a sequence such that $$J(u_n)\,\,\, {\rightarrow}\, \inf_{{\varphi}\, {\in}\,X}J(\varphi)\,.$$ Then for \(n\) large enough, we obtain \begin{eqnarray*}0\, {\geq}\, J(u_n) & = & {\dfrac{1}{p}}\int_{\Omega}{\vert{\Delta}u_n\vert}^p\,{\omega}_1\, dx + {\dfrac{1}{q}}\int_{\Omega}{\vert{\Delta}u_n\vert}^q\,{\nu}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert {\nabla}u_n\vert}^p\,{\omega}_2\, dx + {\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}u_n{\vert}^s\, {\nu}_2\, dx\\ & - & \int_{\Omega}f\, u_n\, dx - \int_{\Omega}{\langle}G , {\nabla}u_n{\rangle}\, dx, \end{eqnarray*} and we have \begin{eqnarray*} & & {\dfrac{1}{p}}\int_{\Omega}{\vert{\Delta}u_n\vert}^p\,{\omega}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert{\nabla}u_n\vert}^{p}\, {\omega}_2\, dx\end{eqnarray*} Hence, by Theorem 4 (with \({\eta}=1\)), Remark 3(a) and (6), we obtain \begin{eqnarray*} {\Vert} u_n{\Vert}_X^p &=& \int_{\Omega}{\vert {\Delta}u_n\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert {\nabla}u_n\vert}^p\,{\omega}_2\, dx\\ & & {\leq}\, p {\bigg(}\int_{\Omega}f\, u_n\, dx + \int_{\Omega}{\langle}G , {\nabla}u_n{\rangle}\, dx{\bigg)}\\ & & {\leq} \, p\,{\bigg(}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}\, {\Vert u_n \Vert}_{L^p(\Omega , {\omega}_2)} + {\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)} {\Vert\,\vert {\nabla}u_n\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\,{\bigg)}\\ & & {\leq}\, p\, {\bigg(}C_{\Omega}\, {\bigg\Vert}{\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}{\Vert\,\vert {\nabla}u_n\vert\,\Vert}_{L^p(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}{\Vert\,\vert{\nabla}u_n\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}{\bigg)}\\ & & {\leq}\, p\, {\bigg(}C_{\Omega}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{q'}(\Omega , {\nu}_2)}{\bigg)}{\Vert}u_n{\Vert}_X. \end{eqnarray*} Hence, $$\displaystyle {\Vert}u_n{\Vert}_X\, {\leq}\, {\bigg[}p\, {\bigg(}\,C_{\Omega}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2\, {\bigg\Vert} {\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}{\bigg)}{\bigg]}^{1/(p-1)}.$$ Therefore \(\{u_n\}\) is bounded in \(X\). Since \(X\) is reflexive, there exists a subsequence, still denoted by \(\{u_n\}\), and a function \(u\,{\in}\,X\) such that \(u_n{\rightharpoonup}\, u\) in \(X\). Since, $$ X\, {\ni}\,{\varphi}\, \mapsto \, \int_{\Omega}\, f\, {\varphi}\, dx + \int_{\Omega}{\langle}G,{\nabla}{\varphi}{\rangle}\, dx,$$ and $$ X\,{\ni}\,{\varphi}\, \mapsto \,{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , {\omega}_1)}^p + {\Vert{\Delta}{\varphi}\Vert}_{L^q(\Omega , {\nu}_1)}^q + {\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}^p + {\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}^s,$$ are continuous then \(J\) is continuous. Moreover since \(1< q,s< p< {\infty}\) we have that \(J\) is convex and thus lower semi-continuous for the weak convergence. It follows that $$J(u)\, {\leq}\, \liminf_{n} J(u_n)\, = \, \inf_{{\varphi}\, {\in}\,X}J(\varphi),$$ and thus \(u\) is a minimizer of \(J\) on \( X \) (see Theorem 25.C and Corollary 25.15 in [14]). For any \({\varphi}\, {\in}\,X\) the function \begin{eqnarray*} {\lambda}\, \mapsto \, & & {\dfrac{1}{p}}\int_{\Omega}{\vert}{\Delta}(u+{\lambda}{\varphi}){\vert}^p\, {\omega}_1\, dx + {\dfrac{1}{q}}\int_{\Omega}{\vert}{\Delta}(u+{\lambda}{\varphi}){\vert}^q\, {\nu}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert{\nabla}(u+{\lambda}\, {\varphi})\vert}^p\,{\omega}_2\, dx \\ & &+ {\dfrac{1}{s}} \, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}{\varphi}){\vert}^s\, {\nu}_2\, dx - \int_{\Omega}(u+{\lambda}\, {\varphi})\, f\, dx - \int_{\Omega}{\langle}G , {\nabla}(u+{\lambda}\, {\varphi}){\rangle}\,dx \end{eqnarray*} has a minimum at \({\lambda}=0\). Hence, $${\dfrac{d}{d{\lambda}}}{\bigg(}J(u+{\lambda}\,{\varphi}){\bigg)} {\bigg\vert}_{{\lambda}=0} = 0, \ {\forall}\, {\varphi}\, {\in}\, X.$$ We have $${\dfrac{d}{d\,{\lambda}}} \, {\bigg(}{\vert}\,{\nabla}(u+{\lambda}\, {\varphi}){\vert}^p\, {\omega}_2{\bigg)} = p\, \{{\vert}{\nabla}(u+{\lambda}\, {\varphi}){\vert}^{p-2} ({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\, {\vert}{\nabla}{\varphi}{\vert}^2)\}\,{\omega}_2,$$ and $${\dfrac{d}{d\,{\lambda}}} \, {\bigg(}{\vert}\,{\Delta}(u+{\lambda}\, {\varphi}){\vert}^p\, {\omega}_1{\bigg)} = p\, {\vert{\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{p-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\,{\omega}_1,$$ and we obtain \begin{eqnarray*} 0 & = & {\dfrac{d}{d{\lambda}}}{\bigg(}J(u+{\lambda}\,{\varphi}){\bigg)} {\bigg\vert}_{{\lambda}=0} = {\bigg[}{\dfrac{1}{p}}{\bigg(}p\, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}\,{\varphi}){\vert}^{p-2}({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\,{\vert}{\nabla}{\varphi}{\vert}^2)\,{\omega}_2\, dx\end{eqnarray*} \begin{eqnarray*} && + p\,\int_{\Omega}{\vert {\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{p-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\, {\omega}_1\, dx{\bigg)} + {\dfrac{1}{s}}{\bigg(}s\, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}\,{\varphi}){\vert}^{s-2}({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\,{\vert}{\nabla}{\varphi}{\vert}^2)\,{\nu}_2\, dx{\bigg)}\nonumber\\ && + {\dfrac{1}{q}}{\bigg(} q\,\int_{\Omega}{\vert {\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{q-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\, {\nu}_1\, dx{\bigg)} - \int_{\Omega}{\varphi}\, f \, dx- \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx {\bigg]} {\bigg\vert}_{{\lambda}=0}\\ & = & \int_{\Omega} {\vert{\Delta}u\vert}^{p-2}{\Delta}u\,{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^{p-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx + \int_{\Omega} {\vert{\Delta}u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\, dx \nonumber\\ &&+ \int_{\Omega}{\vert}{\nabla}u{\vert}^{s-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx - \int_{\Omega} f\, {\varphi}\, dx - \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx. \end{eqnarray*} Therefore \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u\vert}^{p-2}{\Delta}u\,{\Delta}{\varphi}\, {\omega}_1\,dx + \int_{\Omega} {\vert}{\nabla}u{\vert}^{p-2}{\langle}{\nabla}u\,{\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\,dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^{s-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx\\&& = \int_{\Omega} f\, {\varphi}\, dx + \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} for all \({\varphi}\, {\in}\, X\), that is, \(u\, {\in}\, X\) is a solution of problem \((P)\).
