1. Introduction

In this paper , we deal with attraction-repolsion chemotaxis system

in a \(d\)-dimensional box \(\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\) is a bounded domain with smooth boundary \(\alpha,\beta,\mu,\chi,\xi,\beta,\gamma,\eta>0\). In the model (1) \(u_{1}\), \(u_{2}\) and \(u_{3}\) represent the cell density, the concentration of the chemoattractant (attractive signal) and the concentration of the chemorepellent (repulsive signal) respectively, \(\mathbf g(u)\) is logistic source. The classical Keller-Segel system can be obtained by setting \(d_{i} = 1 ,(i=1,2,3), \xi = 0, u_{3}\equiv0, \mathbf g(u)\equiv 0 \) in (1) which models the mechanism of chemotaxis and has been extensively studied since 1970, we refer to [1, 2, 3, 4] and the references therein. Apart form the aforementioned system a source of logistic type is included in (1) to describe the spontaneous growth of cells. The effect of preventing ultimate growth has been widely studied.

Chemotaxis is a chemosensitive movement of species which may detect and respond to chemical substances in the environment. The first model about chemotaxis was proposed by Keller and Segel [5]

which describes the aggregation process of the slime mold formation in Dictyostelium Discoidium, where \(v\) denotes the chemical concentration and u is the concentration of species. For this system, there have been abundant results. Osaki and Yagi [6] found that when \(n = 1\), all the solutions are global and bounded. When \(n \geq 2\), blow-up may happen (see Horstmann and Wang [7]; Herrero et al., [8]; Winkler et al., [9]). A detailed introduction into the mathematic of the Keller-Segel model for chemotaxis is presented in Horstmann [1, 10, 11].

In the study of chemotaxis-diffusion-growth models, the pattern dynamics is another mathematically challenging and physically important research project (see Tello and Winkler [12], Aida and Yagi [13], Kurata et al., [14], Painter and Hillen [15], Okuda and Osaki [16], Kuto et al., [17] and Banerjee et al., [18]. Guo and Hwang [19] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Their result can be interpreted as a rigorous mathematical characterization for pattern formation in the Keller-Segel model. By using the similar method, Fu and Liu [20] proved that the linear unstable positive constant equilibrium in the Keller-Segel model with a logistic source is also unstable in the full nonlinear sense. The emergence of patterns is a phenomenon frequently observed in the physical world [21].

Many authors have investigated the formation of patterns by using self-diffusive reaction-diffusion models [21, 22, 23, 24, 25]. Recently, some researchers made attempts to discover the effect of cross-diffusion on the pattern formation, and found that with appropriate cross-diffusion coefcients, linear reaction terms are sufficient to produce pattern formation [26, 27, 28], but there is only few attention having been paid to this direction. Therefore, based on the model (1): First, we analyse criteria of linear stability and instability of the positive constant equilibrium \({\mathbf w_{c}}\) (see Theorem 1). Second, by applying the higher-order energy estimates, the embedding theorem and the Guo-Strauss' bootstrap technique (see Guo and Strauss [29]), it is proved that for given any general perturbation of magnitude \(\delta\), its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of \(\ln{\frac{1}{\delta}}\) (see Theorem 2). We assert further that the positive constant equilibrium point \({\mathbf w_{c}}\) is nonlinearly unstable in the above conditions (Corollary 1). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a quantitative characterization for the nonlinear evolution of early-stage spatiotemporal pattern formation in the model (1).

The organization of this paper is as follows: in Section 2, we first prove Turing instability does not take place in the absence of chemotactic effect. Second, we give linear stability and instability criterions for the model (1), and discuss some properties of solutions for the corresponding linearized system. In Section 3, we consider the growing modes of (1), and prove the Bootstrap lemma. In Section 4, quantitative characterization for pattern formation and proof of nonlinear instability are given.

2. Linear stability and instability criterions

In this section, we study in detail linear Stability,linear instability of positive constant equilibrium point \(\mathbf w_{c}=(1,\frac{\alpha}{\beta},\frac{\gamma}{\eta})\) to the model (1) in a \(d\)-dimensional box \(\Omega =\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\), and \(\mathbf{g(u)}=\mu \mathbf{u_{1}(1-u_{1})}\).

2.1. Stability of positive constant equilibrium point for (1) without chemotaxis

We consider the stability of \({\mathbf w_{c}}\) for the corresponding system (1) without chemotaxis For sake convenience, take \(\mathbf{ w(x,t)= (U_{1}(x,t),U_{2}(x,t),U_{3}(x,t))^T}\) and \[\begin{array}{ll} G({w})=\left(% \begin{array}{ccc} g_{1}({w}) & \\ g_{2}({w}) &\\ g_{3}({w})) \end{array}% \right) = \left(% \begin{array}{ccc} \mu{u_{1}}(1-u_{1}) & \\ \alpha{u_{1}-\beta{u_{2}}} &\\ \gamma{u_{1}-\eta{u_{3}}}) \end{array}% \right), \end{array} \] then \[\frac{\partial{G}}{\partial{w}}|_{ w_{c}} \equiv{G_{w}({w_{c}})}= \left(% \begin{array}{ccc} -\mu & 0 & 0\\ \alpha & -\beta & 0 \\ \nu & 0 & -\eta \end{array} \right). \]

Lemma 1. The positive equilibrium point \(\mathbf w_{c}\) of (3) is locally asymptotically stable.

Proof. Let \(0=k_{1}< k_{2}< k_{3}< \cdot\cdot\cdot\) be the eigenvalues of the operator \(-\Delta\) on \(\mathbb{T}^{d}\) with the homogeneous Neumann boundary condition, and \(E(k_{i})\) be the eigenspace corresponding to \(k_{i}\) in \(H^{1}(\mathbb{T}^{d})\). Let \({\mathbf X}=[H^{1}(\mathbb{T}^{d})]^{3}\) and \({\mathbf X}_{ij}=\left\{c\cdot\phi_{ij} | c\in \mathbb{R}^{3}\right\}\), where \(\left\{\phi_{ij}, j=1,\cdot\cdot\cdot,\dim E(k_{i}) \right\} \) is an orthonormal basis of \(E(k_{i})\). Then \({\mathbf X}=\oplus_{i=1}^\infty {\mathbf X}_i\), \({\mathbf X}_i=\oplus_{j=1}^{\dim E(\mu_i)}{\mathbf X}_{ij}\). Let \(D= diag(d_{1}, d_{2}, d_{3})\). The linearization of (3) at \(\mathbf w_{c}\) is $$ {\mathbf{w}}_{t}=\left(D\Delta+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\right){\mathbf w}. $$ For each \(i\geq1\), \({\mathbf X}_i\) is invariant under the operator \(D\Delta+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\), and \(\lambda\) is an eigenvalue of this operator on \({\mathbf{X}}_i\) if and only if it is an eigenvalue of the matrix \( -k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\). The characteristic polynomial of \(-k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\) is given by $$\begin{array}{ll} det (\lambda{I}-( -k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})))= \left(% \begin{array}{ccc} \lambda+ k_{i}d_{1} +\mu& 0 & 0\\ -\alpha & \lambda+ k_{i}d_{2}+\beta & 0 \\ -\gamma & 0 & \lambda+ k_{i}d_{3}+\eta \end{array} \right) = 0 \end{array} $$ implies \(\Psi(\lambda)= (\lambda+ k_{i}d_{1}+\mu)(\lambda+ k_{i}d_{2}+\beta)(\lambda+ k_{i}d_{3}+\eta)=0\), then \(\lambda_{1}=-(k_{i}d_{1}+\mu) \), \(\lambda_{2}=-(k_{i}d_{2}+\beta)\) and \(\lambda_{3} = -(k_{i}d_{3}+\eta)\). So all the eigenvalues are negative, hence \( \mathbf w_{c}\) is locally asymptotically stable, this complete the proof.

2.2. Criteria of linear stability and instability

Let \(\hat{u}_{1}({\mathbf x},t)=u_{1}({\mathbf x},t)-u_{1c}, \hat{u}_{2}({\mathbf x},t)=u_{2}({\mathbf x},t)-u_{2c}, \hat{u_{3}}({\mathbf x},t)=u_{3}({\mathbf x},t)-u_{3c}\) be nonlinear evolution of a perturbation around \((u_{1c},u_{2c},u_{3c})=(1,\frac{\alpha}{\beta},\frac{\nu}{\eta})\), and omitting the symbol \(`` \wedge"\), then we rewrite (3) with The corresponding linearized system can be written as Let \({\mathbf w}({\mathbf x},t)\equiv(u_{1}({\mathbf x},t),u_{2}({\mathbf x},t),u_{3}({\mathbf x},t))^{\mathrm{T}}\), \({\mathbf q}=(q_{1},\ldots,q_{d})\in \mathbb{N}^{d}\) and \(e_{{\mathbf q}}({\mathbf x})=\prod^{d}\limits_{i=1}\cos(q_{i}x_{i})\). Then \(\{e_{{\mathbf q}}({\mathbf x})\}_{{\mathbf q}\in\mathbb{N}^{d}}\) forms a basis of the space of functions in \(\mathbb{T}^{d}\) that satisfy the homogeneous Neumann boundary condition. We try to find a normal mode to the linearized system (5) of the following form where \({\mathbf r_{{\mathbf q}}}\) is a vector depending on \({\mathbf q}\). Substituting (6) into (5), we have \[ \lambda_{{\mathbf q}}{\mathbf r_{{\mathbf q}}}=\left(% \begin{array}{ccc} -d_{1}q^2-\mu & \chi q^2 & -\xi q^2\\ \alpha & -d_{2}q^2 -\beta & 0\\ \gamma &0 & -d_{3}q^2-\eta \end{array}% \right){\mathbf r_{{\mathbf q}}}:={\mathbf L_{q}}{\mathbf r_{{\mathbf q}}}, \] where \(q^2=|{\mathbf q}|^{2}=\sum^{d}\limits_{i=1}q^2_{i}\). Then the corresponding characteristic equation of \({\mathbf L_{q}}\) is where and In order to consider instability of \({\mathbf w_{c}}\), we make the following basic assumptions:

• (H\(_{1}\)) There exists \({\textbf q}\in \mathbb{N}^{d}\) such that the matrix \({\textbf L_{q}}\) has at least one eigenvalue with positive real part;
• (H\(_{2}\)) \(d_{1}, d_{2}, d_{3}>0\) and \(d_{i}\neq d_{j}\), \(i\neq j\), \(i,j=1,2,3\).

It is know that a first necessary condition for Turing instability to happen is that \(d_{i}\neq d_{j} (i\neq j)\), implying that \(u_{1}, u_{2}\) and \(u_{3}\) must move with different diffusion constants.
For every \(\lambda_{1}({\mathbf q}), \lambda_{2}({\mathbf q}), \lambda_{3}({\mathbf q})\) be the solutions of \(\det({ \lambda_{\textbf q}}\mathrm{I}-{\mathbf L_{q}})=0\). It will be state by Lemma 3 that there exist finitely many values \({\mathbf q}\in \mathbb{N}^{d}\) such that \[\max\left\{\mathrm{Re}\lambda_{1}({\mathbf q}), \mathrm{Re}\lambda_{2}({\mathbf q}), \mathrm{Re}\lambda_{3}({\mathbf q})\right\}>0.\] Hence there exists one \(q^{2}\) having the largest eigenvalue

• (H\(_{3}\)) At \(\mathbf{\overline{q}}=(\overline{q}_{1},\cdot\cdot\cdot,\overline{q}_{d})\in \mathbb{N}^{d}\) which attains \(\lambda_{\max}=\mathrm{Re}\lambda_{i}({\mathbf {\overline{q}}})\), we assume that the Jordan canonical form of the matrix \({\mathbf L_{\overline{q}}}={\mathbf G}_{\mathbf w}({\mathbf w_{c}})+{\mathbf Q}(\overline{q}^2)\) is \(J=\mathrm{diag}(\lambda_{1}({\mathbf{\overline{q}}}),\lambda_{2}({\mathbf{\overline{q}}}),\lambda_{3}({\mathbf{\overline{q}}}))\), where \(\overline{q}^2=\sum^{d}\limits_{i=1}\overline{q}^2_{i}\) and \[ {\mathbf Q}(\overline{q}^2):=\left(% \begin{array}{ccc} -d_{1}\overline{q}^2 &\chi\overline{q}^2 & -\xi\overline{q}^2\\ 0 & -d_{2}\overline{q}^2 & 0\\ 0 & 0 & -d_{3}\overline{q}^2 \end{array}% \right). \]

Let us carry on discussion on the characteristic equation (7). Denote \[A:=\bar{B}^{2}_{2}-3\bar{B}_{1}, B:=\bar{B}_{2}\bar{B}_{1}-9\bar{B}_{0}, C:=\bar{B}^{2}_{1}-3\bar{B}_{2}\bar{B}_{0}\] and \begin{eqnarray*}\Delta&=& B^{2}-4AC=3\left\{4\bar{B}^{3}_{1}+4\bar{B}^{3}_{2}\bar{B}_{0}+27\bar{B}^{2}_{0}-\bar{B}^{2}_{2}\bar{B}^{2}_{1}-18\bar{B}_{2}\bar{B}_{1}\bar{B}_{0}\right\}\\ &:=& Q_{6}q^{12}+Q_{5}q^{10}+Q_{4}q^{8}+Q_{3}q^{6}+Q_{2}q^{4}+Q_{1}q^{2}+Q_{0}, \end{eqnarray*} where \begin{eqnarray*} Q_{6}&=&3\left\{4C^{3}_{21}C_{01}+27C_{01}-C^{2}_{21}C^{2}_{11}-18C_{21}C_{11}C_{01}\right\},\\ Q_{5}&=&6\left\{27C_{01}C_{02}+2C^{3}_{21}C_{02}+6C^{2}_{21}C_{22}C_{01}-C^{2}_{21}C_{11}C_{12}-C_{21}C^{2}_{11}C_{22}\right.\\ &&\left.-9C_{21}C_{11}C_{02}-9C_{21}C_{12}C_{01}-9C_{22}C_{11}C_{01}\right\}, \end{eqnarray*} \begin{eqnarray*} Q_{4}&=&3\left\{27C_{02}+54C_{01}C_{03}+4C^{2}_{21}C_{03}+12C^{2}_{21}C_{22}C_{02}+12C_{21}C^{2}_{22}C_{01}+4C^{2}_{11}\right.\\ &&\left.-C^{2}_{21}C^{2}_{12}-2C_{11}C_{13}C^{2}_{21}-4C_{21}C_{22}C_{11}C_{03}-C^{2}_{22}C^{2}_{11}-18C_{21}C_{11}C_{03}\right.\\ &&\left.-18C_{21}C_{12}C_{02}-18C_{21}C_{13}C_{01}-18C_{22}C_{11}C_{02}-18C_{22}C_{12}C_{01}\right\},\\ Q_{3}&=&6\left\{27C_{01}C_{04}+27C_{02}C_{03}+2C^{3}_{21}C_{04}+12C^{3}_{22}C_{01}+6C^{2}_{21}C_{22}C_{03}\right.\\ &&\left.+6C_{21}C^{2}_{22}C_{02}+4C_{11}C_{12}-C^{2}_{21}C_{12}C_{13}-C_{21}C_{22}C^{2}_{12}-2C_{21}C_{22}C_{11}C_{13}\right.\\ &&\left.-C^{2}_{22}C_{11}C_{12}-9C_{21}C_{11}C_{04}-9C_{21}C_{12}C_{03}-9C_{21}C_{13}C_{02}\right.\\ &&\left.-9C_{22}C_{11}C_{03}-9C_{22}C_{12}C_{02}-9C_{22}C_{13}C_{01}\right\},\\ Q_{2}&=&3\left\{27C_{03}+54C_{02}C_{04}+4C^{3}_{22}C_{02}+12C^{2}_{21}C_{22}C_{04}+4C^{2}_{12}+12C_{21}C^{2}_{22}C_{03}\right.\\ &&\left.+8C_{11}C_{13}-C^{2}_{21}C^{2}_{13}-4C_{21}C_{22}C_{12}C_{13}-C^{2}_{22}C^{2}_{12}-2C^{2}_{22}C_{11}C_{13}-18C_{21}C_{12}C_{04}\right.\\ &&\left.-18C_{21}C_{13}C_{03}-18C_{22}C_{11}C_{04}-18C_{22}C_{13}C_{02}-18C_{22}C_{12}C_{03}\right\},\\ Q_{1}&=&6\left\{27C_{03}C_{04}+2C^{3}_{22}C_{03}+6C_{21}C^{2}_{22}C_{04}+4C_{12}C_{13}-C_{21}C_{22}C^{2}_{13}-C_{12}C_{13}C^{2}_{22}\right.\\ &&\left.-9C_{21}C_{13}C_{04}-9C_{22}C_{12}C_{04}-9C_{22}C_{13}C_{03}\right\},\\ Q_{0}&=&3\left\{ 4C^{3}_{22}C_{04}+4C^{2}_{13}+27C^{2}_{04}-C^{2}_{22}C^{2}_{13}-18C_{22}C_{13}C_{04}\right\}. \end{eqnarray*} The derivative of \(\psi(\lambda_{{\mathbf q}})\) is \(\psi'(\lambda_{{\mathbf q}})=3\lambda^{2}_{{\mathbf q}}+2\bar{B}_{2}\lambda_{{\mathbf q}}+\bar{B}_{1}.\) Obviously, equation \(\psi'(\lambda_{{\mathbf q}})=0\) has two roots as follows Next, let us give one result concerning the cubic equation in Hu et al., [30] (which was first introduced in Fan [31]), which is used to discuss the linear stability and instability of positive constant equilibrium solution for the model (1).