(II) Uniqueness. If \(u_1, u_2\, {\in}\, X\) are two weak solutions of problem \((P)\), we have \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1\,{\Delta}{\varphi}\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1\,{\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert}{\nabla}u_1{\vert}^{p-2}\,{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx\\&& + \int_{\Omega}{\vert}{\nabla}u_1{\vert}^{s-2}\,{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle} G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} and \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2\,{\Delta}{\varphi}\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2\,{\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert}{\nabla}u_2{\vert}^{p-2}\,{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx \\&&+ \int_{\Omega}{\vert}{\nabla}u_2{\vert}^{s-2}\,{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle} G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} for all \({\varphi}\in X\). Hence \begin{eqnarray*} & &\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1 - {\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)}{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1 - {\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2{\bigg)}{\Delta}{\varphi}\, {\nu}_1\, dx\\ & & + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle} - {\vert}{\nabla}u_2{\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}{\bigg)}\,{\omega}_2\, dx + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle} \\&&- {\vert}{\nabla}u_2{\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}{\bigg)}{\nu}_2\, dx = 0. \end{eqnarray*} Taking \({\varphi}= u_1-u_2\), and using Lemma 5 (b) there exist positive constants \({\alpha}_p, {\tilde{\alpha}_p}, {\alpha}_q, {\alpha}_s\) such that \begin{eqnarray*} 0 & = & \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)} (\,{\Delta}u_1 - {\Delta}u_2)\, {\omega}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{q-2} {\Delta}u_2{\bigg)}(\,{\Delta}u_1 - {\Delta}u_2)\, {\nu}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}u_1 - {\nabla}u_2{\rangle} - {\vert}{\nabla}u_2{\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}u_1-{\nabla}u_2{\rangle}{\bigg)}\,{\omega}_2\, dx\\ && + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}u_1 - {\nabla}u_2{\rangle} - {\vert}{\nabla}u_2{\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}u_1-{\nabla}u_2{\rangle}{\bigg)}\,{\nu}_2\, dx\\ & = & \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)} (\,{\Delta}u_1 - {\Delta}u_2)\, {\omega}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{q-2} {\Delta}u_2{\bigg)}(\,{\Delta}u_1 - {\Delta}u_2)\, {\nu}_1\, dx\\ && + \int_{\Omega}{\langle}\, {\vert}{\nabla}u_1{\vert}^{p-2}{\nabla}u_1 - {\vert}{\nabla}u_2{\vert}^{p-2}{\nabla}u_2, {\nabla}u_1 - {\nabla}u_2{\rangle}\, {\omega}_2\, dx\end{eqnarray*}\begin{eqnarray*} && + \int_{\Omega}{\langle}\, {\vert}{\nabla}u_1{\vert}^{s-2}{\nabla}u_1 - {\vert}{\nabla}u_2{\vert}^{s-2}{\nabla}u_2, {\nabla}u_1 - {\nabla}u_2{\rangle}\, {\nu}_2\, dx\\ & {\geq} & {\alpha}_p \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{p-2} {\vert {\Delta}u_1 - {\Delta}u_2\vert}^2\,{\omega}_1\, dx + \, {\tilde{\alpha}_p} \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{p-2}{\vert {\nabla}u_1 - {\nabla}u_2 \vert}^2\, {\omega}_2\, dx \\ && + {\alpha}_q\, \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{q-2} {\vert {\Delta}u_1 - {\Delta}u_2\vert}^2\,{\nu}_1\, dx + {\alpha}_s \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{s-2}{\vert {\nabla}u_1 - {\nabla}u_2 \vert}^2\, {\nu}_2\, dx\\ & {\geq} & {\alpha}_p \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{p-2} {\vert {\Delta}u_1 - {\Delta}u_2\vert}^2\,{\omega}_1\, dx + {\tilde{\alpha}_p} \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{p-2}{\vert {\nabla}u_1 - {\nabla}u_2 \vert}^2\, {\omega}_2\, dx. \end{eqnarray*} Therefore \({\Delta}u_1={\Delta}u_2\) and \({\nabla}u_1 = {\nabla}u_2\) a.e. and since \(u_1,u_2\, {\in}\,X\), then \(u_1 = u_2\) a.e. (by Remark 1).
(III) Estimate for \({\Vert u \Vert}_X\).
In particular, for \({\varphi}=u\, {\in}\, X\) in Definition 6 we have \begin{eqnarray*} \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^q\,{\nu}_1\, dx + \int_{\Omega}{\vert {\nabla}u\vert}^p\,{\omega}_2\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^s\, {\nu}_2\, dx= \int_{\Omega}f\, u\, dx + \int_{\Omega}{\langle}G , {\nabla}u{\rangle}\, dx. \end{eqnarray*} Then, by Theorem 4 and Remark 3(a), we obtain \begin{eqnarray*} {\Vert u \Vert}_X^p & = & \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\nabla}u\vert}^p\,{\omega}_2\, dx\\ &{\leq} & \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^q\,{\nu}_1\, dx + \int_{\Omega}{\vert {\nabla}u\vert}^p\,{\omega}_2\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^s\, {\nu}_2\, dx\\ & = & \int_{\Omega}f\,u\,dx + \int_{\Omega}{\langle}G, {\nabla}u{\rangle}\, dx\\ & {\leq} & {\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p'}(\Omega , {\omega}_2)}{\Vert u \Vert}_{L^p(\Omega , {\omega}_2)} + {\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}{\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\leq} & C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)}{\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^p(\Omega , {\omega}_2)} + M_2 {\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)} {\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}\\ & {\leq} & {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s'}(\Omega , {\nu}_2)}{\bigg)}{\Vert u \Vert}_X. \end{eqnarray*} Therefore, $${\Vert u \Vert}_X {\leq}\,{\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$