Lemma 2. Let equation \(x^{3} + bx^{2} +cx + d = 0\), where \(b, c, d \in \mathbb{R}\). Let further \(A=b^{2}-3c, B=bc-9d, C=c^{2}-3bd\) and \(\Delta= B^{2 }- 4AC\). Then the equation has three real roots if and only if \(\Delta \leq 0\); the equation has one real root and a pair of conjugate complex roots if and only if \(\Delta>0\). Furthermore, the conjugate complex roots are \(w=\frac{-2b+Y^{1/3}_{1}+Y^{1/3}_{2}}{6}\pm i\frac{\sqrt{3}\left({Y^{1/3}_{1}-Y^{1/3}_{2}}\right)}{6}\), where \(Y_{1,2}=bA+\frac{3\left(-B\pm\sqrt{B^{2}-4AC}\right)}{2}.\)

According to Lemma 2, on the one hand, if \(\Delta\leq0\), then (7) has three real roots \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\), and denote \(\lambda_{1}({\mathbf q})\leq\lambda_{2}({\mathbf q})\leq\lambda_{3}({\mathbf q})\). From this, we further infer that \(\lambda^{\ast}_{1,2}({\mathbf q})\) also are real. Moreover, recall that \(\bar{B}_{2}=-(\lambda_{1}({\mathbf q})+\lambda_{2}({\mathbf q})+\lambda_{3}({\mathbf q}))>0\), it means that (7) has at least one eigenvalue with negative real part. On the other hand, if \(\Delta>0\), then Equation (7) has one real root \(\lambda_{1}({\mathbf q})\) and a pair of conjugate complex roots \[\lambda_{2,3}({\mathbf q})=\frac{-2\bar{B}_{2}+Y^{1/3}_{1}+Y^{1/3}_{2}}{6}\pm \mathrm{i}\frac{\sqrt{3}\left({Y^{1/3}_{1}-Y^{1/3}_{2}}\right)}{6}\] with \[Y_{1,2}=\bar{B}_{2}A+\frac{3\left(-B\pm\sqrt{B^{2}-4AC}\right)}{2}.\] Notice by the Routh-Hurwitz criterion that \({\mathbf q}=0\), in the case of \(C_{22}C_{13}>C_{04}\), then (8) has three negative roots. So we consider the case \({\mathbf q}\neq0\) in the sequel. In this section, our first main purpose is to give criteria for linear stability and instability of \({\mathbf w_{c}}\).

Theorem 1. (Linear stability and instability). Let \({\mathbf w_{c}}\) be positive constant equilibrium solution of (1). Assume that \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) are three roots of \(\psi(\lambda)=\lambda^{3}+\bar{B}_{2}\lambda^{2}+\bar{B}_{1}\lambda+\bar{B}_{0}=0\), and that \(\lambda^{\ast}_{1}\) and \(\lambda^{\ast}_{2}\) are two roots of \(\psi'(\lambda)=3\lambda^{2}+2\bar{B}_{2}\lambda+\bar{B}_{1}=0\), then we have the following conclusions:

• (1) If one of the following conditions holds, then \({\mathbf w_{c}}\) is linearly stable.
  • H\(_{S1}\) \(\Delta\leq0\), \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{1}< \lambda^{\ast}_{2}< 0\).
  • H\(_{S2}\) \(\Delta>0\), \(\bar{B}_{0}>0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}< 0\), \(\mathrm{Re}\lambda_{3}< 0\).
• (2) If one of the following conditions holds, then \({\mathbf w_{c}}\) is linearly unstable.
  • H\(_{U1}\) \(\Delta\leq0\), and one of the following conditions holds:
  • H\(_{U11}\) \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{2}>\lambda^{\ast}_{1}>0\).
  • H\(_{U12}\) \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{2}>0>\lambda^{\ast}_{1}\).
  • H\(_{U13}\) \(\bar{B}_{0}< 0\) and \(\lambda^{\ast}_{2}>0>\lambda^{\ast}_{1}\).
  • H\(_{U14}\)\(\bar{B}_{0}< 0\) and \(\lambda^{\ast}_{1}< \lambda^{\ast}_{2}< 0\).
  • H\(_{U2}\) \(\Delta >0\), and one of the following conditions holds:
  • H\(_{U21}\) \(\bar{B}_{0}>0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}>0\), \(\mathrm{Re}\lambda_{3}>0\).
  • H\(_{U22}\) \(\bar{B}_{0}< 0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}< 0\), \(\mathrm{Re}\lambda_{3}< 0\).

Here \(\Delta= B^{2}-4AC\), \(A:=\bar{B}^{2}_{2}-3\bar{B}_{1}\), \(B:=\bar{B}_{2}\bar{B}_{1}-9\bar{B}_{0}\), \(C:=\bar{B}^{2}_{1}-3\bar{B}_{2}\bar{B}_{0}\), in particular, \(\bar{B}_{0}=\psi(0)=-\lambda_{1}\lambda_{2}\lambda_{3}\).

Proof. Let \(\Delta\leq0\). By Lemma 2, the equation \(\psi(\lambda)=\lambda^{3}+\bar{B}_{2}\lambda^{2}+\bar{B}_{1}\lambda+\bar{B}_{0}=0\) has three real roots \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) and assume \(\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\). Moreover, the equation \(\psi'(\lambda)=3\lambda^{2}+2\bar{B}_{2}\lambda+\bar{B}_{1}=0\) has also two real roots \(\lambda^{\ast}_{1}\) and \(\lambda^{\ast}_{2}\) with \(\lambda^{\ast}_{1}\leq\lambda^{\ast}_{2}\), and \[ \begin{array}{ll} \psi'(\lambda)>0, \forall \lambda\in(-\infty, \lambda^{\ast}_{1})\cup(\lambda^{\ast}_{2}, +\infty),\\ \psi'(\lambda)< 0, \forall \lambda\in(\lambda^{\ast}_{1}, \lambda^{\ast}_{2}). \end{array} \] Therefore, \[\psi(\lambda^{\ast}_{1})\geq0, \psi(\lambda^{\ast}_{2})\leq0\] and \[\lambda_{1}\in(-\infty, \lambda^{\ast}_{1}], \lambda_{2}\in[\lambda^{\ast}_{1}, \lambda^{\ast}_{2}], \lambda_{3}\in[\lambda^{\ast}_{2},+\infty). \] Let condition ( H\(_{U11}\)) hold. If \(\lambda^{\ast}_{1}>0\), then \(\lambda_{2}>0\), \(\lambda_{3}>0\). Since \(\psi(\lambda)\) is increasing for all \(\lambda\in(-\infty, \lambda^{\ast}_{1}]\) and \(\psi(0)=\bar{B}_{0}>0\), one has \(\lambda_{1}< 0\). If \(\lambda_{1}>0\), this contradicts \(\bar{B}_{2}>0\). Hence, \({\mathbf w_{c}}\) is linearly unstable. Under the condition ( H\(_{U12}\)), if \(\lambda^{\ast}_{1}< 0\), \(\lambda^{\ast}_{2}>0\), then \(\lambda_{1}< 0\). Since \(\psi(\lambda)\) is decreasing for all \(\lambda\in(\lambda^{\ast}_{1}, \lambda^{\ast}_{2})\) and \(\bar{B}_{0}>0\), we have \(\lambda_{2}>0\), \(\lambda_{3}>0\). This means that \({\mathbf w_{c}}\) is linearly unstable.
Similarly, it is proved that when condition ( H\(_{U13}\)) or ( H\(_{U14}\)) holds, eigenvalues \(\lambda_{1}< 0\), \(\lambda_{2}< 0\) and \(\lambda_{3}>0\), that is, \({\mathbf w_{c}}\) is linearly unstable.
In the case ( H\(_{S1}\)), By monotonicity of \(\psi(\lambda)\) for all \(\lambda\in(\lambda^{\ast}_{2},+\infty)\), it holds \(\lambda_{1}< 0\), \(\lambda_{2}< 0\) and \(\lambda_{3}< 0\). Hence, \({\mathbf w_{c}}\) is linearly stable.
We now let \(\Delta>0\). In view of Lemma 2, \(\psi(\lambda)=0\) has one real root \(\lambda_{1}\) and a pair of conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\). If condition ( H\(_{U21}\)) holds, then it follows from \(\bar{B}_{0}>0\) that real root \(\lambda_{1}< 0\). Therefore, \({\mathbf w_{c}}\) is linearly unstable based on \(\mathrm{Re}\lambda_{2}>0\), \(\mathrm{Re}\lambda_{3}>0\). Similarly, we can also prove that if condition ( H\(_{U22}\)) holds, then \({\textbf w_{c}}\) is linearly unstable. If condition (H\(_{S2}\)) holds, it is easily to obtain that \({\textbf w_{c}}\) is linearly stable. This completes the proof.

2.3. Some properties of solutions of the linearized system (5)

Lemma 3. If \({\mathbf q}\in \mathbb{N}^{d}\) and \(q^2\) sufficiently large, then all eigenvalues of \({\textbf L_{q}}\) have negative real parts.

Proof. Notice that \(C_{21}\), \(C_{22}\), \(C_{11}\), \(C_{13}\), \(C_{01}\), \(C_{04}\) and \(\bar{B}_{2}\) are all positive, where the parameters are mentioned in (8) and (9). In addition, \(\bar{B}_{2}\), \(\bar{B}_{1}\), \(\bar{B}_{0}\) and \(\bar{B}_{2}\bar{B}_{1}-\bar{B}_{0}\) are positive if \({\mathbf q}\in \mathbb{N}^{d}\) sufficiently large. It is follows from the Routh-Hurwitz criterion that all eigenvalues of \({\mathbf L_{q}}\) have negative real parts for \({\mathbf q}\in \mathbb{N}^{d}\) sufficiently large.

For given \({\mathbf q}\in \mathbb{N}^{d}\), let \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) be the eigenvalues of \({\mathbf L_{q}}\) and the corresponding eigenvectors by \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\), \({\mathbf r}_{3}({\mathbf q})\). According to eigenvectors, we divide \({\mathbf q}\) into the following four cases to analyze:
Case 1: \({\mathbf q}\in \mathbb{N}^{d}_{R1}\): \({\mathbf L_{q}}\) has three real eigenvalues \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\) and \(\lambda_{3}({\mathbf q})\), and three corresponding linearly independent eigenvectors \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\) and \({\mathbf r}_{3}({\mathbf q})\). In the case we arrange \(\lambda_{1}({\mathbf q})\leq\lambda_{2}({\mathbf q})\leq\lambda_{3}({\mathbf q})\).
Case 2: \({\mathbf q}\in \mathbb{N}^{d}_{R2}\): \({\mathbf L_{q}}\) has a single root \(\lambda_{1}({\mathbf q})=\lambda_{s}({\mathbf q})\) and a double root \(\lambda_{2}({\mathbf q})=\lambda_{3}({\mathbf q})=\lambda_{d}({\mathbf q})\) (or \({\mathbf L_{q}}\) has three repeated real root \(\lambda_{s}({\mathbf q})=\lambda_{d}({\mathbf q})\)), meanwhile, there are only two linearly independent real eigenvectors \({\mathbf r}_{s}({\mathbf q})\) and \({\mathbf r}_{d}({\mathbf q})\). In this case we need find another independent vector \({\mathbf r}'_{d}({\mathbf q})\) satisfying \[({\mathbf L_{q}}-\lambda_{d}({\mathbf q}) \mathrm{I}){\mathbf r}'_{d}({\mathbf q})={\mathbf r}_{d}({\mathbf q}). \] Case 3: \({\mathbf q}\in \mathbb{N}^{d}_{R3}\): (7) has a triple eigenvalue \(\lambda({\mathbf q})\) which only corresponding one linearly independent eigenvector \({\mathbf r}({\mathbf q})\). In this case, we need to supplement another two independent vectors \({\mathbf r}'({\mathbf q})\) and \({\mathbf r}''({\mathbf q})\), which satisfy \[({\mathbf L_{q}}-\lambda({\mathbf q}) \mathrm{I}){\mathbf r}'({\mathbf q})={\mathbf r}({\mathbf q}), ({\mathbf L_{q}}-\lambda({\mathbf q}) \mathrm{I}){\mathbf r}''({\mathbf q})={\mathbf r}'({\mathbf q}).\] Case 4: \({\mathbf q}\in\mathbb{N}^{d}_{C}=\mathbb{N}^{d}-(\mathbb{N}^{d}_{R1}\bigcup\mathbb{N}^{d}_{R2}\bigcup\mathbb{N}^{d}_{R3})\): The characteristic equation (7) has one real root and a pair of conjugate complex roots. The eigenvalues and the corresponding eigenvectors are denoted by \(\lambda_{r}({\mathbf q})\), \(\mathrm{Re}\lambda_{c}({\mathbf q})+i\mathrm{Im}\lambda_{c}({\mathbf q})\), \(\mathrm{Re}\lambda_{c}({\mathbf q})-i\mathrm{Im}\lambda_{c}({\mathbf q})\) and \({\mathbf r}({\mathbf q})\), \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})+i\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\), \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})-i\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\), respectively. Notice that \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\) and \(\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\) are linearly independent vectors. Given any initial perturbation \({\mathbf w}({\mathbf x}, 0)\), it can be expressed as where \(w_{i}({\mathbf q}), w_{d}({\mathbf q}), w'_{d}({\mathbf q}), w_{s}({\mathbf q}), w({\mathbf q}), w'({\mathbf q}), w''({\mathbf q}), w^{\mathrm{Re}}({\mathbf q}), w^{\mathrm{Im}}({\mathbf q}), w_{r}({\mathbf q})\in\mathbb{R}\), \(i=1,2,3\) and Thus, the unique solution \({\mathbf w}({\mathbf x}, t)\) to the linearized system (5) can be written in the following form. Recall that \[\lambda_{\max}= \max\limits_{{\mathbf q}\in \mathbb{N}^{d}}\max\limits_{1\leq i\leq3} \mathrm{Re}\lambda_{i}({\mathbf q})>0,\] where \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are the solutions of (7). Denote By the assumption ( H\(_{3}\)), the largest eigenvalue \(\lambda_{\max}\) can be obtained, provided that \({\mathbf q}\) belongs to \(\mathbb{N}^{d}_{R1}\) or \(\mathbb{N}^{d}_{C}\).
In the sequel, we define \[I=\{i | 1 \leq i \leq 3\}, I_{1}=\{i | \lambda_{i}({\mathbf q})=\lambda_{\max}, 1 \leq i \leq 3\},\] and \[ \begin{array}{ll} \Lambda_{R1}=\mathbb{N}^{d}_{R1}\cap{\mathbb{N}^{d}}_{\max}, \Lambda_{C}=\mathbb{N}^{d}_{C}\cap{\mathbb{N}^{d}}_{\max},\\ \Lambda_{C1}=\{{\mathbf q}\in\Lambda_{C} | \mathrm{Re}\lambda_{c}({\mathbf q})=\lambda_{\max}\},\\ \Lambda_{C2}=\{{\mathbf q}\in\Lambda_{C} | \lambda_{r}({\mathbf q})=\lambda_{\max}\},\\ \Lambda_{C3}=\{{\mathbf q}\in\Lambda_{C} | \mathrm{Re}\lambda_{c}({\mathbf q})=\lambda_{\max}, \lambda_{r}({\mathbf q})=\lambda_{\max}\}. \end{array} \] Let \(e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})\) be the dominant part of the solution \(e^{\mathfrak{L}t}{\mathbf w}_{0}({\mathbf x})\) of the linearied system (5) and \begin{eqnarray*} e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})&=&\sum\limits_{{\mathbf q}\in\Lambda_{R1}}\sum\limits_{i\in I_{1}}w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q})e^{\lambda_{\max}t}e_{{\mathbf q}}({\mathbf x})\notag\\ && +\sum\limits_{{\mathbf q}\in\Lambda_{C1}}\left[w^{\mathrm{Re}}({\mathbf q})\left(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\cos[(\mathrm{Im}\lambda_{c}({\mathbf q}))t] -\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\sin[(\mathrm{Im}\lambda_{c}({\mathbf q}))t]\right)\right.\notag\\ &&\left.+w^{\mathrm{Im}}({\mathbf q})\left(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\sin[(\mathrm{Im}\lambda_{c}({\mathbf q}))t] +\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\cos[(\mathrm{Im}\lambda_{c}({\mathbf q}))t]\right)\right]e^{\lambda_{\max}t}\notag\\ && +\sum\limits_{{\mathbf q}\in\Lambda_{C2}}w_{r}({\mathbf q}){\mathbf r}_{r}({\mathbf q})e^{\lambda_{\max}t}e_{{\mathbf q}}({\mathbf x})\notag \end{eqnarray*} Since \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are the roots of (7), let \(\beta_{i}({\mathbf q})=\frac{1}{q^{2}}\lambda_{i}({\mathbf q})\), then \(\beta_{1}({\mathbf q})\), \(\beta_{2}({\mathbf q})\), \(\beta_{3}({\mathbf q})\) are the three roots of \({\mathbf F}_{q}(\beta_{{\mathbf q}})=\det\left(\beta_{{\mathbf q}}\mathrm{I}-\frac{1}{q^{2}}{\mathbf L_{q}}\right)=0\) and \begin{eqnarray*} {\mathbf F}_{q}(\beta_{{\mathbf q}})&=&\det\left(% \begin{array}{ccc} \beta_{{\mathbf q}}+d_{1}+\frac{\mu}{q^2}& -\chi &\xi\\ -\alpha & \beta_{{\mathbf q}}+d_{2}+\frac{\beta}{q^2} & 0\\ -\gamma & 0&\beta_{{\mathbf q}}+d_{3}+\frac{\eta}{q^2} \end{array}% \right)\\ &=&\beta^{3}_{{\mathbf q}}+\bar{b}_{2}({\mathbf q})\beta^{2}_{{\mathbf q}}+\bar{b}_{1}({\mathbf q})\beta_{{\mathbf q}}+\bar{b}_{0}({\mathbf q}) \end{eqnarray*} with Moreover, One can define a function \({\mathbf F}^{\ast}(\beta_{{\mathbf q}})\) of the form \[{\mathbf F}^{\ast}(\beta_{{\mathbf q}}):=\beta^{3}_{{\mathbf q}}+\bar{b}_{2}\beta^{2}_{{\mathbf q}}+\bar{b}_{1}\beta_{{\mathbf q}}+\bar{b}_{0} =(\beta_{{\mathbf q}}+d_{1})(\beta_{{\mathbf q}}+d_{2})(\beta_{{\mathbf q}}+d_{3}).\] It is clear from the assumption ({\textbf H\(_{2}\)}) that the equation \({\textbf F}^{\ast}(\beta_{{\mathbf q}})=0\) has different negative roots \(-d_{1}\), \(-d_{2}\), \(-d_{3}\). For \(q^{2}\) sufficiently large, it follows from Lemma 3 that \(\mathrm{Re}\beta_{i}({\mathbf q})< 0\), \(\forall 1\leq i\leq 3\). Thus and For \(q^{2}\) large enough, by (18) and (19), we have Again combining (18) and (20) yields for \(q^{2}\) sufficiently large Applying (21) and (22), for every sequence \(\{{\mathbf q}_{m}\}\in \mathbb{N}^{d}\), there exists a subsequence of \(\{{\mathbf q}_{n}\}\) such that for \(1 \leq i\leq 3\) there exist limits \[\lim\limits_{n\rightarrow\infty}\mathrm{Re}\beta_{i}({\mathbf q}_{n}), \lim\limits_{n\rightarrow\infty}\mathrm{Im}\beta_{i}({\mathbf q}_{n}).\] Hence Notice by (18) and (23) that This means that \(\{\beta_{1},\beta_{2},\beta_{3}\}\) is a permutation of \(\{-d_{1},-d_{2},-d_{3}\}\). So for every sequence \(\{{\mathbf q}_{n}\}\in\mathbb{N}^{d}\), there exists a subsequence \(\{{\mathbf q}_{n_{j}}\}\) such that \[\lim\limits_{j\rightarrow\infty}\beta_{i}({\mathbf q}_{n_{j}})=\beta_{i}.\] Hence we can assume that \[ \lim\limits_{q^2\rightarrow\infty}\beta_{i}({\mathbf q})=-d_{i}, \forall 1 \leq i \leq3, \] or equivalently Using the similar arguments of Lemma 4 in Hoang [1], the following lemma can be derived.

Lemma 4. If \({\mathbf q}\in \mathbb{N}^{d}\) and \(q^2\) sufficiently large, then \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are real numbers and \(\lambda_{i}({\mathbf q})\neq\lambda_{j}({\mathbf q})\), \(i\neq j, i,j=1,2,3\).

Proof. It follows from the assumptions ({\textbf H\(_{2}\)}) and (25) that \(\mathrm{Re}\lambda_{i}({\mathbf q})\neq\mathrm{Re}\lambda_{j}({\mathbf q}), i\neq j.\) If there exists a sequence \(\{{\mathbf q}_{n}\}\in\mathbb{N}^{d}\) such that the sequence \(\lambda_{i_{n}}({\mathbf q}_{n})\notin\mathbb{R}\), then we can choose a subsequence \(\{n_{m}\}\) of \(\{n\}\) and an integer \(j, 1 \leq j\leq 3\) such that \(i_{n_{m}}\equiv j\). Hence \[\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\lambda_{j}({\mathbf q}_{n_{m}})=-d_{j},\] and \[\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\overline{\lambda_{j}({\mathbf q}_{n_{m}})}=-d_{j},\] where \(\overline{\lambda_{j}({\mathbf q}_{n_{m}})}\) is the complex conjugation of \(\lambda_{j}({\mathbf q}_{n_{m}})\). Notice that \(\overline{\lambda_{1}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{2}({\mathbf q}_{n_{m}}), \lambda_{3}({\mathbf q}_{n_{m}})\right\}\), \(\overline{\lambda_{2}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{1}({\mathbf q}_{n_{m}}), \lambda_{3}({\mathbf q}_{n_{m}})\right\}\) and \(\overline{\lambda_{3}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{1}({\mathbf q}_{n_{m}}), \lambda_{2}({\mathbf q}_{n_{m}})\right\}\), then there exists a subsequence of \(\{n_{m}\}\), still denoted by \(\{n_{m}\}\) and \(1 \leq l\leq 3, l\neq j\) such that \(\overline{\lambda_{j}({\mathbf q}_{n_{m}})}=\lambda_{l}({\mathbf q}_{n_{m}})\), one can obtain \[-d_{j}=\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\overline{\lambda_{j}({\mathbf q}_{n_{m}})} =\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\lambda_{l}({\mathbf q}_{n_{m}})=-d_{l}, \forall m\in \mathbb{N}.\] So \(d_{j}=d_{l}\) and \(j\neq l\), in contradiction to the assumption ({\textbf H\(_{2}\)}). Therefore, for \(q^{2}\) sufficiently large \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are real numbers, and we deduce by \(\mathrm{Re}\lambda_{i}({\mathbf q})\neq\mathrm{Re}\lambda_{j}({\mathbf q})\) that \(\lambda_{i}({\mathbf q})\neq\lambda_{j}({\mathbf q})\) whenever \(i\neq j\), which completes the proof.

3. Growing modes and Bootstrap lemma

3.1. Growing modes in the model (1)

For convenience we will always denote universal positive constants depending on \(d_{i}\), \(\chi\), \(\xi\), \(\mu\), \(\alpha\), \(\beta\), \( \gamma \), \(\eta\) \((i=1,2,3)\) by \(C_{k}(k=1,2,\cdot\cdot\cdot)\). Norm in \(L^{2}(\mathbb{T}^{d})\) is denoted by \(\|\cdot\|\).

Lemma 5. Suppose that ({\textbf H\(_{1}\)}) and ({\textbf H\(_{3}\)}) hold, and \({\mathbf w}({\mathbf x},t)\equiv{e^{\mathfrak{L}t}{\mathbf w}_{0}({\mathbf x})\) is a solution to the linearized system (5) with initial condition \({\mathbf w}_{0}({\mathbf x})\). Then there exists a constant \(\hat{C}_{1}>0\) depending on \(d_{i}\), \(\chi\), \(\xi\), \(\mu\), \(\alpha\), \(\beta\), \( \gamma\), \(\eta\) \((i=1,2,3)\) such that}

Proof. We will proceed in the following two cases.
Case 1: For \({\mathbf t\geq 0}\) ,\({\mathbf q}\in \mathbb{N}^{d}\), \(q^{2}\) sufficiently large. By Lemma 4, for \(q^{2}\) sufficiently large, the matrix \({\mathbf L_{q}}\) has three distinct eigenvalues \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) and the corresponding linearly independent eigenvectors \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\), \({\mathbf r}_{3}({\mathbf q})\). We first look for eigenvector \({\mathbf r}_{1}({\mathbf q})\) such that \[{\mathbf r}_{1}({\mathbf q})=(1, r_{12}({\mathbf q}), r_{13}({\mathbf q}))^{\mathrm{T}},\] where \(r_{12}({\mathbf q})\), \(r_{13}({\mathbf q})\) are the solutions of the linear system \[\begin{array}{ll} (-d_{2}q^2-\beta -\lambda_{1}({\mathbf q}))r_{12}({\mathbf q})+ 0=-\alpha,\\ 0+(-d_{3}q^2-\eta -\lambda_{1}({\mathbf q}))r_{13}({\mathbf q})=-\gamma. \end{array} \] \[\begin{array}{ll} r_{12}({\mathbf q})=\frac{\alpha}{(d_{2}q^2+\beta +\lambda_{1}({\mathbf q}))}, \end{array} \] \[\begin{array}{ll} r_{13}({\mathbf q})=\frac{\gamma}{(d_{3}q^2+\eta +\lambda_{1}({\mathbf q}))}, \end{array} \] \[\begin{array}{ll} \displaystyle\lim\limits_{q^2\rightarrow\infty}r_{12}=0,\\ \displaystyle\lim\limits_{q^2\rightarrow\infty}r_{13}=0,\end{array}\] hence

Let \({\mathbf r}_{2}({\mathbf q})=(r_{21}({\mathbf q}), 1, r_{23}({\mathbf q}))^{\mathrm{T}}\), \({\mathbf r}_{3}({\mathbf q})=(r_{31}({\mathbf q}), r_{32}({\mathbf q}), 1)^{\mathrm{T}}\) be eigenvectors corresponding to the eigenvalues \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\), respectively. Then \[\lim\limits_{q^2\rightarrow\infty}r_{21}({\mathbf q})q^{2}=\frac{\chi}{(d_{2}-d_{1})}, \lim\limits_{q^2\rightarrow\infty}r_{23}({\mathbf q})q^{2}=0,\] and \[\lim\limits_{q^2\rightarrow\infty}r_{31}({\mathbf q})q^{2}=\frac{-\xi}{(d_{3}-d_{1})}, \lim\limits_{q^2\rightarrow\infty}r_{32}({\mathbf q})q^{2}=0.\] Therefore By (27) and (28), we deduce that there exists a constant \(C_{1}>0\) such that For \(q^{2}\) sufficiently large, it is follows from (13) that \({\mathbf w}_{{\mathbf q}}=\sum\limits^{3}_{i=1}w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q}).\) Based on Cramer's Rule and Hadamard inequality, we have In terms of (27) and (28), one can obtain Applying (30) and (31) yields where \( C_{2}:=\max\left\{1, \sqrt{(\frac{\chi}{d_{2}-d_{1}})^{2}+1}, \sqrt {(\frac{\xi}{d_{2}-d_{3}})^{2}+1}\right\}>0\). Then, using (29), (32) and \( \lambda_{i}({\mathbf q})\leq \lambda_{\max}\), this shows that for \(q^{2}\) sufficiently large there exists a constant \(C_{3}>0\) independent of \({\mathbf q}\) such that \[\left|w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q})e^{\lambda_{i}({\mathbf q})t}\right|\leq C_{1}C_{2}e^{\lambda_{\max}t}|{\mathbf w}_{{\mathbf q}}|,\] which leads to Case 2: For \(t\leq1\). It is sufficiently to derive standard estimate in \({\mathbf L^{2}}\). From Neumann boundary condition , we can multiplying the first equation in (6) by \(u_{1}\), the second equation by \(k u_{2}\) and the third by \(u_{3}\), adding them together, and integrating the result in \(\mathbb{T}^{d}\), we have \[\begin{array}{ll} \displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{T}^{d}}\{|u_{1}|^2+k|u_{2}|^2+|u_{3}|^2\}{\mathbf d x}+\int_{\mathbb{T}^{d}}\{d_{1}|\nabla{u+{1}}|^2+kd_{2}|\nabla{u_{2}}|^2+d_{3}|\nabla{u_{3}}|^2- \chi (\nabla{u_{1}}\nabla{u_{2}}) \displaystyle+\xi (\nabla{u_{1}}\nabla{u_{3}})\}{\mathbf d x}\\ \displaystyle=-\mu\int_{\mathbb{T}^{d}}u_{1}^2{\mathbf d x}-k\beta\int_{\mathbb{T}^{d}}u_{2}^2{\mathbf d x}- \eta\int_{\mathbb{T}^{d}}u_{3}^2{\mathbf d x}+\alpha k\int_{\mathbb{T}^{d}}u_{1}u_{2}{\mathbf d x}+\gamma \int_{\mathbb{T}^{d}}u_{1}u_{3}{\mathbf d x}. \end{array} \] where \(k=\frac{\chi^{2}d_{3}}{d_{1}d_{2}d_{3}+d_{2}\xi^{2}}\). Then the integrand of the second integral can be estimated as follows Using Young inequality, we deduce that Then \[\begin{array}{ll} \displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{T}^{d}}\{|u_{1}|^2+k|u_{2}|^2+|u_{3}|^2\}{\mathbf d x}\leq \displaystyle\max(-\mu+\frac{k\alpha^{2}}{2\beta}+\frac{\gamma^2}{2\eta},-\frac{\beta}{2}, \frac{\eta}{2})\int_{\mathbb{T}^{d}}(|u_{1}|^2+ k|u_{2}|^2+|u_{3}|^2){\mathbf d x}. \end{array}\] By Grownwall inequality, we can obtain \(\|{\mathbf w}({\mathbf \cdot},t)\|\leq{\hat{C}_{1}e^{\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|},\) where \(\hat{C}_{1}= \max(-\mu+\frac{k\alpha^{2}}{2\beta}+\frac{\gamma^2}{2\eta},-\frac{\beta}{2},\frac{\eta}{2})\). This completes the proof.

3.2. Bootstrap lemma and \(H^{2}\)-estimate in the model (1)

Denote \[\partial_{ x_{i} x_{j}}u=\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}, \partial_{ x_{i}}u=\frac{\partial u}{\partial x_{i}}, D^\alpha u=\frac{\partial^{|\alpha|}u}{\partial x^{\alpha_1}_{1}\cdots\partial x^{\alpha_d}_{d}},\] where \(\alpha=(\alpha_1,\cdots,\alpha_d)\), \(|\alpha|=\sum^{d}\limits_{i=1}\alpha_i\), \(i,j=1,\cdots,d\). Let us introduce By standard theory of parabolic equation, we can establish the existence of local solutions for the model (4).

Lemma 6. (Local existence). For \(s\geq1 (d=1)\) and \(s\geq2 (d=2,3)\), there exist a \(T_{0}>0\) such that the problem (4) with \(u_{1}(\cdot,0), u_{2}(\cdot,0), u_{3}(\cdot,0)\in{H^{s}(\mathbb{T}^{d})}\) has a unique solution \({\mathbf w}(\cdot,t)\) on \((0,T_{0})\) which satisfies \[ \|{\mathbf w}(t)\|_{H^{s}(\mathbb{T}^{d})}\leq{C}{\|{\mathbf w}(0)\|_{H^{s}(\mathbb{T}^{d})}}, \] where \(C\) is a positive constant depending on \(d_{i}, \xi, \chi, \alpha, \beta, \gamma, \eta (i=1,2,3)\).

Lemma 7. Let \({\mathbf w}({\mathbf x},t)=(u_{1}({\mathbf x},t),u_{2}({\mathbf x},t),v({\mathbf x},t))^{\mathrm{T}}\) be a solution of the nonlinear perturbation systemare the generic constants > (3). Then \begin{eqnarray*} &&\frac{1}{2}\frac{d}{dt}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{|D^{\alpha}u_{1}|^2+k|D^{\alpha}u_{2}|^2 +|D^{\alpha}u_{3}|^2\right\}d{\mathbf x}\\ &&+\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{\frac{d_{1}}{4}|\nabla (D^{\alpha}u_{1})|^2+\frac{kd_{2}}{2}|\nabla (D^{\alpha}u_{2})|^2 +\frac{3d_{3}}{2}|\nabla( D^{\alpha}u_{3})|^2\right\}d{\mathbf x}\\ &&+\frac{\beta{k}}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{2}|^2d{\mathbf x}+\frac{\eta}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{3}|^2d{\mathbf x}\\ &&\leq\hat{C}_2\|{\mathbf w}\|_{H^2(\mathbb{T}^{d})}\|\nabla^3{\mathbf w}\|^2+\hat{C}_{3}\|u_{1}\|^2, \end{eqnarray*} where \(\hat{C}_{2}\) and \(C_{0}\) are the generic constants and \(\hat{C}_{3}=(\frac{\alpha^{2}\eta k +\nu ^{2}}{8 \beta\eta a^{2}})c_{0}\).

Proof. Let \({\mathbf w}({\mathbf x},t)\) be a solution of (4). It is not hard to verify that if \({\tilde{\mathbf w}}({\mathbf x},t)=(\tilde{u}_{1}({\mathbf x},t),\tilde{u}_{2}({\mathbf x},t),\tilde{u}_{3}({\mathbf x},t))^{\mathrm{T}}\) is the even extension of \({\mathbf w}({\mathbf x},t)\) on \(2\mathbb{T}^{d}=(-\pi,\pi)^{d} (d=1,2,3)\). The \({\tilde{\mathbf w}}({\mathbf x},t)\) is also the solution of (4) with the homogeneous Neumann boundary conditions and periodical boundary conditions on \(2\mathbb{T}^{d}\). Therefore,

Using Young inequality, we get The nonlinear term \(J_1\) is bounded by Recalling that the Sobolev imbedding \(H^{2}(\mathbb{T}^{d})\hookrightarrow L^{\infty}(\mathbb{T}^{d})\) for \(d\leq3\), we have Notice that Moreover, if \(g\in H^{1}(2\mathbb{T}^{d})\) with \(\int_{2\mathbb{T}^{d}} g=0\), then It follows from (43) and (44) that \[\|\partial_{x_{i}}g\|\leq C_{9}\|\nabla(\partial_{x_{i}}g)\|, \|\partial_{x_{i}x_{j}}g\|\leq C_{9}\|\nabla(\partial_{x_{i}x_{j}}g)\|\] and Together with (40) and (45), we further get Then as a consequence of(40) and (45), one can obtain where \(C_{12}:=C_{4}+(1+2d)C_{9}\).
Applying interpolation, we can deduce that for all \(\varepsilon>0\), By the choice of \(\varepsilon>0\) in (48) such that \(\left(\frac{\alpha^{2}k\eta+\beta\nu^{2}}{2\beta\eta}\right)C_{0}\varepsilon=d_{1}/4\), then Then as a consequence of(40) ,(41),(42)and (45), one can obtain Substituting (47), (49)-(50) into (37), we have \begin{eqnarray*} &&\frac{1}{2}\frac{d}{dt}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{|D^{\alpha}u_{1}|^2+k|D^{\alpha}u_{2}|^2 +|D^{\alpha}u_{3}|^2\right\}d{\mathbf x}\\ &&+\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{\frac{d_{1}}{4}|\nabla (D^{\alpha}u_{1})|^2+\frac{kd_{2}}{2}|\nabla (D^{\alpha}u_{2})|^2 +\frac{3d_{3}}{2}|\nabla( D^{\alpha}u_{3})|^2\right\}d{\mathbf x}\\ &&+\frac{\beta{k}}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{2}|^2d{\mathbf x}+\frac{\eta}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{3}|^2d{\mathbf x}\\ &&\leq\hat{C}_2\|{\mathbf w}\|_{H^2(\mathbb{T}^{d})}\|\nabla^3{\mathbf w}\|^2+\hat{C}_{3}\|u_{1}\|^2, \end{eqnarray*} where \(\hat{C}_{2}\) and \(C_{0}\) are the generic constants and \(\hat{C}_{3}=(\frac{\alpha^{2}\eta k +\gamma^{2}}{8 \beta\eta a^{2}})c_{0}.\)This completes the proof of Lemma 7.

Lemma 8. Let \({\textbf w}({\textbf x},t)\) be a solution to the system (4) such that for \(0\leq t\leq T\),

and Then for \(0\leq{t}\leq{T}\), where \(\hat{C}_4=\max\{(1+C^2_{9})k, 4\hat{C}^2_1[1+ \hat{C}_3(1+C^2_{9})/(2\lambda_{\max})]\}\geq1\), if \(k\geq1\). \(\hat{C}_4=\max\{(1+C^2_{9})/k, 4\hat{C}^2_1[1+\hat{C}_3(1+C^2_{9})/(2\lambda_{\max}k)]\}\geq 1\), if \(k<1\).

Proof. It follows from (45) that

So By Lemma 7 and (51), we infer Integrating (57) and using (52), we conclude We first consider the case \(k\geq1\). By (57), we have \[\sum\limits_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},t)\|^2\leq k\sum\limits_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},0)\|^2 +\frac{4\hat{C}^2_1\hat{C}_3}{\lambda_{\max}}e^{2\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|^2.\] We see from this estimate and (55) that where \(\hat{C}_4:=\max\left\{(1+C^2_{9})k, 4\hat{C}^2_1\left[1+\frac{\hat{C}_3(1+C^2_{9})}{\lambda_{\max}}\right]\right\}\). On the other hand, for \(K< 1\), we deduce by (57) that \[ \sum_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},t)\|^2\leq\frac{1}{K}\left(\sum_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},0)\|^2 +\frac{4\hat{C}^2_1\hat{C}_3}{\lambda_{\max}}e^{2\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|^2\right). \] This estimate, combined with (52) and (55) gives where \(\hat{C}_4:=\max\left\{\frac{1+C^2_{9}}{k}, 4\hat{C}^2_1\left[1+\frac{\hat{C}_3(1+C^2_{9})}{\lambda_{\max}k}\right]\right\}\). This completes the proof of Lemma 8.