Corollary 8. Under the assumptions of Theorem 7 with \(2\, {\leq}\,q,s < p < {\infty}\). If \(u_1,u_2\, {\in}\, X\) are solutions of \[ (P_1)\left\{ \begin{array}{lll} & Lu_1(x) = f(x) - {\textrm{div}}(G(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_1(x) = {\Delta}u_1(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] and \[ (P_2)\left\{ \begin{array}{lll} & Lu_2(x) = {\tilde{f}}(x) - {\textrm{div}}({\tilde{G}}(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_2(x) = {\Delta}u_2(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] then $${\Vert u_1 - u_2\Vert}_X\, {\leq}\, {\dfrac{1}{{{\gamma}}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}\, {\bigg\Vert} {\dfrac{f-{\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert} {\dfrac{{\vert G - {\tilde{G}} \vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)},$$ \noindent where \({\gamma}\) is a positive constant, \(C_{\Omega}\) and \(M_2\) are the same constants of Theorem 7.

Proof. If \(u_1\) and \(u_2\) are solutions of \((P1)\) and \((P2)\) then for all \({\varphi}\, {\in}\, X\) we have \begin{eqnarray*} \int_{\Omega}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1\,{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1\, {\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert{\nabla}u_1\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx \end{eqnarray*}

In particular, for \({\varphi}= u_1 - u_2\), we obtain in (7).
(i) By Lemma 5(b) and since \(2\, {\leq}\, q,s< p< {\infty}\), there exist two positive constants \({\alpha}_p\) and \({\alpha}_q\) such that \begin{eqnarray*} &&\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1 - {\vert{\Delta}_2\vert}^{p-2}{\Delta}u_2{\bigg)}\, {\Delta}(u_1 - u_2)\, {\omega}_1\, dx \geq {\alpha}_p\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} +{\vert{\Delta}u_2\vert}{\bigg)}^{p-2}\, {\vert}{\Delta}u_1 - {\Delta}u_2{\vert}^2\, {\omega}_1\, dx\\ & & {\geq}\, {\alpha}_p\, \int_{\Omega}{\vert {\Delta}u_1 - {\Delta}u_2\vert}^{p-2}{\vert}{\Delta} u_1 - {\Delta}u_2{\vert}^2\, {\omega}_1\, dx = {\alpha}_p \int_{\Omega}{\vert{\Delta}(u_1 - u_2)\vert}^p\, {\omega}_1\, dx, \end{eqnarray*} and analogously $$\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1 - {\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2{\bigg)}\, {\Delta}(u_1 - u_2)\, {\nu}_1\, dx \, {\geq}\, {\alpha}_q\int_{\Omega}{\vert{\Delta}(u_1 - u_2)\vert}^q\, {\nu}_1\, dx\, {\geq}\, 0.$$ (ii) Since \(2\, {\leq}\, q,s< p< {\infty}\) and by Lemma 5(b), there exit two positive constants \({\tilde{\alpha}}_p\) and \({\alpha}_s\) such that \begin{eqnarray*} & & \int_{\Omega} {\bigg(}{\vert{\nabla}u_1\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}(u_1 - u_2){\rangle} - {\vert{\nabla}u_2\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}(u_1 - u_2){\rangle}{\bigg)}\, {\omega}_2\, dx\\ & & = \int_{\Omega}{\langle} {\vert{\nabla}u_1\vert}^{p-2}{\nabla}u_1 - {\vert{\nabla}u_2\vert}^{p-2}{\nabla}u_2 , {\nabla}(u_1 - u_2){\rangle}\, {\omega}_2\, dx\\ & & {\geq}\, {\tilde{\alpha}}_p\,\int_{\Omega}({\vert{\nabla}u_1\vert}+{\vert{\nabla}u_2\vert})^{p-2}{\vert{\nabla}u_1 - {\nabla}u_2\vert}^2\, {\omega}_2\, dx\\ & & {\geq}\,{\tilde{\alpha}}_p\int_{\Omega}{\vert{\nabla}u_1 - {\nabla}u_2\vert}^{p-2}\, {\vert{\nabla}u_1 - {\nabla}u_2\vert}^2\, {\omega}_2\, dx = \ {\tilde{\alpha}}_p \int_{\Omega}{\vert {\nabla}(u_1 - u_2)\vert}^p\, {\omega}_2\, dx, \end{eqnarray*} and analogously, \begin{eqnarray*} \int_{\Omega} {\bigg(}{\vert{\nabla}u_1\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}(u_1 - u_2){\rangle} - {\vert{\nabla}u_2\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}(u_1 - u_2){\rangle}{\bigg)}\, {\nu}_2\, dx {\geq}\, {\alpha}_s \int_{\Omega}{\vert {\nabla}(u_1 - u_2)\vert}^s\, {\nu}_2\, dx\, {\geq}\, 0. \end{eqnarray*} (iii) By Remark 3(a) we have \begin{eqnarray*} & & {\bigg\vert}\int_{\Omega} (f - {\tilde{f}})\, (u_1 - u_2)\, dx + \int_{\Omega}{\langle} G - {\tilde{G}}, {\nabla}(u_1 - u_2){\rangle}\, dx{\bigg\vert}\\ & & {\leq} \ {\bigg(}C_{\Omega} {\bigg\Vert} {\dfrac{f - {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert}{\dfrac{{\vert G - {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}\, {\Vert u_1 - u_2\Vert}_X. \end{eqnarray*} Hence, with \({\gamma} = \min\{ {\alpha}_p, {\tilde{\alpha}}_p\}\), we obtain \begin{eqnarray*} & & {\gamma}\, {\Vert u_1 - u_2 \Vert}_X^p\, {\leq}\, {\alpha}_p\int_{\Omega}{\vert{\Delta}(u_1 - u_2)\vert}^p\, {\omega}_1\, dx + {\tilde{\alpha}}_p\int_{\Omega}{\vert{\nabla}(u_1 - u_2)\vert}^p\, {\omega}_2\, dx\\ & & {\leq}\,{\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f - {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G - {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}\, {\Vert u_1 - u_2\Vert}_X. \end{eqnarray*} Therefore, $${\Vert u_1 - u_2\Vert}_X\, {\leq}\, {\dfrac{1}{{\gamma}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f - {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G - {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$