4. Main result

Assume \(\theta\) be a small fixed constant. For \(\delta>0\) arbitrary small, we define the escape time \(T^{\delta}\) by where \(\lambda_{\max}\) is the dominant eigenvalue which is the maximal growth rate (see (10)). Obviously, Our main result in this paper is as follows:

Theorem 2. Suppose that (H\(_{1}\)),(H\(_{2}\))and(H\(_{3}\)) are satisfied. Let \({\textbf w}_{0}({\textbf x})\in{H^2(\mathbb{T}^{d})}\) with \(\|{\mathbf w}_0({\mathbf x})\|=1\). Then there exist constants \(\delta_0>0, \hat{C}>0\), and \(\theta>0\) depending on \(d_{i}, \chi, \xi, \mu, \alpha, \beta, \eta, \gamma, (i=1,2,3)\) such that \(\forall 0< \delta\leq\delta_0\), if the initial perturbation of the steady state \({\mathbf w_{c}}\) is \({\mathbf w}^{\delta}({\mathbf \cdot},0)=\delta{\mathbf w}_0\), then its nonlinear evolution \({\mathbf w}^{\delta}({\mathbf \cdot},t)\) satisfies

for \(0\leq{t}\leq{T^{\delta}}\), and \(\rho>0\) is the gap between the largest growth rate \(\lambda_{\max}\) and the rest of \(\mathrm{Re}\lambda_{i}({\mathbf q})\) in (7), \(e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})\) defined in (16) is the dominant part of the solution of the linearized system (5).

Proof. Let \({\mathbf w}^{\delta}({\mathbf x},t)\) be the solutions to (4) with initial data \({\mathbf w}^{\delta}({\mathbf \cdot},0)=\delta{\mathbf w}_0\). Define

From the definition of \(T^{\ast}\) and Lemma 5, for \(\forall 0\leq t\leq{T^{\ast}}\), we can obtain Furthermore, by Lemma 8 and the bootstrap argument, we possess Applying Duhamel's principle, we know that the solution of (4) It follows from Lemma 5, (40), (44) and Lemma 8 that for \(0\leq{t}\leq\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\), where \(\hat{C}_{5}=\max C^2_{9}\{\chi+\chi\frac{C_{4}}{C^2_{9}}+\xi+\xi\frac{C_{4}}{C^2_{9}}+\mu C_{1}\}\). By (66) and (68), we see that for \(t\leq\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\), We now prove that if \(\delta_{0}\) and \(\theta\) are chosen such that and as well as then \(T^{\delta}=\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\) for \(\delta\leq\delta_{0}\).
If \(T^{\ast\ast}\) is the smallest, we can let \(t=T^{\ast\ast}\leq{T^{\delta}}\) in (67). By (70) and (71) we have \[\begin{array}{ll} \displaystyle\left\|{\mathbf w}^{\delta}(T^{\ast\ast})\right\|_{H^2(\mathbb{T}^{d})} \leq\sqrt{\hat{C}_4}\left\|{\mathbf w}^{\delta}_0\right\|_{H^2(\mathbb{T}^{d})}+\sqrt{C_4} \theta \displaystyle< \frac{1}{\hat{C}_{2}}\min\left\{\frac{d_{1}}{4}, \frac{d_{2}}{4}, \frac{d_{3}K}{2}\right\}, \end{array}\] for \(\delta\) sufficiently small and \(\hat{C}_4\geq1\), in contradiction to the definition of \(T^{\ast\ast}\). On the other hand, if \(T^{\ast}\) is the minimum, we can let \(t=T^{\ast}\) in (67), so that \[\begin{array}{ll} \displaystyle\left\|{\mathbf w}^{\delta}({\mathbf \cdot},T^{\ast})-\delta{e}^{\mathfrak{L}T^{\ast}}{\mathbf w}_0\right\| \leq \hat{C}_1\hat{C}_4\hat{C}_5\left\{\frac{\delta\|{\mathbf w}_0\|^{2}_{H^2(\mathbb{T}^{d})}}{\lambda_{\max}} +\frac{\theta}{\lambda_{\max}}\right\}\delta e^{\lambda_{\max}T^{\ast}} \displaystyle< \frac{\hat{C}_1}{2}\delta e^, \end{array} \] for sufficiently small \(\delta_{0}\) in (73) and \(\hat{C}_5/{\hat{C}}_{2}\leq1\). This again contradicts the definition of \(T^{\ast}\). Therefore, the desired assertion follows. Finally, we prove the inequality (62). Notice by (14) that We next estimate each term \(J_{i} (i=6,7,8,\cdots,12)\) on the right-hand sides of (73). It is not difficult to know that there are finitely many values \({\mathbf q}\in \mathbb{N}^{d}\) satisfying \(\mathrm{Re}\lambda_{i}({\mathbf q})=\lambda_{\max}\) and \(|{\mathbf q}|\) is bounded for each \({\mathbf q}\in{\mathbb{N}^{d}}_{\max}\). For each \({\mathbf q}\in \mathbb{N}^{d}\), \(q^{2} < N\) there exists a constant \(C_{*}>0\) such that By the similar method to prove (32), using (74) and (13) there exists a constant \(C_{**}>0\) such that and By (29), (32),(74), (75) and \(\|{\mathbf w}_{0}\|=1\), there exists a constant \(\hat{C}_{6}>0\) such that \[\displaystyle J^{2}_7\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t}\left(\frac{\pi}{2}\right)^{d}\sum\limits_{{\mathbf q}\in\Lambda_{R1}}|{\mathbf w}_{{\mathbf q}}|^{2}\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t}\|{\mathbf w}_{0}\|^{2}\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t},\] that is, Moreover, Similarly, there exists a constant \(\hat{C}_{7}>0\) such that Substituting (69), (77)-(81) into (73) yields \begin{eqnarray*} \displaystyle\left\|{\mathbf w}^{\delta}({\mathbf \cdot},t)-\delta{e}^{\mathfrak{M}t}{\mathbf w}_0\right\| &\leq&\hat{C}_1\hat{C}_4\hat{C}_5\left\{\frac{\delta\|{\mathbf w}_{0}\|^2_{H^2}}{\lambda_{\max}}+\frac{\delta e^{\lambda_{\max}t}}{\lambda_{\max}}\right\}\delta e^{\lambda_{\max}t} +\hat{C}_6\delta e^{(\lambda_{\max}-\rho)t}+\delta e^{(\lambda_{\max}-\rho)t}+5\hat{C}_7 \delta e^{(\lambda_{\max}-\rho)t}\\ &\leq&\left\{(1+\hat{C}_{6}+5\hat{C}_7)e^{-\rho t}+\frac{\hat{C}_1\hat{C}_4\hat{C}_5}{\lambda_{\max}}\left(\delta\|{\mathbf w}_0\|^2_{H^2(\mathbb{T}^{d})} +\delta e^{\lambda_{\max}t}\right)\right\}\delta e^{\lambda_{\max}t}\\ &\leq&{\hat{C}}\left\{e^{-\rho t}+\delta\|{\mathbf w}_0\|^2_{H^2(\mathbb{T}^{d})}+\delta e^{\lambda_{\max}t}\right\}\delta e^{\lambda_{\max}t}, \forall 0\leq{t}\leq{T^{\delta}}, \end{eqnarray*} where \(\hat{C}:=\max\{1+\hat{C}_{6}+5\hat{C}_7, \frac{\hat{C}_1\hat{C}_4\hat{C}_5}{\lambda_{\max}}\}\) and thereby completes the proof.

Corollary 1.(Nonlinear instability). Let the conditions ( \({\mathbf H_{1}}\)),(\({\mathbf H_{2}}\)) and (\({\mathbf H_{3}}\) ) are holds. Then the positive constant equilibrium point \({\mathbf w_{c}}\) of the problem (1) is nonlinearly unstable in the sense of the \(L^{2}\)-norm.

Proof. Notice that \({\mathbf L}_{{\mathbf q_{0}}}\) has an eigenvalue \(\mathrm{Re}\lambda_{{\mathbf q_{0}}} = \lambda_{\max}\), if there exists \({\mathbf q}_0=(q_{01},\ldots,q_{0d})\in{\mathbb{N}^{d}}_{\max}\), and denote the corresponding eigenvector by \({\mathbf r}_{{\mathbf q_{0}}}\). Assume \[{\mathbf w}_0({\mathbf x})=\kappa\frac{{\mathbf r}({\mathbf q}_0)}{|{\mathbf r}({\mathbf q}_0)|}e_{{\mathbf q}_0}({\mathbf x})\] with \(\kappa=1/\|e_{{\mathbf q}_0}\|=\sqrt{(2/\pi)^{d}}\) so that \(\|{\mathbf w}_{0}(x)\|=1\). In addition, if \(t=T^{\delta}\) then for \(\delta\) sufficiently small, we require

It follows from Theorem 2 and(80) that Notice that the dominant part of the solution of the linearized system (5) satisfies By (81) and (82), we deduce that \[\|{\mathbf w}^{\delta}({\mathbf \cdot},T^{\delta})\|>\frac{1}{2}\theta>0.\]

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

Living organisms consist of one or more tiny components of several types and shapes termed cells on which molecules move in continuous random motion. These cells and the molecules can be considered as subdivisions of a 2-dimensional plane on which particles randomly move. The plane can be much wider, but considering that each molecule moving on the plane has a starting cell, we can restrict this movement to a few groups of cells. The knowledge obtained from this small group of cells can be extended to improve our understanding of the molecules movement on a larger scale. The shape of the cells dictates the different random possibilities of a molecule movement to neighboring cells from the starting cell.

A cell can assume different shapes, including square and hexagonal shapes. In both square and hexagonal tilings assumed by a cell, the set of all possibilities of a molecule moving towards a neighboring cell can be seen as a Markov chain \(\{X_t , t > 0\}\) [1]. This Markov chain is driven by a parameter \(p\) which represents the probability for the molecule under study to move from one cell to a neighboring cell. A Markov chain can be discrete or continuous depending on whether the time considered is discrete or continuous [2].

A recent study made in [1] on this topic considered discrete time process. It was demonstrated that the molecule is faster in the hexagonal tiling than in the square tiling.

In this paper, we will look at the continuous process and compare the result with those found in the discrete process. We will examine how the probability impacts the movement of a molecule from cell \(i\) to cell \(j\). When a molecule moves from a cell \(i\) to \(j\), the possible next step of the movement depends on the number of cells enclosing it. For example, from the central cell of the square tiling , a molecule has four possibilities to move to while there are six possibilities in the hexagonal tiling.

In the aforementioned study ([1]), two starting positions were considered: the central cell and the surrounding ones. We only consider the central cell to be the starting position of the molecule since each cell (even the border cells) can be considered as central by enlarging the plane.

Infinitesimal generators in continuous time will replace the transition matrices in discrete time to describe the movement of the molecule. In this study, the space is discrete.

Sometimes, the transition matrices can be very large and almost impossible to handle for doing computations. In order to reduce the calculations, we will use the state symmetries after identifying the non-equivalent cells in each tiling, then we will lump states with ²the same properties [3]. Symmetric groups afford a precise definition of structural equivalence for Markov chains states in aggregating them to making a partition of the original Markov process in small subsets that conserve all the previous properties [4]. This aggregation results in a new Markov chain (aggregated chain) with fewer number of states such that the finite probabilities of aggregated states equals the finite probabilities of the corresponding states of the initial Markov chain [5].

The specific questions we want to address include:

• (1) What is the effect of the discrete or continuous nature of time in the oscillatory movement of the molecule?
• (2) What is the effect of the probability, the time and the shape of the tiling in the attainment of the equilibrium in continuous Markov process under consideration ?

2. Markov chains

2.1. Definitions

Definition 1. A sequence of random variables \(\{X_n\}_{n_{\geq 0}}\) in a countable space E is called stochastic process. E is called states space whose elements will be written \textit{i, j, k, ...}.

When \(X_n = i\), the process is in the state \(i\) or visits the state \(i\) at the time \(n\). Sequences of random variables which are independent and identically distributed are stochastic process but they do not take into account the dynamic of evolution of systems due to their independence. To introduce this dynamic, one must take into account the influence of the past, which Markov chains do, like the equation of recurrence in deterministic systems [2]. Then we introduce the following:

Definition 2. For all \(n\in \mathbb{N}\) and all states \(i_0 , i_1, i_2, i_3,...,i_{n-1},i,j \in E\),

then the process \(\{X_n\}_{n_{\geq 0}}\) is called Markov chain.

The Equation (1) is called Markov property. The matrix \(P=\{p_{ij}\}_{i,j\in E},\) where \begin{equation*} p_{ij}=P(X_{n+1}=j\arrowvert X_n =i) \end{equation*} is the probability to move from \(i\) to \(j\), is called transition matrix of the chain.
Since all \(p_{ij}\) are probabilities and the transition happens from a state \(i\) to a state \(j\), one has \(p_{ij}\geq 0\) and \[\sum_{k\in E} p_{ik}=1,~~ \forall i,j.\] A matrix indexed by E and satisfying the above properties is a stochastic matrix.
A Markov chain is said to be \textit{discrete time} if the state space of the possible outcomes of the process is finite.

2.2. Continuous-time Markov chains

Definition 3. A continuous-time Markov chain \(X(t)\) is defined by two components: a jump chain, and a set of holding time parameters \(\lambda_i.\) The jump chain consists of a countable set of states \(S\subset\{0,1,1,...\}\) along with transition probabilities \(p_{ij}\). We assume \(p_{ii}=0\), for all non-absorbing states \(i\in S\). We assume that:

• 1) If \(X(t)=i\), the time until the state changes has exponential \((\lambda_i)\) distribution;
• 2) If \(X(t)=j\), the next state will be in \(j\) with probability \(p_{ij}\).

The process satisfies the Markov property 1.

For a continuous Markov chain, the Equation (1) can be rewritten as follows: This chain is homogeneous if the second member of (2) does not depend on the time \(t\). If (2) is a system of differential equations that does not depend on \(t\), it is said to be an autonomous system ([6]) whose stability depends on the signs of its eigenvalues. We can then define the transition matrix, \(P(t)\).
Assuming the states are \(1, 2,..., r\), then the state transition matrix for any \(t\geq 0\) is given by Let \(X(t)\) be a continuous-time Markov chain with transition matrix \(P(t)\) and state space \(S=\{0,1,2,...\}\). A probability distribution \(\pi\) on \(S\) i.e, a vector \(\pi =[\pi_1, \pi_2,\pi_3,..]\), where \(\pi \in [0,1]\) and $$\sum_{i} \pi_i = 1$$ is said to be a stationary distribution for \(X(t)\) if The intuition here is exactly the same as in the case of discrete-time chains. If the probability distribution of \(X(0)\) is \(\pi\), then the distribution of \(X(t)\) is also given by \(\pi\), for any \(t\geq 0\). The Equation (3) is solution to the so called backward Chapman-Kolmogorov equation below [7] Calculation of Equation (5) may be cumbersome and tedious. This hindrance can be overcome by using lumpability if the transition matrix satisfies some conditions (see [8], [9] and [3]).
The following definition from [9] is important for the suit of this paper.

Definition 4. Let \(\{X_t\}\) be a Markov chain with state space \(S=\{1,2,\cdots,r\}\) and initial vector \(\pi\). Given a partition \(\bar{S}=\{E_1, E_2, \cdots, E_v\}\) of the space \(S\), a new chain \(\bar{X}_n\) can be defined as follows: At the \(jth\) step, the state of a new chain is the set \(E_k\) when \(E_k\) contains the state of the \(jth\) step of the original chain.

Precisely, a continuous Markov chain is said to be lumpable with respect to the partition \(\bar{S}\) if for \(i,j \in E_\eta\), According to [8], a Markov chain X whose transition probability matrix from state \(i\) to state \(j\) denoted by \(p_{ij}\) is lumpable with respect to the partition \(\bar{S}\) if and only if for every pair of sets \(E_\eta\) and \(E_\theta\), \(\sum_{k} \in E_\eta p_{ik}\) has the same value for every \(e_i\) in \(E_\theta\). These common values form the transition probabilities \(p_{\eta, \theta}\) for the lumped chain. Moreover, one has the following theorem from [9].

Theorem 1. Let X(t) be an irreducible continuous-time Markov chain with stationary distribution \(\pi\). If it is lumpable with respect to a partition of the state space, then the lumped chain also has a stationary distribution \(\bar{\pi}\) whose components can be obtained from \(\pi\) by adding corresponding components in the same cell of partition.

Infinitesimal Generator of Continuous-time Markov chains

The infinitesimal generator matrix, usually shown by G, gives us an alternative way of analyzing continuous-time Markov chains. Consider a continuous-time Markov chain \(X(t)\). Assume \(X(0)=i\). The chain will jump to the next state at time \(T_1\), where \(T_1\sim Exponential(\lambda_i)\). In particular, for a very small \(\delta \geq 0\), we can write \begin{equation*} P(T_1 \leq \delta)=1-e^{-\lambda_i \delta}\simeq 1-(1-e^{-\lambda_i \delta}) =\lambda_i\delta. \end{equation*} Thus, in a short interval of length, \(\delta\) the probability of leaving state \(i\) is approximately \(\lambda_i \delta\). For this reason, \(\lambda_i\) is often called the transition rate out of state \(i\). Formally, we can write More details and the following definition may be found in [10].

Definition 6. For a continuous-time Markov chain, we define the generator matrix G. The (i,j)th entry of the transition matrix is given by

An infinitesimal generator always satisfies the equation For an infinitesimal generator to be lumpable, it must satisfy the condition contained in the following definition that the reader can check in [9].

Definition 6. We say that an infinitesimal generator G is lumpable if

First hitting times

Definition 7. Let (\(X_t)_{t\ge 0}\) be a Markov chain with generator matrix G. The hitting time of a subset A\(\subset \)S is the random variable $$\tau^A (\omega )=inf\{t\ge 0|X_t(\omega)\in A\}$$ with the usual convention \(inf\emptyset =\infty\).

Theorem 2. The vector of mean hitting times \(k^A=\{ k_i^A|i\in S\}\) is the minimal nonnegative solution of

The reader can find out more about this in [11].

3. Investigation of the movement on small tiling in continuous time

In this section we investigate the motion of a molecule in two small tilings: the square tiling and the hexagonal one. This movement from a cell \(i\) to a cell \(j\) is considered as being an homogeneous Markov chain. States with the same stochastic behavior are lumped together using symmetries of states deduced from groups acting on the cellular complexes. According to [12], the group acting on a polygon is a dihedral group. In the particular case of the small square tiling, we have the symmetric group \(S_9\) and for the hexagonal tiling we have \(S_7\). Thanks to these groups, we will use the technique of lumpability. This lumpability is effective in forming new chains from the old ones without losing the primitive properties and simplifying tedious calculations.
At each step, the molecule is supposed to leave the central cell and move into the surrounding cells. In [1], it is shown that the movement of biological molecule on tilings (either square or hexagonal) can be modeled by a (discrete time) Markov chain. We will extend this movement of biological molecule on small tiling in continuous time.

3.1. Continuous-time process in small square tilng

We already have important results from previous works on discrete-time Markov process in small cell complexes ([1]). We want to extend this study to the continuous case especially in the square tiling. We will assume a discrete space throughout this study.

As already highlighted, a molecule is supposed to be at the central cell (cell 1-1 on Figure 1) at the beginning of the motion. When coming from this position, the molecule can immediately move to one of the following neighboring cells: 2-1, 2-3, 2-5 and 2-7. Thus, the probability of moving to each one of them is the same. However,to move to the cells at the corners, the molecule will move in two steps: the first is the transit to the surrounding cell and the second to the corner. This means that there is also the same probability to move to each corner cell. But this probability differs from the preceding. In the paragraph below, we analyze this to show how to reduce calculations of the infinitesimal generator.

3.1.1. Infinitesimal generator and probability matrix

The molecule has four possibilities of moving to neighboring state with, assume, probability \(p\). All cells can be reached in one step from the center except those located at the corner (corner cells) of the tiling. Therefore, the infinitesimal generator, \(G\) for a square tiling takes the form where all \(\alpha \geq 0\) is the transition rate. This matrix corresponds to an irreducible chain because it is always possible to go from one state to another (see [13] for further details).
We now compute the probability matrix \(P(t)\) defined by Equation (3) by using the Chapman-Kolmogorov backward equation (see Equation (5)).
A direct computation of Equation (5) will be tedious because of the size of Equation (12). We therefore lump the symmetric states as depicted on Figure 2. This figure shows that the symmetric group \(S_9\) is a partition of the proposed Markov chain.

The original Markov chain is lumped as \textcolor{magenta}{(1-1)}\textcolor{cyan}{(2-1 2-3 2-5 2-7)}\textcolor{red}{(2-2 2-4 2-6 2-8)}.
The new infinitesimal generator is obtained from The original Markov chain and the new infinitesimal generator satisfy all the hypotheses of the Definitions 4 and 6. Substituting each parameter by its value in the Equation (13), we get the new infinitesimal generator, Substituting Equation (14) into Equation (5), we get for \begin{equation*} P(t)= \begin{pmatrix} p_{11}& p_{12} & p_{13} \\ p_{21}& p_{22} & p_{23} \\ p_{31}& p_{32} & p_{33} \end{pmatrix}, \end{equation*} where all \(p_{ij}\) (\(i,j\in\{1,2,3\}\)) are functions depending on the same variable \(t\)); the following system: where \(p'_{ij}\) indicates the derivative of \(p_{ij}\) (\(i,j\in \{1,2,3\}\)). The multiplication of the right part of the equality yields: This system is made of equivalent equations. Thus, instead of solving the whole system, we just solve one of the systems with three equations. We can either solve or or again Algebraic computations show that the matrix associated to any of the subsystems (i.e. Equation (17), Equation (18), and Equation (19) ) has three eigenvalues : \(\lambda_1 =-6\alpha,~~ \lambda_2 = -3\alpha\) and \(\lambda_3 =0\) and the corresponding eigenvectors: $$v_1 =\begin{pmatrix} 1\\ \frac{-1}{2}\\ \frac{1}{4} \end{pmatrix},~~~ v_2 =\begin{pmatrix} 1\\ \frac{1}{4}\\ \frac{-1}{2} \end{pmatrix}, ~~~ v_3 =\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}.$$ The general solution of each subsystem can be written as where \(c_i\) are constants. We then have successively Since \(P(0)=I_3\), after substitution and computations, we get %Equation \ref{Solution matrix expsmallsquare} can be found by computing directly the exponential of the product of \(t\) and the infinitesimal generator \(G\).

3.1.2 Stationary distribution and limiting probability

A stationary distribution of a Markov chain is a probability distribution that remains unchanged in the Markov chain as time progresses. Typically, it is represented as a row vector \(\pi\) whose entries are probabilities summing to \(1\) and, given the transition matrix \(P\), it satisfies the Equation (4). It can be shown (see [14])that the Equation (4) is equivalent to with \(G\) the infinitesimal generator of the chain. Considering \(\pi\), the stationary distribution associated to the lumped chain above reduces the relation Equation (25) as Equation (26) together with \(\sum\limits_{i} \pi_i =1\) yields Solving this system (Equation (27)), we find the stationary distribution \(\pi =\left(\frac{1}{9},\frac{4}{9},\frac{4}{9}\right).\) This stationary distribution found is exactly the same as the one associated with the original chain, i.e. non-lumped system. Another parameter relating to stationary distribution is the limiting distribution.

Definition 8. The limiting distribution of a Markov chain seeks to describe how the process behaves a long time after.

For the limiting distribution to exist, the following limit must exist for any states \(i\) and \(j\) Furthermore, for any state \(i\), the following sum must be \(1\). This ensures that the numbers obtained do, in fact, constitute a probability distribution. Provided these two conditions are met, then the limiting distribution of a Markov chain with \(X_0 =i\) is the probability distribution given by \(l=(L_{ij})_{states~j}\). For any time-homogeneous Markov chain that is aperiodic and irreducible, \(\lim_{n\longrightarrow\infty} \mathbf{P}^n\) converges to a matrix with all rows identical and equal to \(\pi\).
For time-homogeneous Markov chains, any limiting distribution is a stationary distribution [15]. The relation Equation (28) applied on the matrix Equation (24) provides the following matrix: $$ \mathbf{P}_{\pi}=\begin{pmatrix} \frac{1}{9}&\frac{4}{9} & \frac{4}{9} \\ \frac{1}{9}&\frac{4}{9} & \frac{4}{9} \\ \frac{1}{9}& \frac{4}{9} & \frac{4}{9} \end{pmatrix}, $$ which is the limiting distribution of the Markov chain deduced from the square tiling. It is a stochastic matrix. It satisfies the Equation (29) as expected.

3.1.3. Calculation of the mean hitting times

In this section, we want to compute the mean value of the time to be spent by the molecule in a cell for the first time by using the Equation (11). For \(A=\{1\}\), we have the following system: \begin{equation*} \left\lbrace \begin{array}{rrr} k_1^A &=&0,\\ g_{21}k_1^A +g_{22}k_2^A +g_{23}k_3^A &= &-1,\\ g_{31}k_1^A +g_{32}k_2^A +g_{33}k_3^A &=&-1,\\ \end{array} \right. \end{equation*} whose solution is \( \begin{pmatrix} 0 \\ \\ \frac{2}{\alpha} \\ \\ \frac{5}{2\alpha} \end{pmatrix} \) after substituting all the \(g_{ij}\) with their corresponding values in Equation (14). In the same way, we respectively have for \(A=\{2\}\) and \(A=\{3\}\) the following vectors: \( \begin{pmatrix} \frac{1}{4\alpha} \\ \\ 0 \\ \\ \frac{1}{2\alpha} \end{pmatrix} \) and \( \begin{pmatrix} \frac{7}{8\alpha} \\ \\ \frac{5}{8\alpha} \\ \\ 0 \end{pmatrix}. \) The undermentioned matrix H summarizes the findings for the mean hitting times:

3.1.5. R simulation of effect of probability and time on the movement in square tiling

Figure 3 and 4 represent the transition rate against time. The Figure 3, in particular, shows how the variation in the transition rates affects the attainment of the equilibrium. By comparing graph 3a and graph 3c, we can see that the variation in the \(\alpha\) parameter value affects the oscillation of the state curves. This means that the variation of the transition rates influences the attainment of the equilibrium. We can notice that on the graph 3c where the probability value is the smallest, the equilibrium state is reached quicker than on the other two graphs of the same figure.

On the other hand, Figure 4, also represents the curve behavior in time variation for a fixed value of the transition rates. Hence, by comparing Figure 4a and Figure 4c we find that the slope of the state curves reaches stability at almost the fifteenth unit of time. This explains why, considering a larger time interval, the equilibrium status seems to be reached very early.
For example, if we choose the second as unit of time, we can note that from graph Figure 4a starts the equilibrium phase almost at the eighth second. Considering a larger interval of time (as 100 at Figure 4b or Figure 4c), the equilibrium attainment time is still the fifteenth second. It can be seen that the starting state curve is less steep on Figure 4a where the time is 30 than in graph 4c where the time is 150.

From these two panels of graphs, we see that the fastness in the attainment of the equilibrium is not dictated by the duration of the movement but by the value of the probability (transition rates).

4. Continuous-time process in small hexagonal tiling

In this section, we examine how some parameters influence the behavior of the motion in the hexagonal tiling. We will consider the aggregated complex cell for reducing the computations. The unique starting position is the central cell 1-1.

4.1. Infinitesimal generator and probability matrix

Let us consider the small hexagon depicted on Equation (5). From the central cell, the molecule (the system) has six possible equiprobable destinations which are its neighboring cells. Based on this information, we then produce the following infinitesimal generator

On Figure 5 we have two kinds of equivalent cells: The central cell and the surrounding ones. Thus, we can make a partition of the chain in two states instead of seven. The Figure 6 summarizes what happens exactly in lumping the equivalent cells.

The new infinitesimal generator may be written in the following way: The probability matrix \(P(t)\) is solution to the Kolmogorov Equation (5) and can be written as

4.2. Stationary distribution and limiting probability

The stationary distribution of the lumped chain is the vector \(\pi(\pi_1, \pi_2)\) such that Doing necessary substitution, we get: $$ \begin{pmatrix} \pi_1 & \pi_2 \end{pmatrix} \begin{pmatrix} -6\alpha & 6\alpha \\ \alpha & -\alpha \end{pmatrix} = \left\lbrace \begin{array}{cc} -6\alpha\pi_1 +\alpha \pi_2 &=0, \\ 6\alpha\pi_1 -\alpha\pi_2 & = 0. \end{array} \right. $$ This relation together with \(\pi_1 + \pi_2 = 1\) yields \(\pi =\left(\frac{1}{7}, \frac{6}{7}\right).\) The second component of the stationary distribution is made up of the sum of the stationary probabilities of all the six aggregated states. To compute the limiting distribution, we are going to use again the formula given in Equation (28). We then have \(\lim\limits_{t\rightarrow \infty} P(t)=\begin{pmatrix} \frac{1}{7}& \frac{6}{7} \\ \frac{1}{7}& \frac{6}{7} \end{pmatrix} \) as expected.

4.3. Calculation of the mean hitting time

It is easy to check that the matrix of the mean hitting time for the movement of the particle in the hexagonal tiling is

4.4. Simulation of the effects of probability and time on the movement

Figure 7 and Figure 8 plot the impact of the probability and the time on the attainment of the equilibrium in a hexagonal tiling when we consider continuous time.

The collection of graphs illustrated on Figure 7 and Figure 8 depicts how fast the molecule reaches the equilibrium in the hexagonal tiling in continuous time. The main factor which affects the attainment of the equilibrium is the transition rate. The transition rate \(\alpha=\frac{1}{6}\) is a critical value which affects particularly the motion of the molecule in the hexagonal tiling. For this value, if the molecule quits the central cell, it will never come back into it. A quick substitution in Equation (35) and a glance on Figure 8b can allow to verify it.

5. Discussion of results and conclusion

Under continuous-time conditions, we have checked the same results. In fact, Equation (30) and Equation (35) show that the average transition time from state 1 to state 2 is greater in the square tiling than in the hexagonal tiling. A glance at the panels of graphs depicted above shows that the greater the probability (transition rate), the later the equilibrium is reached in both square and hexagonal tilings. However, when comparing the movement in both tilings, we realize that the equilibrium is quickly reached in hexagonal tiling than in the square one. Increasing the value of the transition rate leads to a quick or late attainment of the equilibrium.

In this paper, the movement of a molecule in two kinds of tilings has been studied: the square tiling and the hexagonal one. Its has been established that only two parameters, among the four considered, have an impact on the quick or late attainment of the equilibrium. The parameters under consideration in this study were the nature of the time (discrete or continuous), the probability (so called transition rate), the time and the shape of the tiling.

In [1], the movements of the molecules in the tilings were modelled using discrete-time Markov chains. It was established that this motion reaches the equilibrium point faster in the hexagonal tiling than in the square one. This same finding is established in continuous-time Markov chains. It is to be deduced that the nature of time does not have an impact on reaching the equilibrium point.However, the shape of the tiling is a core parameter for the attainment of the equilibrium. That is, the molecule is faster in hexagonal tiling than in the square one.

Another important parameter is the transition rate in the infinitesimal generator. During this study, it has been demonstrated that for both hexagonal and square tilings, the rapidity to attain the equilibrium depends also upon the transition rate under consideration. Hence, the smaller the transition rate, the faster the molecule is, in reaching the equilibrium position and vice versa. To put it in a nutshell, this study has proven the influence of transition rate and the shape of the tiling were important for the rapidity of the movement. Other parameters do not have considerable impact on the movement.

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 well known "Growth" phenomenon is one of the most important phenomena of asymptotic behavior, where many researches omit from its study especially when it comes from the evolution problems. It gives us very important information to know the behavior of equation when time arrives at infinity, it differs from global existence and blow up in both mathematically and in applications point of view. Although the interest of the scientific community for the study of delayed problems is fairly recent, multiple techniques have already been explored in depth.

In this direction, we are concerned with the delayed damped system

where \(\omega,b, \mu_{1}\) are positive constants, \(p>2\) and \(\tau_{1}, \tau_{2}\) are the time delay with \(0\leq\tau_{1}< \tau_{2}\) and \(\mu_{2}\) is bounded function and \(\varpi\) is a differentiable function.

It is well known that viscous materials are the opposite of elastic materials which have the capacity to store and dissipate mechanical energy. As the mechanical properties of these viscous substances are of great importance when they appear in many applications of other applied sciences. Many searchers have paid attention to this problem.

In the absence of the strong damping \(\Delta u_{t}\), that is for \(w=0\) and in absence of the distributed delay term. Our problem (1) has been investigated by many authors and results on the local/global existence, and stability have been established. See for example [1, 2, 3, 4]. In [5], the authors looked into the following system

where decay result of an exponential rate was showed.

In [6], Song and Xue considered with the following viscoelastic equation with strong damping: The authors showed, under suitable conditions on \(g\), that there were solutions of (3) with arbitrarily high initial energy that blow up in a finite time. For the same Problem (3), in [7], Song and Zhong showed that there were solutions of (3) with positive initial energy that blew up in finite time. In [8], Zennir considered with the following viscoelastic equation with strong damping: They proved the exponential growth result under suitable assumptions.
In [9] the authors considered the following problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms They proved a blow up result if \(p>m\) and established the global existence.
In this article, we investigated Problem (1), in which all the damping mechanism have been considered in the same time, these assumptions make our problem different form those studied in the literature, specially the Exponential Growth of solutions. We will prove that if the initial energy \(E(0)\) of our solutions is negative (this means that our initial data are large enough), then our local solutions in bounded and as \(t\) tends to \(\infty\), used idea in [10, 11, 12, 13].
Our aim in the present work is to extend the existing exponential growth results to strong damping for a viscoelastic problem with distributed delay under the following assumptions:

• (A1) \(\varpi \in(\mathbb{R}_{+}, \mathbb{R}_{+})\) is decreasing function so that
  \begin{equation} \varpi(t)\geq0\hspace{0.3cm}, \ 1-\int_{0}^{\infty }\varpi\left( q\right) dq=l>0. \label{A1} \end{equation}

  (7)
• (A2) There exists a constant \(\xi>0\) such that
  \begin{equation} \varpi^{\prime }\left( t\right) \leq -\xi \varpi\left( t\right) \hspace{0.3cm}, \ t\geq 0. \label{A2} \end{equation}

  (8)
• (A3) \(\mu _{2}:[\tau_{1}, \tau_{2}]\rightarrow\mathbb{R}\) is bounded function so that
  \begin{equation} \Big(\frac{2\delta-1}{2}\Big)\int_{\tau_{1}}^{\tau_{2}}\vert\mu _{2}(q)\vert dq\leq\mu _{1}\hspace{0.3cm}, \ \delta>\frac{1}{2}.\label{A3} \end{equation}

  (9)

2. Main results

First, as in [14], we introduce the new variable \begin{equation*} y(x, \rho, q, t)=u_{t}(x, t-q\rho), \end{equation*} then we obtain Let us denote by Therefore, Problem (1) takes the form % with initial and boundary conditions % where $$(x, \rho, q, t)\in \Omega\times(0, 1)\times (\tau_{1}, \tau_{2})\times(0, \infty).$$ We state without proof the local existence theorem that can be established by combining arguments of [15].

Theorem 1. Assume (7), (8) and (9) holds. Let

Then for any initial data $$(u_{0},u_{1},f_{0})\in \mathcal{H}\hspace{0.2cm}/\hspace{0.2cm}\mathcal{H}= H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\times L^{2}(\Omega\times(0, 1)\times(\tau_{1}, \tau_{2})),$$ with compact support, Problem (13) has a unique solution $$u\in C([0, T]; \mathcal{H}),$$ for some \(T>0\).

In the next theorem we give the global existence result, its proof based on the potential well depth method in which the concept of so-called stable set appears, where we show that if we restrict our initial data in the stable set, then our local solution obtained is global in time. One can make use of arguments in [16].

Theorem 2. Suppose that (7), (8), (9) and (14) hold. If \(u_{0}\in W\), \(u_{1}\in H^{1}_{0}(\Omega)\) and

where \(C_{*}\) is the best Poincare's constant. Then the local solution \(u(t, x)\) is global in time.

Lemma 1. Assume (7), (8), (9) and (14) hold, let \(u(t)\) be a solution of (12), then \(\mathcal{E}(t)\) is non-increasing, that is

satisfies

Proof. By multiplying the Equation (12)\(_{1}\) by \(u_{t}\) and integrating over \(\Omega\), we get

and, we have Then, we get By (18) and (19), we get (16). Further using Young's inequality, (7), (8) and (9) in (20), we obtain (17).

Now we are ready to state and prove our main result. For this purpose, we define

Theorem 3. Suppose that (7)-(9) and (14). Assume further that \(E(0)< 0\) holds. Then the unique local solution of problem (12) grows exponentially.