Corollary 9. Assume \(2\,{\leq}\, q, s < p< {\infty}\). Let the assumptions of Theorem 7 be fulfilled, and let \(\{f_m\}\) and \(\{G_m\}\) be sequences of functions satisfying \(\displaystyle {\dfrac{f_m}{{\omega}_2}}\,{\rightarrow}\, {\dfrac{f}{{\omega}_2}}\) in \(L^{p\,'}(\Omega , {\omega}_2) \) and \(\displaystyle {\Bigg\Vert {\dfrac{\vert G_m - G \vert}{{\nu}_2}} \Bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\rightarrow}\,0\) as \(m\to\infty\). If \(u_m\,{\in}\, X\) is a solution of the problem \[ (P_m)\left\{ \begin{array}{lll} & Lu_m(x) = f_m(x) - {\textrm{div}}(G_m(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_m(x) = {\Delta}u_m(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] then \(u_m {\rightarrow}\, u\) in \(X\) and \(u\) is a solution of problem \((P)\).

Proof. By Corollary 8 we have $${\Vert u_m - u_r\Vert}_X\, {\leq}\, {\dfrac{1}{{\gamma}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f_m - {f}_r}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G_m - {G}_r\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$ Therefore \(\{u_m\}\) is a Cauchy sequence in \(X\). Hence, there is \(u\, {\in}\, X\) such that \(u_m\,{\rightarrow}\, u\) in \(X\). We have that \(u\) is a solution of problem \((P)\). In fact, since \(u_m\) is a solution of \((P_m)\), for all \({\varphi}\, {\in}\, X\) we have

where
\(I_1 =\int_{\Omega}{\bigg(}{\vert{\Delta} u\vert}^{p-2}{\Delta}u - {\vert{\Delta}u_m\vert}^{p-2}{\Delta}u_m{\bigg)}\, {\Delta}{\varphi}\,{\omega}_1\,dx,\)
\(I_2 = \int_{\Omega}{\bigg(}{\vert{\Delta} u\vert}^{q-2}{\Delta}u - {\vert{\Delta}u_m\vert}^{q-2}{\Delta}u_m{\bigg)}{\Delta}{\varphi}\, {\nu}_1\,dx,\)
\(I_3 = \int_{\Omega} {\bigg(}{\vert {\nabla}u\vert}^{p-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle} -{\vert{\nabla}u_m\vert}^{p-2}{\langle}{\nabla}u_m, {\nabla}{\varphi}{\rangle}{\bigg)} \,{\omega}_2\,dx,\)
\( I_4 = \int_{\Omega} {\bigg(}{\vert {\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle} - {\vert{\nabla}u_m\vert}^{s-2}{\langle}{\nabla}u_m , {\nabla}{\varphi}{\rangle}{\bigg)}\,{\nu}_2\,dx.\)
We have that:
(1) By Lemma 5 (a) there exists \(C_p>0\) such that \begin{eqnarray*} {\vert I_1 \vert} & {\leq} & \int_{\Omega}{\big\vert} {\vert{\Delta}u\vert}^{p-2}{\Delta}u - {\vert{\Delta}u_m\vert}^{p-2}{\Delta}u_m{\big\vert}\, {\vert{\Delta}{\varphi}\vert}\, {\omega}_1\, dx\\ & {\leq} & \, C_p\, \int_{\Omega}{\vert{\Delta}u - {\Delta}u_m\vert}\,({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{p-2}\, {\vert{\Delta}{\varphi}\vert}\, {\omega}_1\, dx. \end{eqnarray*} Let \(r = p/(p-2)\). Since \(\displaystyle {\dfrac{1}{p}} + {\dfrac{1}{p}} + {\dfrac{1}{r}} = 1\), by the Generalized Hölder's inequality we obtain \begin{eqnarray*} {\vert I_1\vert} & \leq& C_p\, {\bigg(}\int_{\Omega}{\vert{\Delta}u - {\Delta}u_m\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p} {\bigg(}\int_{\Omega}({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{(p-2)r}\, {\omega}_1\, dx{\bigg)}^{1/r}\end{eqnarray*}\begin{eqnarray*} {\leq} \, C_p {\Vert u - u_m\Vert}_X\, {\Vert \varphi\Vert}_X {\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^p(\Omega , {\omega}_1)}^{(p-2)}. \end{eqnarray*} Now, since \(u_m{\rightarrow}\, u\) in \(X\), then exists a constant \(M>0\) such that \({\Vert u_m\Vert}_X\, {\leq}\, M\). Hence, Therefore, \begin{eqnarray*} {\vert I_1 \vert} & {\leq} & C_p\, (2M)^{p-2}\, {\Vert u - u_m \Vert}_X\, {\Vert \varphi \Vert}_X = C_1 \, {\Vert u - u_m\Vert}_X\, {\Vert{\varphi}\Vert}_X. \end{eqnarray*} Analogously, there exists a constant \(C_3\) such that $${\vert I_3 \vert}\, {\leq}\, C_3 {\Vert u - u_m\Vert}_X\, {\Vert \varphi \Vert}_X.$$ (2) By Lemma 5 (a) there exists a positive constant \(C_q\) such that \begin{eqnarray*} {\vert I_2 \vert} & {\leq} & \int_{\Omega}{\big\vert} {\vert{\Delta}u\vert}^{q-2}{\Delta}u - {\vert{\Delta}u_m\vert}^{q-2}{\Delta}u_m{\big\vert}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx\\ & {\leq} & C_q\, \int_{\Omega}{\vert{\Delta}u - {\Delta}u_m\vert}\,({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{q-2}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx. \end{eqnarray*} Let \({\alpha} = q/(q-2)\) (if \(2< q< p< {\infty}\)). Since \(\displaystyle {\dfrac{1}{q}} + {\dfrac{1}{q}} + {\dfrac{1}{\alpha}} = 1\), by the Generalized Hölder's inequality we obtain \begin{eqnarray*} {\vert I_2\vert} & \leq &C_q\, {\bigg(}\int_{\Omega}{\vert{\Delta}u - {\Delta}u_m\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q} {\bigg(}\int_{\Omega}({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{(q-2){\alpha}}\, {\nu}_1\, dx{\bigg)}^{1/{\alpha}}\\ & =& C_q\, {\Vert {\Delta}u - {\Delta}u_m\Vert}_{L^q(\Omega,{\nu}_1)}\, {\Vert {\Delta}\varphi\Vert}_{L^q(\Omega , {\nu}_1)} {\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^q(\Omega , {\nu}_1)}^{q-2}. \end{eqnarray*} Now, by Remark 3(a) and (9) we have \begin{eqnarray*} {\vert I_2\vert} & {\leq}& C_q\ M_1 {\Vert {\Delta}u - {\Delta}u_m\Vert}_{L^p(\Omega,{\omega}_1)}\,M_1\, {\Vert {\Delta}\varphi\Vert}_{L^p(\Omega , {\omega}_1)} M_1^{q-2}\,{\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^p(\Omega , {\omega}_1)}^{q-2}\\ & {\leq} & C_q \ M_1^q {\Vert u - u_m \Vert}_X {\Vert \varphi \Vert}_X \ (2M)^{q-2}\\ & = & C_2 \, {\Vert u - u_m\Vert}_X \ {\Vert \varphi \Vert}_X. \end{eqnarray*} Analogously, if \(2< s< p< {\infty}\), there exists a positive constant \(C_4\) such that $${\vert I_4 \vert}\, {\leq}\, C_4\, {\Vert u - u_m \Vert}_X \ {\Vert \varphi \Vert}_X.$$ In case \(q=2\) and \(s=2\), we have \({\vert I_2 \vert}, {\vert I_4\vert}\, {\leq}\,M_1^2\, {\Vert u -u_m\Vert}_X\, {\Vert \varphi \Vert}_X\).
Therefore, we have \(I_1,I_2, I_3, I_4{\rightarrow}\, 0\) when \(m{\rightarrow}\, {\infty}\).
(3) We also have \begin{eqnarray*} {\bigg\vert}\int_{\Omega}(f_m - f)\,{\varphi}\, dx + \int_{\Omega}{\langle} G_m - G, {\nabla}{\varphi}{\rangle}\, dx{\bigg\vert} {\bigg(} C_{\Omega}{\bigg\Vert}{\dfrac{f_m - f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert}{\dfrac{{\vert G_m - G \vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega, {\nu}_2)}{\bigg)}{\Vert \varphi \Vert}_X {\rightarrow}\, 0, \end{eqnarray*} when \(m{\rightarrow}\, {\infty}\).
Therefore, in (8), we obtain when \(m{\rightarrow}\, {\infty}\) that \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta} u\vert}^{p-2}{\Delta}u\, {\Delta}{\varphi}\,{\omega}_1\,dx + \int_{\Omega}{\vert{\Delta} u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\,dx + \int_{\Omega} {\vert {\nabla}u\vert}^{p-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\,{\omega}_2\,dx\\ & & + \int_{\Omega} {\vert {\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\,{\nu}_2\,dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle}G, {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} i.e., \(u\) is a solution of problem \((P)\).