Proof. From (16), we have

Hence and We set where \(\varepsilon>0\) to be specified later.
Multiplying \((12)_{1}\) by \(u\) and taking derivative of (25), we obtain Using and We obtain, from (26), Therefore, using (23) and by setting \(\delta_{1}\) so that, \(\dfrac{1}{4\delta_{1}c_{1}}=\kappa \), substituting in (29), we get For \(0< a< 1\), from (21) Substituting in (30), we get Using Poincare's inequality, we obtain At this point, we choose \(a>0\) so small that \begin{equation*} \alpha_{1}=\frac{p(1-a)}{2}-1>0, \end{equation*} and assume then we choose \(\kappa\) so large that \begin{equation*} \alpha_{2}=\Big(\frac{p(1-a)}{2}-1\Big)-\int_{0}^{t}\varpi(q)dq\Big(\frac{p(1-a)-1}{2}\Big)- \frac{c}{4c_{1}\kappa}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)>0. \end{equation*} Once \(\kappa\) and \(a\) are fixed, we pick \(\varepsilon\) so small enough so that \begin{equation*} \alpha_{4}=1-\varepsilon\kappa>0, \end{equation*} and Thus, for some \(\beta>0\), estimate (33) becomes and Next, using Young's and Poincare's inequalities, from (25) we have For some \(c>0\). Since, \(H(t) > 0\), we have from (12) then \begin{eqnarray} \frac{1}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}&<&\frac{b}{p}\Vert u\Vert_{p}^{p}< \frac{b}{p}\Vert u\Vert_{p}^{p}+(\varpi o\nabla u)\label{sys2.1.1}+\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\notag \end{eqnarray} In the other hand, using (7), to get Consequently, Inserting (41) into (38), to see that there exists a positive constant \(k_{1}\) such that From inequalities (36) and (42) we obtain the differential inequality where \(\lambda>0\), depending only on \(\beta\) and \(k_{1}\). A simple integration of (43), we obtain From (24) and (35), we have By (44) and (45), we have \begin{equation*} \Vert u\Vert_{p}^{p}\geq C e^{(\lambda t)}, \forall t>0. \end{equation*} Therefore, we conclude that the solution in the \(L_{p}\)-norm growths exponentially. This completes the proof.

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

Measles disease has been recognized as one of the world's most contagious diseases, which has the potential to be extremely severe. The transmission mode of measles disease is by person-to-person through the air by infectious droplets which has over \(90\%\) attack rates among its susceptible persons [1]. Measles disease is caused by the measles virus, an infectious agent belonging to the virus family called Paramyxoviridae that causes an infection of the respiratory system with a common symptom of  red rash on the infected person's skin [2]. Infectious diseases have been a serious concern for both human and animals populations globally. Control and prevention measures are therefore important tasks both from the human survival and economic point of views because public health issue is a major concern to the world. For effective intervention measures, a complete understanding of disease transmission like measles, and how burden they can be to a human, is therefore necessary. Even though the disease is possible and entirely preventable by taking two doses of a safe and effective vaccine, world-wide vaccination coverage with the first dose of measles vaccine still stands at \(85\%\) presently. This is still not close to the \(95\%\) proposed by World Health Organization (WHO) that will be needed to prevent outbreaks. Consequently, many people in some communities are still at risk of contracting the disease. According to WHO, the second dose of vaccination coverage, though increasing, still stands at \(67\%\) currently. But to be in a higher level of safety of not contracting the disease, the first and second doses of the vaccine must be taken.

Several factors are responsible for people not to be vaccinated against the disease. The wrong perceptions about immunization have caused some not to go for vaccination and release their wards for immunization [3]. Lack of access to health care facilities is another major factor, particularly among the rural people, and can cause them to mix out on vaccination programmes being organized by their governments.

Researchers have attributed the main causes of transmission to inappropriate education about the measles disease and low detection rate at early stages. Financial constrains is also a major factor as some individual will opt for treatment of the disease locally or traditionally than going to hospitals for professional treatments. Though there are campaigns coupled with regular and targeted measles vaccination coverage across the 36 states and Federal Capital Territory (FCT) in Nigeria, measles disease is still very prevalence with evidence of sickness and death from measles outbreaks. There is a measles outbreak in Nigeria during the first quarter of 2019. With almost six thousands cases of measles which nearly doubled the cases reported in 2018, and 15 deaths recorded as at March 2019 [4]. Evidently, measles disease is one of the infectious diseases invading Nigeria, so also some other developing countries in Africa.

Mathematical modelling of infectious diseases has proven to be powerful and useful tools. They are good in proposing and testing theories, and in comparing, planning, implementing and evaluating various detection, prevention, therapy and control intervention programs for infectious diseases. Right from the beginning of \(20^{th}\) century, researchers have been using epidemiological models to modelled infectious diseases [5]. Examples of such models are seen in [6, 7, 8]. Only a few of these models described by [5] focused on modelling childhood epidemics as measles. Roberts [9] carried out a study on predicting and preventing measles disease epidemics in New Zealand. In the work, they used a compartmental SIR model to model the dynamics of measles disease under varied immunization strategies in a population taking into consideration size and age structure. Momoh [10] developed a mathematical epidemiological model for control of measles disease. They used compartmental SEIR model for varying population size which best describes a population dynamics of developing countries like Africa. From previous literature, it has been ascertained that vaccination protects susceptibles individuals against infectious diseases such as measles by producing herd (crowd) immunity. Examples of such works can be found in [8, 11, 12, 13, 14] for instances. Sowole et al., [15], modelled measles disease using SEIR model. Taking Senegal as a case study, they looked at the effect a control measure will have on the exposed class over the entire population dynamics. Their model realized that if more people at the latent period go for treatment and therapy before they can be able to transmit the disease, the disease will be eradicated within a short period of time.

Our goal is to model measles disease transmission in Nigeria and come up with the control measures in reduction (and by extension, elimination) of the transmission of the disease in the country with control in the Susceptibles and Exposed compartments. Finding a threshold condition that will determine whether an infectious disease in a population will continue to spread or will die out with time in a given population is one of the fundamental questions of epidemiological modelling. So, we derived a fundamental epidemiological quantity, \(R_0\), called the basic reproductive number, which is the threshold parameter. This work uses a compartmental Susceptible-Exposed-Infective-Recovered epidemic model to formulate mathematical measles disease model and its corresponding mathematical analysis and numerical simulations were well presented.

The rest of the paper is structured as follow: the model is described and formulated in Section 2, model simulation is given in Section 3 and the conclusions are given in Section 4.

2. Model description and formulation

The model divides the total human population at any given time into four sub-populations to explains the transmission dynamics of the measles disease in a given human population at a given time.

The total human population \(N(t)\) is divided into sub-populations of Susceptible class\(S(t)\), Exposed compartment \(E (t)\), Infective compartment \(I(t)\) and Recovered class \(R(t)\). So we have:

The population under our consideration is homogeneous and interacting which reflect the demography of a typical developing countries [16]. This model fit well as it presents experiments of an exponentially increasing dynamics of the measles disease.

Figure 1, shows flow diagram of measles disease in a deterministic population. Where the model variables and their meaning are presented in Table 1.

Table 1. State variables used and their meanings.

Variable	Meaning
\(S(t)\)	The number of susceptible individuals at  a given  time, t
\(E(t)\)	The number of exposed individuals at a given time, t
\(I(t)\)	The number of infective individuals at a given time, t
\(R(t)\)	The number of recovered individuals at a given time, t

Susceptible compartment includes individuals who are at risk of developing measles infection if they had contact with infected individuals. Exposed class consists of individuals that had the infection but are not showing the symptoms of the measles disease and could not transmit the disease to others. Infective compartment consists of individuals that are showing the symptoms of the disease and can infect others.

The recovered human compartment comprises of individuals who have recovered from the disease and had permanent immunity.

The susceptible class \(S\) is increasing by recruitment (birth and/or immigration rate) which we denoted by \(b\). It is decreasing by susceptible class who have been immunized by the rate \(v\), is decreasing by infection if there is a contact with infected individuals at a rate of \(\beta\), and is diminishing by leaving (normal death and emigration) rate which we denoted by \(\mu\) so that:

For the second class, Exposed \(E\), the individuals here is formed by direct contact with infected individuals at a rate of \(\beta\). We define parameters and their meaning in Table 2 below.

Table 2. State parameters and their meaning.

Parameter	Meaning
\(b\)	Recruitment rate ( by birth and/or immigrants)
\(v\)	vaccination rate for the susceptible class who later got vaccinated
\(\mu\)	Leaving rate (by death  and/or emigrants)
\(\beta\)	The contact rate
\(\gamma\)	The rate at which an infective individuals recovered per unit time
\(\sigma\)	The rate of exposed individuals who have undergone testing and therapy
\(\alpha\)	The rate at which an exposed become infective

The class \(E\) is decreasing by individuals who have undergone testing and measles therapy at a rate of \(\sigma\), the individuals who progresses into infected class at a rate of \(\alpha\) and also this class is diminishing by leaving rate of \(\mu\) so that:

The third class \(I\), of infective individuals is formed by individuals who progresses from exposed class to this class at a rate \(\alpha\). This class is decreasing by individuals who are recovering from infection at a rate of \(\gamma\) and is diminishing by leaving rate of \(\mu\). We then have that: From the SEIR model, it is assumes that Susceptible-immunized individuals, Exposed-Recovered individuals and Infected-Recovered individuals become immune to the measles disease permanently, i.e. you can only be infected once with the disease. We use this assumption to generates the fourth class \(R\) of individuals who have complete protection against the disease. This class \(R\) of recovered individuals is diminishing by leaving rate of \(\mu\). So that: This SEIR model is now represented by the following system of first order differential equations and shows the transitions between the four compartments of the model:

2.1. Model assumptions

The following are the assumptions for the compartmental SEIR model for measles disease which we modeled in this work:

• (i) The recruitment rate (which is through newborns and/or migrants) are assumed to be susceptible to the disease.
• (ii) We assumed that every person in the population under consideration is susceptible to the measles disease.
• (iii) Every Individual is equally likely to be infected by the infectious individual (s) in a case of contact except for those who are immune against the measles disease.
• (iv) Infectious individuals are detected early and isolated for immediate treatment and education
• (v) The population is homogeneously mixed. By homogeneously mixed we mean a population that interacts among themselves in such a uniformly manner.
• (vi) The population is a varying population where recruitment rate and leaving rate are differ within a given time steps.
• (vii) There is no treatment failure, a patient will either recover or die.
• (viii) Recovered individuals are permanently immune against the disease.

2.2. Properties of the model

The basic properties of our model are that of the properties of ``feasible solution'' and ``positivity of the solution''. The feasible solution of the model equations shows the region in which the solution of the equations are biologically significant and the positivity of the solutions tells the non-negativity of the solutions of the model equations.

2.3. Feasible solution

Here, the deterministic SEIR model is used to model infectious disease in a human population. It will be reasonable to assumed that the parameters used and variables in all classes are non negative, that is \(t \geq 0\). The show that all variables of the model are non-negative for all given non-negative initial conditions are provided below. The feasible solution region which is positively invariant set of the model is given by: The following lemma established this claim.

Lemma 1. The set \(\Omega\) is positively invariant and attracts all solution in \(\mathbb{R}_{+}^{4}\).

Proof. Since \(N(t) = S(t) + E(t) + I(t) + R(t)\). Adding Equations (2) to (5) together gives us the rate of change of the total population: \begin{align*} \frac{dN}{dt} &= \frac{dS}{dt} + \frac{dE}{dt} + \frac{dI}{dt} + \frac{dR}{dt} \\ &= b - \mu(S+E+I+R)\\ &= b - \mu N \hspace{1cm} (since \hspace{0.1cm} N \hspace{0.1cm} = S+E+I+R).\end{align*} This is a first order linear differential equation and a first order linear differential equation of the form

can be solved by introducing integrating factor. Here, \(\mu\) and \(b\) are both constants, so the integrating factor \(I.F\) is given as: \(I.F= \exp \bigg(\int \mu dt \bigg).\) Now multiplying both sides of Equation (8) with \(\exp(\int \mu dt)\) given, The left hand side of Equation (9) is \( \frac{d}{dt} \bigg[ N(t). \exp \bigg(\int\mu dt \bigg)\bigg]\), therefore \[ \frac{d}{dt} \bigg[ N(t). \exp \bigg (\int\mu dt \bigg) \bigg] = b . \exp \bigg( \int \mu dt \bigg) \] Integrating both sides, we have \[ N(t). \exp (\mu dt) = \frac{b}{\mu} \exp (\mu dt) + K,\] where \(K\) is a constant. So that \[N(t) = \frac{b}{\mu} + K\exp(-ut) . \] When \(t=0\), we have that \[ N(0) = \frac{b}{\mu} + K. \] Therefore, \[ K = N(0) - \frac{b}{\mu}\,. \] Substituting the value of \(K\), the solution (with simplification) of this linear differential equation will becomes: \[ N(t) = N(0)\exp^{-\mu t}+ \frac{b}{\mu} (1- \exp(-\mu t)). \] Taking the limit as \(t \rightarrow \infty\), we have \[ N(t) \leq \frac{b}{\mu}.\] Therefore, we have established that \(\Omega\) is positively invariant and attracts all solution in \(\mathbb{R}_{+}^{4}\).

2.3.1 Positivity of results

In this section we prove that all variables in the SEIR model Equations (2) to (5) are non-negative.

Lemma 2. Let the initial data set be \((S, E, I, R) (0) \geq 0 \in \Omega\), then the solution set \((S(t), E(t), I(t), R(t) ) \) of the equations (2) to (5) is positive \(\forall\]t > 0\).

Proof. From Equation (2) if we assumed that: \[\frac{dS}{dt} = b - \beta SI - (v + \mu) S \geq -(\beta SI + v + \mu S ).\] That is \[\frac{dS}{dt} \geq -(\beta I + v + \mu )S \hspace{0.5cm} or \hspace{0.5cm} \frac{dS}{S} \geq -(\beta I + v + \mu )dt \hspace{1cm} (by \hspace{0.1cm} separation \hspace{0.1cm} of \hspace{0.1cm} variables, \hspace{0.1cm} since \hspace{0.1cm} S \ne 0) .\] Integrating both sides of the inequalities, we have \[In \big( S(t) \big) \geq -(\beta I + v + \mu )t + K.\] So that \[S(t) \geq K \exp \bigg( -(\beta I + v + \mu )t\bigg). \] At \(t = 0\); this becomes: \[S(t) \geq S(0) \exp \bigg( -(\beta I + v + \mu )0\bigg) \geq 0 \hspace{1cm} (since \hspace{0.1cm} (\beta I + v + \mu ) > 0).\] That is \[ S(t) > 0.\] Similarly from Equation (3), we have \[\frac{dE}{dt} = \beta SI - (\mu + \alpha + \sigma)E \geq - (\mu + \alpha + \sigma)E \] i.e., \[\frac{dE}{dt} \geq - (\mu + \alpha + \sigma)E \] or \[\frac{dE}{E} \geq - (\mu + \alpha + \sigma)dt \hspace{1cm} (by \hspace{0.1cm} separation \hspace{0.1cm} of \hspace{0.1cm} variables, \hspace{0.1cm} since \hspace{0.1cm} E \ne 0) .\] Integrating both sides of the inequalities we have; \[ E(t) \geq K \exp \bigg(- (\mu + \alpha + \sigma)t \bigg).\] At \(t = 0\), we have that \[E(t) \geq E(0) \exp \bigg(- (\mu + \alpha + \sigma\bigg)0 \geq 0, \hspace{1cm} (since \hspace{0.1cm} (\mu + \alpha + \sigma) > 0.\] That is \( E(t) > 0.\) Also from Equation (4), we have \[\frac{dI}{dt} = \alpha E - (\mu + \gamma)I \geq - (\mu + \gamma)I.\] That is \[\frac{dI}{dt} \geq - (\mu + \gamma)I.\] Then we have that: \[\frac{dI}{I} \geq - (\mu + \gamma)dt.\] On integrating we obtains \[ In \big( I(t) \big) \geq - (\mu + \gamma)t + K,\] that is \[I(t) \geq K \exp \bigg(- (\mu + \gamma)t \bigg).\] At \(t=0\), we have \(I(t) \geq I(t) \exp \bigg(- (\mu + \gamma)0 \bigg).\) So that \( I(t) > 0.\) Finally from Equation (5), we have \[\frac{dR}{dt} = v S + \gamma I + \sigma E - \mu R > \sigma E - \mu R.\] That is, \[\frac{dR}{dt} > \sigma E - \mu R.\] Which has an integrating factor \( I.F = \exp (-\mu t).\) Then we have: \[\frac{dR}{dt}.\exp (-\mu t) > \sigma E. \exp (-\mu t) - \mu R . \exp (-\mu t).\] Integrating at constant \(K\), we have \[ R(t) > \frac{\sigma E}{\mu} + K \exp (-\mu t).\] When \(t=0\) we obtained that \(R(0) >\frac{\sigma E}{\mu} + K.\) The solution then becomes \[R(t) > R(0)\exp (-\mu t) +\frac{\sigma E}{\mu} (1- \exp (-\mu t)).\] That is \(R(t) > 0.\) Hence, we have proved that all variables are positive \(\forall t > 0\).

2.4. Existence and uniqueness of solution for the SEIR model

The general first-order ODE is in the form: One will be interested in asking the following questions:

• (1) Under what conditions, the solution to Equation (10) exists?
• (2) Under what conditions, there is a unique solution to Equation (10)?

To answers these, let \begin{equation*} \begin{cases} f_1 = b - \beta SI - \mu S,\\ f_2 = \beta SI - (\mu + \alpha + \sigma)E,\\ f_3= \alpha E - (\mu + \gamma)I,\\ f_4 = \gamma I + \sigma E - \mu R. \end{cases} \end{equation*} We use the following theorem to established the existence and uniqueness of solution for our SEIR model.

Theorem 1.[Uniqueness of Solution] Suppose \(D\) denotes the domain, and

and suppose that \(f (t, x)\) satisfies the Lipschitz condition: then whenever the pairs \((t, x_1)\) and \((t, x_2 )\) belong to the domain \(D\), where \(k\) is used to represent a positive constant, there exist a constant \(\delta > 0\) such that there exists a unique (exactly one) continuous vector solution \(x(t)\) of the system (10) in the interval \(|t - t_0 | \leq \delta\).

It is important to note that condition (12) is satisfied by requirement that \( \bigg \{ \frac {\partial{f_i}}{\partial{x_j}},\hspace{0.2cm} _{ i, j = 1, 2, ..., n},\) is continuous and bounded in the domain \(D\).

Lemma 3. If \(f (t, x)\) has continuous partial derivative \(\frac{\partial{f_i}}{\partial{x_j}}\) on a bounded closed convex domain \(\mathbb{R}\) (i.e, convex set of real numbers), where \(\mathbb{R}\) is used to denotes real numbers, then it satisfies a Lipschitz condition in \(\mathbb{R}\).

Our interest is in the domain so, we look for a bounded solution of the form \(0 < \mathcal{R} < \infty.\) We now prove the following existence theorem.

Theorem [Existence of solution] Let \(D\) denote the domain defined in (11) such that (12) and (13) hold. Then, there exist a solution of model system of Equations (2)-(5) which is bounded in the domain \(D\).