Example 1. Let \({\Omega} = \{ (x,y)\,{\in}\,{\mathbb{R}}^2 \, : \, x^2+y^2 < 1 \}\), \({\omega}_1(x,y) = (x^2+y^2)^{-1/2}\), \({\omega}_2(x,y) = (x^2+y^2)^{-1/4}\) (\({\omega}_i\, {\in}\, A_4\), \(p=4\) and \(q=s=3\)), \({\nu}_1(x,y) = (x^2+y^2)^{-1/3}\), \({\nu}_2(x,y) = (x^2+y^2)^{1/8}\), \(\displaystyle f(x,y) = {\dfrac{\cos(xy)}{(x^2+y^2)^{1/6}}}\) and \(\displaystyle G(x,y) ={\bigg(} {\dfrac{\sin(x+y)}{(x^2+y^2)^{1/6}}},{\dfrac{\sin(xy)}{(x^2+y^2)^{1/6}}}{\bigg)}\). By Theorem 7 , the problem \[ \left\{ \begin{array}{llll} & {\Delta}{\bigg[}(x^2+y^2)^{-1/2}\,{\vert{\Delta}u\vert}^2{\Delta}u + (x^2+y^2)^{-1/3} {\vert{\Delta}u\vert}{\Delta}u{\bigg]}\\ & -\,{\textrm{div}}{\bigg[}(x^2+y^2)^{-1/4}{\vert{\nabla}u\vert}^2{\nabla}u + (x^2+y^2)^{-1/8}{\vert{\nabla}u\vert}{\nabla}u{\bigg]}\\ & = f(x) - {\textrm{div}}(G(x)),\ \ {\textrm{in}} \ \ {\Omega} \\ & u(x) = {\Delta}u = 0, \ \ {\textrm{in}} \ \ {\partial\Omega} \end{array} \right. \] has a unique solution \(u\, {\in}\, W^{2,4}(\Omega , {\omega}_1)\, {\cap}\, W_0^{1,4}(\Omega , {\omega}_2)\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Fixed point theory of special mappings like nonexpansive, asymptotically nonexpansive, contractive and other mappings is an active area of interest and finds applications in many related fields like image recovery, signal processing and geometry of objects [1]. From time to time, some versions of theorems relating to fixed points of functions of special nature keep on appearing in almost in all branches of mathematics. Consequently, we apply them in industry, toy making, finance, aircrafts and manufacturing of new model cars. For example, a fixed-point iteration scheme has been applied in IMRT optimization to pre-compute dose-deposition coefficient (DDC) matrix, see [2]. Because of its vast range of applications almost in all directions, the research in it is moving rapidly and an immense literature is currently. The Construction of fixed point theorems (e.g. Banach fixed point theorem) which not only claim the existence of a fixed point but yield an algorithm, too (in the Banach case fixed point iteration \(x_{n+1}=f(x_n))\). Any equation that can be written as \(x=f(x)\) for some map $f$ that is contracting with respect to some (complete) metric on \(X\) will provide such a fixed point iteration. Mann's iteration method was the stepping stone in this regard and is invariably used in most of the occasions, see [3]. But it only ensures weak convergence, see [4] but more often then not, we require strong convergence in many real world problems relating to Hilbert spaces, see [5]. So mathematician are in search for the modifications of the Mann's process to control and ensure the strong convergence, (see [4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein). Most probably the first noticeable modification of Mann's Iteration process was proposed by Nakajo et al. in [8] in 2003. They introduced this modification for only one nonexpansive mapping in a Hilbert space where as Kim and Xu introduced a modification for asymptotically nonexpansive mapping in the Hilbert space in 2006, see [9]. In the same year Martinez et al. in [10] introduced a modification of the Ishikawa Iteration process for a nonexpansive mapping for a Hilbert space. They also gave modification of Halpern iteration method in Hilbert space. Su et al. in [11] gave a monotone hybrid iteration process for nonexpansive mapping in a Hilbert space. Liu et al. in [12] gave a novel iteration method for finite family of quasi-asymptotically pseudo-contractive mapping in a Hilbert space. Let \(H\) be the fixed notation for Hilbert space and \(C\) be nonempty, closed and convex subset of it. First we recall some basic definitions that will accompany us throughout this paper. Let \(P_c(.)\) be the metric projection onto \(C.\) A mapping \(T:C\rightarrow C\) is said to be non-expensive if \(\|Tx-Ty\|\leq \|x-y\|\) \(\forall\) \(x,y \in C\). And \(T:C\rightarrow C\) is said to be quasi-Lipschitz if

1. \(Fix T\neq \phi,\)
2. For all \(p \in FixT\), \(\| Tx-p\|\leq L\|x-p\|\) where \(L\) is a constant \(1\leq L< \infty.\)

If \(L=1\) then \(T\) is known as quasi-nonexpansive. It is well-known that \(T\) is said to be closed if for \(n\rightarrow \infty\), \(x_n \rightarrow x \) and \(\| Tx_n-x_n\|\rightarrow 0\) implies \(Tx=x.\) \(T\) is said to be weak closed if \(x_n\rightharpoonup x\) and \(\|Tx_n-x_n\|\rightarrow 0\) implies \(Tx=x.\) as \(n\rightarrow \infty.\) It is admitted fact that a mapping which is weak closed should be closed but converse is no longer true. Let \(\{T_n\}\) be a sequence of mappings having non-empty fixed points sets. Then \(\{T_n\}\) is defined to be uniformly closed if for all convergent sequences \(\{Z_n\} \subset C\) with conditions \(\|Zx_n-Z_n\|\rightarrow 0\), \(n\rightarrow \infty\) implies the limit of \(\{Z_n\}\) belongs to \(FixT_i.\) In 1953 [3], Mann proposed an iterative scheme given as: $$x_{n+1} =(1-\alpha_n)x_nn+\alpha_nT(x_n); n=0,1,2,\ldots.$$ Guan et al. in [7] established the following non-convex hybrid iteration algorithm corresponding to Mann iterative scheme: $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)x_n+\alpha_nT_nx_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq (1+(L_n-1)\alpha_n)\|x_n-z\|\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0, \end{array} \right.$$ In [7] Guan et al. established non-convex hybrid iteration algorithm and proved some strong convergence results relating to common fixed points for a uniformly closed asymptotic family of countable quasi-Lipschitz mappings in \(H.\) They applied their results for the finite case to obtain fixed points. In this article, we establish a non-convex hybrid algorithms. We also establish strong convergence theorems about common fixed points related to a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in the realm of Hilbert spaces. An application of this algorithm is also given. We fix \(\overline{co}C_n\) for closed convex closure of \(C_n\) for all \(n\geq 1\), \(A=\{z\in H:\|z-P_Fx_0\|\leq 1\}\), \({T_n}\) for countable quasi-\(L_n\)-Lipschitz mappings from \(C\) into itself, and \(T\) be closed quasi-nonexpansive mapping from \(C\) into itself to avoid redundancy. We also present an application of our algorithm.

2. Main results

In this part we formulate our main results. We start with some basic definitions.

Definition 1. \(\{T_n\}\) is said to be asymptotic, if \(\lim_{n\rightarrow \infty} L_n=1\)

Proposition 2. For \(x\in H\) and \(z\in C\), \(z=P_Cx\) iff we have \(\langle x-z,z-y\rangle\geq 0\) for all \(y\in C\).

Proposition 3. The common fixed point set \(F\) of above said \({T_n}\) is closed and convex.

Proposition 4. For any given \(x_0\in H\), we have \(p=P_Cx_0\) \(\Longleftrightarrow\) \(\langle p-z,x_0-p\rangle\geq 0\), \(\forall z\in C\).

Theorem 5. Suppose that \(\alpha_n\in (0,1]\), and \(\beta_n\in [0,1]\) for all \(n\in N\). Then \(\{x_n\}\) generated by $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)T_nx_n+\alpha_nT_nz_n, & n\geq 0,\\ z_n=(1-\beta_n)+\beta_nT_nx_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq L_n(1+(L_n-1)\alpha_n\beta_n)\|x_n-z\|\}\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0, \end{array} \right.$$ converges strongly to \(P_Fx_0\).