Proof. Let

We shows that \(\frac{\partial{f_i}}{\partial{x_j}},\hspace{0.2cm} i,j=1,2,3,4\) are continuous and bounded. We explored the following partial derivatives for all the model equations. From Equation (14); \[ \bigg|\frac{\partial f_1}{\partial S}\bigg| = \bigg|-\beta I - v - \mu \bigg| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_1}{\partial E}\bigg| = |0| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_1}{\partial I}\bigg| = \bigg|-\beta S \bigg| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_1}{\partial R}\bigg| = |0| < \infty.\] Similarly, from Equation (15): \[ \bigg|\frac{\partial f_2}{\partial S}\bigg| = \bigg|\beta I \bigg| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_2}{\partial E}\bigg| = \bigg|-(\mu + \alpha + \sigma) \bigg| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_2}{\partial I}\bigg| = \bigg|\beta S \bigg| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_2}{\partial R}\bigg| = |0| < \infty.\] Also from Equation (16); \[ \bigg|\frac{\partial f_3}{\partial S}\bigg| = |0| < \infty, \hspace{0.3cm} \bigg|\frac{\partial f_3}{\partial E}\bigg| = |\alpha| < \infty,\hspace{0.3cm} \bigg|\frac{\partial f_3}{\partial I}\bigg| = |- (\mu + \gamma)| < \infty,\hspace{0.3cm} \bigg|\frac{\partial f_3}{\partial R}\bigg| = |0| < \infty.\] Finally, from Equation (17): \[ \bigg|\frac{\partial f_4}{\partial S}\bigg| = |v| < \infty,\hspace{0.3cm} \bigg|\frac{\partial f_4}{\partial E}\bigg| = |\alpha| < \infty,\hspace{0.3cm} \bigg|\frac{\partial f_4}{\partial I}\bigg| = |\gamma| < \infty,\hspace{0.3cm} \bigg|\frac{\partial f_4}{\partial R}\bigg| = |-\mu| < \infty.\] We have clearly established that all these partial derivatives are continuous and bounded, hence, by Theorem \ref{thm1}, we can say that there exist a unique solution of (2) to (5) in the region \(D\).

2.5. Existence of steady states of the system

Under this section we will find the equilibrium points and established asymptotic stability of the SEIR model. In order to obtained the equilibrium points of the system, we equate the system of equations of the model to zeros; i.e.;

2.6. Stability of the SEIR model (local asymptotic stability)

To find the local stability of the model, we first find the equilibrium points (disease free equilibrium) of the system (2) to (5). By equating the system to zeros we have From Equation (19), we have \( b = \beta SI + (v + \mu) S \equiv (\beta I + v + \mu) S\) which implies that \( S = \frac{b}{\beta I + v + \mu}\) but at the initial state, \(v =0\) and \(\beta = 0,\) which implies \( S = \frac{b}{\mu}. \) Now from Equation (20), we have \( \beta SI = (\mu + \alpha + \sigma)E,\) which implies that \( E = \frac{\beta SI}{(\mu + \alpha + \sigma)},\) hence \( E = 0\) (since \(\beta = 0\)). Similarly from Equation (21), we have \( \alpha E = (\mu + \gamma)I,\) which implies that \( I = \frac{\alpha E}{(\mu + \gamma)},\) hence \( I = 0\) (since \(E = 0 \)). Finally from Equation (22), we have \( v S + \gamma I + \sigma E = \mu R,\) which implies that, \( R = \frac{ v S + \gamma I + \sigma E}{\mu}, \) hence \(R = 0\) (since \(v, I\) and \(E = 0\)). Thus we have disease free equilibrium for this SEIR model which is given by \(\mathcal{P}_0 = \bigg(\frac{b}{\mu}, 0, 0, 0 \bigg).\)

Note 1. We assumed \(b \ne \mu\) for this model.

2.7. The basic reproduction number \(R_0\)

The basic reproduction number \(R_0\) of an infection can be thought of as the number of cases one measles infection case can generates on average over the course of its infectious period, in an otherwise uninfected (susceptible) population\footnote{Source: Wikipedia}.

We determined the basic reproduction number \(R_0\) by employing the method of next generation matrix. We considered only the two infected classes in our model; the Exposed \(E\) and the Infectious \(I\) compartments. That is \(m = 2\). Define \(G = F V^{-1}\), therefore \(R_0 = \rho (F V^{-1})\) where \(\rho (F V^{-1})\) is called the spectral radius of \(FV^{-1}\). ({The spectral radius is the maximum of the absolute value of the eigenvalues}.)

Let \(H'(x) = \mathcal{F}(x) - \mathcal{V}(x),\) which gives Taking the Jacobian of \(\mathcal{F}(x)\) and \(\mathcal{V}(x)\) respectively at the disease free equilibrium \(\mathcal{P}_{0}\), we obtained Set \(k_1 = (\mu + \alpha + \sigma)\) and \(k_2 = (\mu + \gamma)\) such that \[ F V^{-1} = \begin{bmatrix} \frac{b \alpha \beta}{\mu k_{1} k_{2}}& \frac{\beta b}{\mu k_2} \\ 0 & 0 \\ \end{bmatrix}\] and \[ |F V^{-1} - \lambda I_{2}| = \begin{vmatrix} \frac{b \alpha \beta}{\mu k_{1} k_{2}}-\lambda & \frac{b \beta }{\mu k_2} \\ 0 & - \lambda\\ \end{vmatrix} = 0\,.\] Solving we get \[ \lambda^2 - \frac{b \beta \alpha}{\mu (\mu + \alpha+\sigma)(\mu + \gamma)}\lambda = 0\,.\] So that We can easily obtain the value of \(R_0\) using the values provided in Table 3. If \(R_0 < 1\), it implies that the disease will be eradicated from the population with time. Conversely, if \(Ro > 1\), the measles disease is endemic.
Let's define \[ R_0 = \frac{b \beta \alpha}{\mu (\mu + \alpha +\sigma)(\mu + \gamma)} \hspace{0.3cm} \text{where:} \hspace{0.2cm} \mu (\mu + \alpha +\sigma)(\mu + \gamma) \ne 0\,.\]

2.8. Interpretation of \(R_0\)

\(R_0\) is used to explained the fact that the transmission rate, that is, the rate at which exposed individuals will become infected and the contact rate, i.e.; the average number of effective contacts with other (susceptible) individuals per infective individuals per unit time which is relative to the rate at which an infectious individuals will recovered per unit time plays a significant role in determining whether or not the measles epidemic will occur in a given human population. Thus, the disease free equilibrium \((\frac{b}{\mu}, 0, 0, 0)\) is locally asymptotically stable given that \(Ro < 1\), that is, \(b \beta \alpha < \mu (\mu + \alpha +\sigma)(\mu + \gamma). \) Alternatively, if \(Ro > 1\), then the disease free equilibrium is unstable, i.e., the system is said to be uniformly persistent, in other words the measles disease is endemic. Hence, \(R_0\) is a threshold parameter for the model that will determines the number of equilibria.

2.9. Endemic Equilibrium

The next thing in our analysis is to shows that an endemic equilibrium: and we also show that if \(Ro > 1\), the measles disease is endemic. From our equilibrium Equations (19) to (22), considering Equations (20) and (21), we have that: \[\beta SI = (\mu + \alpha + \sigma)E,\;\;\;\;(\mu + \gamma)I = \alpha E.\] Which when we divide the first by the second yields, \[S = S^* = \frac{(\mu + \alpha + \sigma) (\mu + \gamma)}{\alpha}.\] That is, \[S^* = \frac{(\mu + \alpha + \sigma) (\mu + \gamma)}{\alpha},\] which is clearly greater than zero. Similarly, adding Equations (19) and (20), we have \[ (v + \mu) S = b - E (\mu + \alpha + \sigma)\] which implies \begin{equation*} S = \frac{ - (\mu + \alpha + \sigma)E + b}{ (v + \mu)}. \end{equation*} But for \(v = 0 \), we have Now, From equation (21), we have From Equation (20), we have Now substituting the values of \(S\) and \(I\) from Equations (27) and (28) into Equation (29), we get \[ E \bigg[ \frac{- \beta (\mu + \alpha + \sigma) \alpha E}{ (v + \mu) (\mu + \gamma)} + \frac{\beta \alpha b}{ \mu (\mu + \gamma)} - (\mu + \alpha + \sigma) \bigg] = 0.\] Hence, either \( E = 0\) or\(\bigg( \frac{- \beta \alpha (\mu + \alpha + \sigma) E}{ \mu (\mu + \gamma)} + \frac{\beta \alpha b}{ \mu (\mu + \gamma)} - (\mu + \alpha + \sigma) \bigg) = 0.\) Now, On dividing both sides of (30) by \(\beta \alpha (\mu + \alpha\ + \sigma)\) and multiplying both sides by \(( \mu (\mu + \gamma)\), we obtain \[ E = \frac{b}{\mu + \alpha + \sigma} - \frac{ \mu (\mu + \gamma)}{\beta \alpha}.\] Therefore, \[ E^* = \frac{b}{\mu + \alpha + \sigma} \bigg[ 1 - \frac{\mu (\mu + \alpha + \sigma)(\mu + \gamma)}{\beta \alpha b} \bigg].\] Recall that, \[R_0 = \frac{\beta \alpha b}{\mu (\mu + \alpha + \sigma)(\mu + \gamma)}\,.\] Now, we have Considering \(I\) from Equation (29), that is \[ I = \frac{\alpha E}{(\mu + \gamma)}\,.\] Substituting \(E^*\) for \(E\), we obtain \[ I = \frac{\alpha }{(\mu + \gamma)} \bigg(\frac{b}{(\mu + \alpha + \sigma)} \bigg[ 1 - \frac{\mu (\mu + \alpha + \sigma)(\mu + \gamma)}{\beta \alpha b} \bigg] \bigg).\] So Finally from Equation (22), \[ v S + \gamma I + \alpha E = \mu R.\] So \[R = \frac{ v S + \gamma I + \alpha E}{\mu} .\] On substituting the values of \(S^*\), \(I^*\) and \(E^*\) for \(S\), \(I\) and \(E\) in above, we obtain \[ R = \frac{v (\mu + \alpha + \sigma) (\mu + \gamma)}{\mu \alpha} + \frac{\gamma \alpha b}{\mu (\mu + \gamma)(\mu + \alpha\ + \sigma)} \bigg[ 1 - \frac{1}{R_0} \bigg] + \frac{\sigma b}{\mu (\mu + \alpha + \sigma)} \bigg[ 1 - \frac{1}{R_0} \bigg].\] At initial state, \(v = 0\), so \begin{align*}R &= \frac{\gamma \alpha b}{\mu (\mu + \gamma)(\mu + \alpha\ + \sigma)} \bigg[ 1 - \frac{1}{R_0} \bigg] + \frac{\sigma b}{\mu (\mu + \alpha + \sigma)} \bigg[ 1 - \frac{1}{R_0} \bigg]\\ &= \frac{b}{\mu (\mu + \alpha + \sigma)} \bigg ( \frac{\alpha \gamma}{\mu + \gamma} + \sigma \bigg) \bigg[ 1 - \frac{1}{R_0} \bigg].\end{align*} Multiplying the right hand side by \(\frac{\mu (\mu + \alpha + \sigma)}{b}\), we have We have shown that \(S^*\), \(E^*\), \(I^*\) and \(R^*\) are all positives meaning that \(P^* = (S^* , E^* , I^* ,R^* ) > 0. P^*\) represent an endemic steady state with constant number of people in the population being infected with the measles disease and if \(Ro > 1\), the measles disease is endemic. This will be biologically reasonable when \(S^* < N\), that is when \( R_0 = \frac{\beta \alpha b}{\mu (\mu + \alpha + \sigma)(\mu + \gamma)} > 1.\) In another words, the necessary and sufficient condition for a unique \(P^*\) to exist in the feasible region \(\Omega\) is that \( 0 < S^* \leq \frac{b}{\mu}.\) or equivalently \(\frac{b}{\mu} \geq 1.\)

2.10. Local asymptotic stability

We established the local stability of the disease-free equilibrium using Jacobian matrix of Equations (2) to (5) and evaluate at disease free equilibrium \(\mathcal{P}_0\). We achieve this by evaluating the Jacobian matrix of Equations (2) to (5) at \(\mathcal{P}_0 = (\frac{b}{\mu}, 0, 0, 0)\). The local stability of the model is determined from the eigenvalues of the Jacobian matrix of the model equations at \(\mathcal{P}_0\). Thus, the Jacobian matrix of Equations (2) to (5) is given by But at initial state \(v = 0\), and at the disease free equilibrium, substituting \(S = \frac{b}{\mu}\), and \(I = 0\) into the Equation (34), we get The disease free equilibrium, \(\mathcal{P}_0\), is locally asymptotically stable if all the eigenvalues of \(J(\mathcal{P}_0)\) are \( < 0\). We will establish this by finding the eigenvalues. To find the eigenvalues, take \(|J(\mathcal{P}_0) - \lambda I_4 | = 0\), then \begin{center} \end{center} implies \[ (\mu + \lambda)^2 \bigg [(\mu + \alpha + \sigma + \lambda)(\mu + \gamma + \lambda + \frac{b \beta \alpha}{\mu}) \bigg] = 0.\] Then Simplify (37) gives \(\lambda_1 = -\mu,\hspace{.2cm} \lambda_2 = -\mu,\hspace{.2cm} \lambda_3 = -(\mu + \sigma + \alpha) , \lambda_4 = -\bigg( \frac{b \beta \alpha}{\mu} + \mu + \gamma \bigg).\) Since the eigenvalues for \(J(\mathcal{P}_0)\) are negative so the disease free equilibrium is locally asymptotically stable.

2.11. Application of \(R_0\) to measles in Nigeria.

A deterministic compartmental mathematical model for measles has been formulated with the aim of the study of the effects of heterogeneous mixing, transmission and control (or elimination) of the disease in Nigeria. Moreover, the basic reproduction number, \(R_0\), has been derived, which is the threshold parameter. Recall that Table 3 gives the model parameters we used for our model and its values and sources.

Table 3. Model parameter values used and Sources.

Parameter	Value	Source
\(b\)	0.03691	[17] and [18]
\(v\)	0.0, 0.25, 0.50 \& 0.75	Assumed
\(\mu\)	0.01241	[19] and [18]
\(\beta\)	0.09091 per day	Immunization Action Coalition
\(\gamma\)	0.125 per day	[20]
\(\sigma\)	0.0, 0.25, 0.50 \& 0.75	Assumed
\(\alpha\)	0.14286 per day	[20]

We will substitute the values of \(\alpha\), \(\beta\), \(b\), \(\mu\), \(\alpha\), \(\sigma\) and \(\gamma\) into \(R_0\) to investigate the behaviour of \(R_0\) with the given parameters in Table 3 which are pertinent to measles in Nigeria. It is easy to check that when \(\sigma = 0\) we have \( R_0 = 1.8105;\) when \(\sigma = 0.25\) we have \( R_0 = 0.69364\); when \(\sigma = 0.50\), We have \( R_0 = 0.42900\) and finally, when \(\sigma = 0.75\) we have \( R_0 = 0.31053 .\)
In summary, when \(\sigma = 0\) we have \(R_0 > 1\) and when \(\sigma \le 1\) we have \(R_0 < 1\). This illustrate the fact that \(\sigma\), which is the rate of exposed individuals who have undergone testing and therapy, will play a significant role in controlling (and eliminating) of the disease. Thus, the disease free equilibrium \((\frac{b}{\mu}, 0, 0, 0)\) is locally asymptotically stable when \(Ro < 1\) for \(\sigma \le 1\) , that is, \(b \beta \alpha < \mu (\mu + \alpha +\sigma)(\mu + \gamma). \) Alternatively, when \(\sigma = 0\), \(Ro > 1\), then the disease-free equilibrium is unstable, i.e., the measles disease will be endemic in the population. Hence, we have established the fact that \(R_0\) is a threshold parameter for the model that will determine the number of equilibria.

3. Numerical results

In this section, we consider the explicit fourth-order Runge-Kutta (RK4) scheme for solving numerically the non-linear first order ordinary differential equations (ODEs) of our SEIR model of Equations (6) with given initial conditions. Here, numerical simulations are used to show the impact of vaccination and therapy using the RK4 scheme on the state equations. Details of the RK4 are discussed in [15]. State equations are solved over a simulated period of time using the RK4 scheme. The parameters used in the simulation are in the Table 3. The simulation results are depicted in Figures 2 to 5.

3. 1. Interpretation of Simulation Results

In Figure 2 we present the dynamics of measles disease when \(\sigma = 0.0\) and \(v = 0 \) and when \(\sigma = 0.25\) and \(v = 0\). We can see in the Figure 2 that, if none of the exposed individuals at the latent period are diagnosed and treated, and there is no control measure been introduced into the susceptible class. It will take a much longer time for exposed individuals to decrease. Likewise, it takes a significantly longer period before we notice any significant improvement for individuals to recovered from the measles disease. Similarly, infective individuals will significantly increase before noticing a drop in the number of infective individuals. On the order hand, when \(25\%\) of the exposed individuals at the latent period are diagnosed and treated, there is a clear improvement in the result.

In Figure 3, we do the simulation of the model when \(\sigma = 0.25\) and \(v = 25\) and when \(\sigma = 0.50\) and \(v = 0.25\). It can be observed from Figure 3 that if \(25\%\) of susceptible individuals are vaccinated in addition to increases of \(25\%\) to \(50\%\) of the exposed individuals at the latent period who are diagnosed and treated. It will take lesser time for exposed individuals to decrease significantly. Also, it takes a significant lesser period before noticing any significant improvement for individuals to recovered from the disease. Similarly, infective individuals will significantly decline over time.

Figure 4, above shows the dynamics of measles disease when \(\sigma = 0.50\) and \(v = 50\) and when \(\sigma = 0.75\) and \(v = 0.50\). From the simulation result of the Figure 4, if \(50\%\) of susceptible individuals are vaccinated in addition to increases of \(50\%\) to \(75\%\) of the exposed individuals at the latent period, who are diagnosed and treated,. There is a sudden decline of exposed individuals with time. This shows a significantly improved result in comparison with what we have in the two figures above. Also, it takes a significant much lesser period for individuals to recover from the disease. Similarly, infective individuals will significantly take a lesser period to go down over time.

Figure 5 shows the simulation of the model when \(\sigma = 0.50\) and \(v = 75\) and when \(\sigma = 0.75\) and \(v = 0.75\). Looking at the Figure 5 closely, we discovered that if \(75\%\) of exposed individuals at the latent period are diagnosed and treated. In addition to increases of \(50\%\) to \(75\%\) of susceptible individuals who are vaccinated,  we have a better result in comparison to the previous three results stated above.

Conclusion

The SEIR model showed significant success in attempting to predict the causes of measles disease transmission within a given population. The model strongly indicated that the spread of disease largely depends on the contact rates of susceptible individuals with infected individuals within a population. With the assumed values for our state variables, we modelled a measles disease in Nigeria using deterministic SEIR model to investigate the impact the control measures can have on susceptibles, as well as exposed individuals at latent period, over the entire population dynamics in controlling and eliminating the disease. Consequently, we established the existence of a solution, uniqueness of the solution, and stability of the solution. The application of \(R_0\) to measles data of Nigeria was also given. Finally, numerical simulations of the model with the RK4 scheme was well presented.