Proof. We give our proof in following steps.
Step 1. We know that \(\overline{co}C_n\) and \(Q_n\) are closed and convex for all \(n\geq 0\). Next, we show that \(F\cap A\subset\overline{co}C_n\) for all \(n\geq 0\). Indeed, for each \(p\in F\cap A\), we have \begin{align*} \|y_n-p\|&=\|(1-\alpha_n)T_nx_n+\alpha_nT_nz_n-p\|\\ &=\|(1-\alpha_n)T_nx_n+\alpha_nT_n((1-\beta_n)+\beta_nT_nx_n)-p\|\\ \nonumber &=\|(1-\alpha_n\beta_n)(T_nx_n-p)+(\alpha_n\beta_n)(T_{n}^2x_n)\|\\ \nonumber &\leq (1-\alpha_n\beta_n)\|T_nx_n-p\|+(\alpha_n\beta_n)\|T_{n}^2x_n\|\\ \nonumber &=L_n(1+(L_n-1)\alpha_n\beta_n)\|x_n-p\| \end{align*} and \(p\in A\), so \(p\in C_n\) which implies that \(F\cap A\subset C_n\) for all \(n\geq 0\). therefore, \(F\cap A\subset\overline{co}C_n\) for all \(n\geq 0\).
Step 2. We show that \(F\cap A\subset\overline{co}C_n\cap Q_n\) for all \(n\geq 0\). it suffices to show that \(F\cap A\subset Q_n\), for all \(n\geq 0\). We prove this by mathematical induction. For \(n=0\) we have \(F\cap A\subset C=Q_0\). Assume that \(F\cap A\subset Q_n\). Since \(x_{n+1}\) is the projection of \(x_0\) onto \(\overline{co}C_n\cap Q_n\), from Proposition 3, we have
\(\langle x_{n+1}-z,x_{n+1}-x_0\rangle\leq 0\), \(\forall z\in \overline{co}C_n\cap Q_n\)
as \(F\cap A\subset\overline{co}C_n\cap Q_n\), the last inequality holds, in particular, for all \(z\in F\cap A\). This together with the definition of \(Q_{n+1}\) implies that \(F\cap A\subset Q_{n+1}\). Hence the \(F\cap A\subset\overline{co}C_n\cap Q_n\) holds for all \(n\geq 0\).
Step 3. We prove \(\{x_n\}\) is bounded. Since \(F\) is a nonempty, closed, and convex subset of \(C\), there exists a unique element \(z_0\in F\) such that \(z_0=P_Fx_0\). From \(x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0\), we have
\(\|x_{n+1}-x_0\|\leq \|z-x_0\|\)
for every \(z\in \overline{co}C_n\cap Q_n\). As \(z_0\in F\cap A\subset\overline{co}C_n\cap Q_n\), we get
\(\|x_{n+1}-x_0\|\leq \|z_0-x_0\|\)
for each \(n\geq 0\). This implies that \(\{x_n\}\) is bounded.
Step 4. We show that \(\{x_n\}\) converges strongly to a point of \(C\) (we show that \(\{x_n\}\) is a cauchy sequence). As \(x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0\subset Q_n\) and \(x_n=P_{Q_n}x_0\) (Proposition 4), we have
\(\|x_{n+1}-x_0\|\geq \|x_n-x_0\|\)
for every \(n\geq 0\), which together with the boundedness of \(\|x_n-x_0\|\) implies that there exists the limit of \(\|x_n-x_0\|\). On the other hand, from \(x_{n+m}\in Q_n\), we have \(\langle x_n-x_{n+m},x_n-x_0\rangle\leq 0\) and hence \begin{align*} \|x_{n+m}-x_n\|^2&=\|(x_{n+m}-x_0)-(x_n-x_0)\|^2\\ \nonumber &\leq\|x_{n+m}-x_0\|^2-\|x_n-x_0\|^2-2\langle x_{n+m}-x_n,x_n-x_0\rangle\\ \nonumber &\leq\|x_{n+m}-x_0\|^2-\|x_n-x_0\|^2\rightarrow0,\ n\rightarrow\infty \end{align*} for any \(m\geq 1\). Therefore \(\{x_n\}\) is a cauchy sequence in \(C\), then there exists a point \(q\in C\) such that \(\lim_{n\rightarrow \infty} x_n=q\).
Step 5. We show that \(y_n\rightarrow q\), as \(n\rightarrow\infty\). Let
\(D_n=\{z\in C:\|y_n-z\|^2\leq\|x_n-z\|^2+L_{n}^2(L_n-1)(L_n+1)\}\).
From the definition of \(D_n\), we have \begin{align*} D_n&=\{z\in C:\langle y_n-z,y_n-z\rangle\leq\langle x_n-z,x_n-z\rangle+L_{n}^2(L_n-1)(L_n+1)\}\\ \nonumber &=\{z\in C:\|y_n\|^2-2\langle y_n,z\rangle+\|z\|^2\leq\|x_n\|^2-2\langle x_n,z\rangle+\|z\|^2+L_{n}^2(L_n-1)(L_n+1)\}\\ \nonumber &=\{z\in C:2\langle x_n-y_n,z\rangle\leq\|x_n\|^2-\|y_n\|^2+L_{n}^2(L_n-1)(L_n+1)\} \end{align*} This shows that \(D_n\) is convex and closed, \(n \in \mathbb{Z^{+}}\cup \{0\}\). Next, we want to prove that \(C_n\subset D_n\),\ \(n\geq 0\).
In fact, for any \(z\in C_n\), we have \begin{align*} \|y_n-z\|^2&\leq[L_n(1+(L_n-1)\alpha_n\beta_n)]^2\|x_n-z\|^2\\ &=\|x_n-z\|^2L_{n}^2+L_{n}^2[2(L_n-1)\alpha_n\beta_n+(L_n-1)^2\alpha_{n}^2\beta_{n}^2]\|x_n-z\|^2\\ &\leq\|x_n-z\|^2L_{n}^2+L_{n}^2[2(L_n-1)+(L_n-1)^2]\|x_n-z\|^2\\ &=\|x_n-z\|^2L_{n}^2+L_{n}^2(L_n-1)(L_n+1)\|x_n-z\|^2. \end{align*}\ From \(C_n=\{z\in C:\|y_n-z\|\leq[L_n(1+(L_n-1)\alpha_n\beta_n)]\|x_n-z\|\}\cap A, n\geq 0\), we have \(C_n\subset A\), \(n\geq 0\). Since \(A\) is convex, we also have \(\overline{co}C_n\subset A\), \(n\geq 0\). Consider \(x_n\in\overline{co}C_{n-1}\), we know that \begin{align*} \|y_n-z\|&\leq\|x_n-z\|^2L_{n}^2+L_{n}^2(L_n-1)(L_n+1)\|x_n-z\|^2\\ &\leq\|x_n-z\|^2+L_{n}^2(l_n-1)(L_n+1). \end{align*} This implies that \(z\in D_n\) and hence \(C_n\subset D_n\), \(n\geq 0\). Sinnce \(D_n\) is convex, we have \(\overline{co}(C_n)\subset D_n\), \(n\geq 0\). Therefore \(\|y_n-x_{n+1}\|^2\leq\|x_n-x_{n+1}\|^2+L_{n}^2(L_n-1)(L_n-1)\rightarrow 0\) as \(n\rightarrow\infty\). That is, \(y_n\rightarrow q\) as \(n\rightarrow\infty\).
Step 6. We show that \(q\in F\). From the definition of \(y_n\), we have \((1+\alpha_n\beta_nT_n)\|T_nx_n-x_n\|=\|y_n-x_n\|\rightarrow 0\) as \(n\rightarrow \infty\). Since \(\alpha_n\in(a,1]\subset[0,1]\),from the above limit we have \(\lim_n\rightarrow\infty\|T_nx_n-x_n\|=0.\)
Since \(\{T_n\}\) is uniformly closed and \(x_n\rightarrow q\), we have \(q\in F\).
Step 7. We claim that \(q=z_0=P_Fx_0\), if not, we have that \(\|x_0-p\|>\|x_0-z_0\|\). There must exist a positive integer \(N\), if \(n>N\) then \(\|x_0-x_n\|>\|x_0-z_0\|\), which leads to
\(\|z_0-x_n\|^2=\|z_0-x_n+x_n-x_0\|^2=\|z_0-x_n\|^2+\|x_n-x_0\|^2+2\langle z_0-x_n,x_n-x_0\rangle\).
It follows that \(\langle z_0-x_n,x_n-x_0\rangle< 0\) which implies that \(z_0\overline{\in} Q_n\), so that \(z_0\overline{\in} F\), this is a contradiction. This completes the proof.

Now, we present an example of \(C_n\) which does not involve a convex subset.

Example 1. Take \(H=R^2\), and a sequence of mappings \(T_n:R^2\rightarrow R^2\) given by \(T_n:(t_1,t_2)\mapsto(\frac{1}{8}t_1,t_2)\), \(\forall(t_1,t_2)\in R^2\), \(\forall n\geq 0\).
It is clear that \(\{T_n\}\) satisfies the desired definition of with \(F=\{(t_1,0):t_1\in(-\infty,+\infty)\}\) common fixed point set. Take \(x_0=(4,0)\),\( a_0=\frac{6}{7}\), we have
\(y_0=\frac{1}{7}x_0+\frac{6}{7}T_0x_0=(4\times\frac{1}{7}+\frac{4}{8}\times\frac{6}{7},0)=(1,0)\).
Take \(1+(L_0-1) a_0=\sqrt{\frac{5}{2}}\), we have
\(C_0=\{z\in R^2:\|y_0-z\|\leq\sqrt{\frac{5}{2}}\|x_0-z\|\}\).
It is easy to show that \(z_1=(1,3)\), \(z_2=(-1,3)\in C_0\). But
\(z^{'}=\frac{1}{2}z_1+\frac{1}{2}z_2=(0,3)\overline{\in} C_0\),
since \(\|y_0-z\|=2\), \(\|x_0-z\|=1\). Therefore \(C_0\) is not convex.

Corollary 6. Assume that \(\alpha_n\in (0,1]\), and \(\beta_n\in [0,1]\) for all \(n\in N\). Then \(\{x_n\}\) generated by $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)Tx_n+\alpha_nTz_n, & n\geq 0,\\ z_n=(1-\beta_n)+\beta_nTx_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq\|x_n-z\|\}\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{C_n\cap Q_n}x_0, \end{array} \right.$$ converges strongly to \(P_{F(T)}x_0\).

Proof. Take \(T_n\equiv T\), \(L_n\equiv 1\) in Theorem 5, in this case, \(C_n\) is convex and closed and , for all \(n\geq 0\), by using Theorem 1.9, we obtain our desired result.

Corollary 7. Assume that \(\alpha_n\in (0,1]\), and \(\beta_n\in [0,1]\) for all \(n\in N\). Then \(\{x_n\}\) generated by $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)Tx_n+\alpha_nTz_n, & n\geq 0,\\ z_n=(1-\beta_n)+\beta_nTx_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq\|x_n-z\|\}\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{C_n\cap Q_n}x_0, \end{array} \right.$$ converges strongly to \(P_{F(T)}x_0\).

3. Application

Here, we give an application of our result for the following case of finite family of asymptotically quasi-nonexpansive mappings \(\{T_n\}_{n=0}^{N-1}\). Let
\(\|T_{i}^{j}x-p\|\leq k_{i,j}\|x-p\|\), \(\forall x\in C\), \(p\in F\),
where \(F\) is common fixed point set of \(\{T_n\}_{n=0}^{N-1}\),\(\lim_j\rightarrow \infty k_{i,j}=1\) for all \(0\leq i\leq N-1\). The finite family of asymptotically quasi-nonexpansive mappings \(\{T_n\}_{n=0}^{N-1}\) is uniformly \(L-Lipschitz\), if
\(\|T_{i}^{j}x-T_{i}^{j}y\|\leq L_{i,j}\|x-y\|\), \(\forall x,y\in C\)
for all \(i\in \{0,1,2,...,N-1\}\), \(j\geq1\), where \(L\geq1\).

Theorem 8. Let \(\{T_n\}_{n=0}^{N-1}: C\rightarrow C\) be a finite uniformly L-Lipschitz family of asymptotically quasi-nonexpansive mappings with nonempty common fixed point set \(F\). Assume that \(\alpha_n\in (0,1]\), and \(\beta_n\in [0,1]\) for all \(n\in N\). Then \(\{x_n\}\) generated by $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)T_{i(n)}^{j(n)}x_n+\alpha_nT_{i(n)}^{j(n)}z_n, & n\geq 0,\\ z_n=(1-\beta_n)+\beta_nT_{i(n)}^{j(n)}x_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq k_{i(n),j(n)}\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1+(k_{i(n),j(n)}-1)\alpha_n\beta)\|x_n-z\|\}\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0, \end{array} \right.$$ converges strongly to \(P_Fx_0\).

We can drive the prove from the following two conclusions.

Conclusion 9. \(\{T_{n=0}^{N-1}\}_{n=0}^{\infty}\) is a uniformly closed asymptotically family of countable quasi-\(L_n\)-Lipschitz mappings from \(C\) into itself.

Conclusion 10. \(F=\bigcap_{n=0}^{N}F(T_n)=\bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), where \(F(T)\) denotes the fixed point set of the mappings \(T\).

Corollary 11. Let \(T: C\rightarrow C\) be a L-Lipschitz asymptotically quasi-nonexpansive mappings with nonempty common fixed point set \(F\). Assume that \(\alpha_n\in (0,1]\), and \(\beta_n\in [0,1]\) for all \(n\in N\). Then \(\{x_n\}\) generated by $$\left\{ \begin{array}{ll} x_0\in C=Q_0, & \text{choosen arbitrarily,}\\ y_n=(1-\alpha_n)T^nx_n+\alpha_nT^nz_n, & n\geq 0,\\ z_n=(1-\beta_n)+\beta_nT^nx_n, & n\geq 0,\\ C_n=\{z\in C:\|y_n-z\|\leq k_n(1+(k_n-1)\alpha_n\beta)\|x_n-z\|\}\cap A, & n\geq 0,\\ Q_n=\{z\in Q_{n-1}:\langle x_n-z,x_0-x_n\rangle\geq 0\},& n\geq 1,\\ x_{n+1}=P_{\overline{co}C_n\cap Q_n}x_0, \end{array} \right.$$ converges strongly to \(P_Fx_0\).

Proof. Take \(T_n\equiv T\) in Theorem 8, we get the desired result.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.