Acknowledgments

We appreciate the support given to us by African Institute for Mathematical Sciences (AIMS), Senegal. All the authors are well appreciated for their contributions.

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

In this paper we consider a coupled system of wave with past history given by

where \(\Omega\) is an open bounded set of \(\mathbb{R}^n\) with smooth boundary \(\Gamma\).

The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an elastic force that attracts one membrane to the other with coefficient \(\alpha >0\). Note that the term \(\int^{\infty}_0g(s)\Delta u(t-s)\;ds\), acts on the first membrane as a stabilizer.

Many interesting physical phenomena such as viscoelasticity, hereditary polarization in dielectrics, population dynamics or heat flow in real conductors, to name some, are modeled by differential equations which are influenced by the past values of one or more variables in play so-called equations with memory. The main problem in the analysis of equations of this kind lies in their nonlocal character, due to the presence of the memory term given by the time convolution of the unknown function against a suitable memory kernel. The memory term may can produce loss of exponential stability for the system, [1]. The history of nonlocal problems with integral conditions for partial differential equations is recent and goes back to [2]. In [3], a review of the progress in the nonlocal models with integral type was given with many discussions related to physical justifications, advantages, and numerical applications.

Coupled wave system has been considered in various contexts. In [4] both wave equations are damped on the boundary and the coupling is effected by compact operator and exponential stability is obtained when the boundary damping is linear. Boundary damping is also considered in [5, 6]. On exact boundary controllability for linearly coupled wave equations, we refer [7]. Uniform exponential stability was given in [8] for wave equations coupled in parallel with coupling distributed springs and viscous dampers due to different boundary conditions and wave propagation speeds.

For weak damping acting only one equation, the optimal polynomial decay to coupled wave equations was studied in [9]. In [10], it was proved that the energy of associated coupled system weakly dissipative decays polynomially with explicit polynomial decay rates for sufficiently smooth solutions. In [11], under new compatibility assumptions, the authors proved polynomial decay for the energy of solutions and optimized previous results by interpolation techniques introduced in [10].

On the asymptotic behavior of the coupled system (1)-(5) we refer the work [12] where the authors proved by method introduced in [11] that the solution has a polynomial rate of decay. The central question of this work is to analyze what is the best decay rate of the system (1)-(5). In this direction, we prove that the associated semigroup decays with rate \(t^{-\frac{1}{2}}\). Moreover we show that the rate is optimal. For what we know in the literature the optimal rate of polynomial decay for coupled wave systems with memory was not previously considered.

The mathematical structure of the paper is organized as follows: In Section 2 we discuss the existence, regularity and uniqueness of strong solutions of the system (1)-(5) by semigroup technique, see [13]. In Section 3 we study the lack of exponential decay using Prüss's results [14]. Finally in section 4 we show that the system is polynomially stable giving an optimal decay rate. That is, this rate cannot be improved. For this we use the recent result due to Borichev and Tomilov [15].

2. Semigroup Setup

Following the approach of Dafermos [16] and Fabrizio and Morro [17], we consider \(\eta =\eta ^{t}(s)\), the relative history of \(u\), defined as Hence, putting \[ \beta_0=1-\int_0^{\infty}g(s)\;ds>0, \] the system (1)-(5) turns into the system

where the third equation is obtained differentiating (6) with respect to \(s\) and the condition (13) means that the history is considered as an initial value.

We study the existence and uniqueness of solutions for the system (7)-(13) using the semigroup techniques. As in [18], we use the following hypotheses on \(g\)

In view of (14), let \(L^2_g(\mathbb{R}^+;H^1_0(\Omega))\) be the Hilbert space of \(H^1_0(\Omega)\)-value functions on \(\mathbb{R}^+\), endowed with the inner product \[ (f,h)_{L^2_g(\mathbb{R}^+,H^1_0(\Omega))}=\int^{\infty}_0g (s)\int_{\Omega}\nabla f(x,s) \cdot \nabla\overline{h(x,s)} \; dx \; ds. \] To give an accurate formulation of the evolution problem we introduce the product Hilbert spaces \[ \mathcal{H}=H^1_0(\Omega)\times L^2(\Omega)\times H^1_0(\Omega)\times L^2(\Omega)\times L^2_g(\mathbb{R}^+;H^1_0(\Omega)) \] endowed with the following inner product where \(U=(u_1,u_2,u_3,u_4,u_5)^T\), \(V=(v_1,v_2,v_3,v_4,v_5)^T\in \mathcal{H}\).
Let \(U=(u,u_t,v,v_t,\eta)^T\) be and we define the operator \(\mathcal{A}:D(\mathcal{A})\subset \mathcal{H}\rightarrow \mathcal{H}\) given by \begin{eqnarray*} \begin{array}{c} \mathcal{A}=\left[ \begin{array}{ccccc} \ 0 & I & 0 & 0 & 0 \\ \beta_0 \Delta & 0 & -\alpha I & 0 & \mathcal{T} \\ \ 0 & 0 & 0 & I & 0 \\ -\alpha I & 0 & \Delta & 0 & 0 \\ \ 0 & I & 0 & 0 & -(\cdot)_s \\ \end{array} \right] \end{array} \end{eqnarray*} with domain \begin{eqnarray*} D(\mathcal{A})&=&\{ (u,\varphi,v,\psi,\eta)^T\in \mathcal{H}; \quad \beta_0 u-\int^{\infty}_0g(s)\eta(s) \; ds \in H^1_0(\Omega)\cap H^2(\Omega),\\ &&\varphi \in H^1_0(\Omega), v\in H^1_0(\Omega)\cap H^2(\Omega), \psi \in H^1_0(\Omega), \eta \in D(\mathcal{T}) \} \end{eqnarray*} where \[ \mathcal{T}\eta = \int^{\infty}_0g(s)\Delta \eta (s) \; ds, \quad \forall \eta \in D(\mathcal{T}) \] with \[ D(\mathcal{T})=\{ \eta \in L^2_g(\mathbb{R}^+;H^1_0(\Omega));\eta_s\in L^2_g(\mathbb{R}^+;H^1_0(\Omega)), \eta(0)=0\}, \] where \(\eta_s\) is the distributional derivative of \(\eta\) with respect to the internal variable \(s\). Therefore, the system (7)-(13) is equivalent to with \(U=(u,u_t,v,v_t,\eta)^T\), \(U_0=(u_0,u_1,v_0,v_1,\eta_0)^T\). With the above notations, we have the following result.

Theorem 1. The operator \(\mathcal{A}\) generate a C\(_0\)-semigroup \(S(t)\) of contraction on \(\mathcal{H}\). Thus, for any initial data \(U_0\in \mathcal{H}\), the problem (7)-(13) has a unique weak solution \(U(t)\in C^0([0,\infty[, \mathcal{H})\). Moreover, if \(U_0\in D(\mathcal{A})\), then \(U(t)\) is strong solution of (7)-(13), that is, \(U(t)\in C^1([0,\infty[,\mathcal{H})\cap C^0([0,\infty[,D(\mathcal{A}))\).

Proof. It is easy to see that \(D(\mathcal{A})\) is dense in \(\mathcal{H}\). Now, for \(U=(u, u_t,v, v_t,\eta)^T\in D(\mathcal{A})\) and using the inner product (15), we get \begin{align*} &\langle {\mathcal{A}}U,U\rangle =\beta_0\int_{\Omega}\nabla u_t\cdot \nabla \overline{u}\;dx+\int_{\Omega}(\beta_0\Delta u-\alpha v+\int^{\infty}_0g(s) \Delta \eta(s)\;ds)\overline{u_t}\;dx\\ &\;\;\;+\int_{\Omega}\nabla v_t\cdot \nabla \overline{v}\;dx+\int_{\Omega}(\Delta v-\alpha u)\overline{v_t}\;dx+\alpha \int_{\Omega}(u_t\overline{v}+v\overline{u_t})\;dx+\int^{\infty}_0g(s)\int_{\Omega}\nabla(u_t-\eta_s(s))\cdot\nabla \overline{\eta} (s)\;dx\;ds \end{align*} from where it follows that \begin{eqnarray*} \langle {\mathcal{A}}U,U\rangle=-\int^{\infty}_0g(s)\int_{\Omega}\nabla \eta_s(s)\cdot \nabla \overline{\eta}(s)\;dx\;ds. \end{eqnarray*} Integrating by parts and using (14), we have \begin{eqnarray*} {\mathcal{R}}e\langle {\mathcal{A}}U,U\rangle=\frac{1}{2}\int^{\infty}_0g'(s)\int_{\Omega}|\nabla \eta(s)|^2\;dx\;ds\leq -\frac{q_1}{2}\int^{\infty}_0g(s)\int_{\Omega}|\nabla \eta(s)|^2\;dx\;ds\leq 0. \end{eqnarray*} Therefore, \(\mathcal{A}\) is a dissipative operator.
Next, we show that \((I-\mathcal{A})\) is maximal. For this, let us consider the equation \[ (I-\mathcal{A})U=F \] where \(U=(u,\varphi,v,\psi,\eta)^T\) and \(F=(f^1,f^2,f^3,f^4,f^5)^T\in \mathcal{H}\). Then, in terms of its components, we can write

Integrating (22), we have Substituting \(\varphi\) and \(\eta\) from (18) and (23) into (19), we get where \[ \beta_g=\beta_0+\int^{\infty}_0g(s)(1-e^{-s})\;ds. \] Note that \(\beta_g\) is a positive constant in virtue of (14). Moreover, it can be shown that the right-had side of (24) is in \(H^{-1}(\Omega)\).
On the other hand, the substitution of \(\psi\) given in (20) into (21) gives us First we prove that \(u, v \in H^1_0(\Omega)\). To do this, let us consider the bilinear form where \(\Phi_1=(u_1,v_1)\) and \(\Phi_2=(u_2,v_2)\). Then, Lax-Milgram theorem (see [19]) provides existence and uniqueness of the solutions \[ u,v\in H^1_0(\Omega). \] From (18) and (20), we have \(\varphi,\psi \in H^1_0(\Omega)\). Now, from (23), we obtain from where it follows that \[ \eta \in L^2_g(\mathbb{R}^+;H^1_0(\Omega)). \] From (19), we get \[ \beta_0 \Delta u+ \int^{\infty}_0g(s)\Delta \eta(s)\;ds \in L^2(\Omega). \] On the other hand, from (22), we obtain \begin{eqnarray*} ||\eta_s||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}\leq C\left(||\varphi||^2_{H^1_0(\Omega)}+||f^5||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))} +||\eta||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}\right)\,. \end{eqnarray*} From where it follows that \[ \eta_s\in L^2_g(\mathbb{R}^+;H^1_0(\Omega)). \] Again from (23), we have \[ \eta(0)=0. \] Thus, \(I-\mathcal{A}\) is maximal. Then, thanks to the Lumer-Phillips theorem (see [13], Theorem 4.3), the operator \(\mathcal{A}\) generates a C\(_0\)-semigroup of contractions \(e^{t\mathcal{A}}\) on \(\mathcal{H}\). The proof is now complete.

3. Lack of exponential decay

Our starting point is to show that the semigroup associated to the system (7)-(13) is not exponential stable. To show this, we assume that \(g(t)=e^{-\mu t}\), with \(t\in \mathbb{R}^+\) and \(\mu>1\). We will use the Prüss's theorem [14] to prove the lack of exponential stability.

Theorem 2. Let \(S(t)=e^{\mathcal{A}t}\) be a C\(_0\)-semigroup of contractions on Hilbert space. Then \(S(t)\) is exponentially stable if and only if \[ \rho(\mathcal{A})\supseteq \{i\beta:\beta\in \mathbb{R}\}\equiv i\mathbb{R} \] and \[ \overline{\lim_{|\beta|\rightarrow \infty}}\|(i\beta-\mathcal{A})^{-1}\|< \infty \] hold, where \(\rho(\mathcal{A})\) is the resolvent set of \(\mathcal{A}\).

To do this, let us consider the spectral problem: where \[ \lim_{m \rightarrow \infty}\lambda_{m}=+\infty. \] The following theorem describes the main results of this section.

Theorem 3. Let \(S(t)\) be C\(_0\)-semigroup of contractions generated by \(\mathcal{A}\). Then \(S(t)\) is not exponentially stable.

Proof. Here we will use the Theorem 2. That is, we will show that there exists a sequence of values \(\lambda_{m}\) such that

It is equivalent to prove that there exist a sequence of data \(F_m\in {\mathcal{H}}\) and a sequence of complex numbers \(\lambda_m\in i\mathbb{R}\), with \(||F_m||_{\mathcal{H}}\leq1\) such that where with \(U_m\) not bounded. To simplify the notation we will omit the subindex \(m\). Then, the Equation (31) becomes Let us consider \(f1=f^3=f^5=0\) and \(f^2=f^4=w_m\) to obtain \(\varphi=i\lambda u\) e \(\psi=i\lambda v\). Then, the system (32) becomes We look for solutions of the form \[ u=aw_m, \quad v=bw_m, \quad \varphi=cw_m,\quad \psi=dw_m,\quad \eta(x,s)=\gamma(s)w_m \] with \(a,b,c,d\in \mathbb{C}\) and \(\gamma(s)\) depend on \(\lambda\) and will be determined explicitly in the sequel. From (33), we get \(a\) and \(b\) satisfy Solving (34)\(_3\) we get Since \(\eta(0)=0\) then \(C=-a\), and (35) becomes Then, from (36) we have where \[ b_0=\int^{\infty}_0g(s)\;ds. \] Now, choosing \(\lambda=\sqrt{\lambda_m}\), using the equation (34)\(_1\) and (34)\(_2\) we obtain \begin{eqnarray*} &&a=\frac{1}{\alpha}, \\ &&b=\frac{\lambda_m(1-\beta_0)}{\alpha^2}-\frac{\lambda_m}{\alpha}\int^{\infty}_0g(s)\gamma(s)\;ds+\frac{1}{\alpha}, \\ &&c=i\frac{\sqrt{\lambda_m}}{\alpha},\\ &&d=i\sqrt{\lambda_m}(\frac{\lambda_m(1-\beta_0)}{\alpha^2} -\frac{\lambda_m}{\alpha}\int^{\infty}_0g(s)\gamma(s)\;ds+\frac{1}{\alpha}). \end{eqnarray*} Recalling that \[ \varphi=cw_m=i\frac{\sqrt{\lambda_m}}{\alpha}w_m \] we get \[ ||\varphi||^2_{L^2(\Omega)}=\frac{\lambda_m}{\alpha^2}. \] Therefore we have \[ \lim_{m\rightarrow \infty}||U_m||^2_{\mathcal{H}}\geq \lim_{m\rightarrow \infty}||\varphi||^2_{L^2(\Omega)}=\lim_{m\rightarrow \infty}\frac{\lambda_m}{\alpha^2}= \infty. \] Using theorem 3 follows that \(S(t)\) is not exponentially stable. The proof is now complete.

4. Polynomial decay and optimally result

In this section we study the polynomial decay associated to the system (7)-(13) and subsequently we find the optimal rate of decay. Then, let us consider the resolvent equation \[ (i\lambda I- \mathcal{A})U=F,\quad \mbox{with}\quad \lambda \in \mathbb{R} \quad \mbox{and}\quad F\in \mathcal{H}, \] that is, In the next step we shall show three lemmas important to proof the main result.

Lemma 1. The solutions of the system (7)-(13), given by the Theorem \ref{teo2.1}, satisfies \begin{eqnarray*} \int_{\Omega}\int^{\infty}_0g(s)|\nabla \eta|^2\; ds \; dx\leq K |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} where \(K\) is a positive constant and \(|\lambda|> 1\).

Proof. Multiplying the equality (39) by \(\overline{\varphi}\) and integrating by parts on \(\Omega\), we get

Substituting \(\varphi\) given in (38) into \(I_1\) and \(I_2\), we have and Now, substituting \(\varphi\) given in (42) into \(I_3\) and integrating by parts, we obtain Substituting (44), (45) and (46) into (43), we get Taking the real part on the left side of the above equality and using the hypotheses (14) on \(g\), our conclusion follows.

Lemma 2. For any \(\epsilon>0\), there exists a positive constant \(K_{\epsilon}\) such that \begin{eqnarray*} &&\beta_0\int_{\Omega}|\nabla u|^2\; dx+\int_{\Omega}|\nabla v|^2\;dx+\alpha \int_{\Omega}(u\overline{v}+v\overline{u})\;dx\\ &&\leq \int_{\Omega}|\varphi|^2\;dx+\int_{\Omega}|\psi|^2\;dx+\epsilon \int_{\Omega}|\nabla u|^2\;dx+K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} where \(K\) is a positive constant.

Proof. Multiplying the equalities (39) and (41) by \(\overline{u}\) and \(\overline{v}\), respectively, integrating by parts on \(\Omega\) and summing up the result, we get

Substituting \(\overline{i\lambda u}\) given in (38) into \(I_4\) and \(\overline{i\lambda v}\) given in (40) into \(I_5\), we find Now, using Poincaré and Young inequalities, we have Using the Lemma 1, our conclusion follows.

Lemma 3. Under the conditions of the previous lemma, we have \begin{eqnarray*} \frac{b_0}{2}\int_{\Omega}|\varphi|^2\; dx&\leq &\epsilon\int_{\Omega}(|\nabla u|^2+|\nabla v|^2)\;dx+ K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} and \begin{eqnarray*} \left(\frac{1}{2}-\frac{K}{|\lambda|^2}\right)\int_{\Omega}|\psi|^2\;dx\leq K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} with \(|\lambda|>1\) large enough.

Proof. Multiplying the equation (42) by \(\int^{\infty}_0g(s)\;ds\overline{\varphi}\) and integrating on \(\Omega\), we find \begin{eqnarray*} \underbrace{i\lambda \int^{\infty}_0g(s)\int_{\Omega}\eta(s)\overline{\varphi}\;dx\;ds}_{:=I_6}-b_0\int_{\Omega}|\varphi|^2\;dx +\int^{\infty}_0g(s)\int_{\Omega}\eta_s(s)\overline{\varphi}\;dx\;ds =\int^{\infty}_0g(s)\int^{\infty}_0f^5(s)\overline{\varphi}\;dx\;ds \end{eqnarray*} where \(b_0=\int^{\infty}_0g(s)\;ds\). On the other hand, noting that \[ \int^{\infty}_0g(s)\int_{\Omega}\eta_s(s)\overline{\varphi}\;dx\;ds=-\int^{\infty}_0g'(s)\int_{\Omega}\eta(s)\overline{\varphi}\;dx\;ds \] and substituting \(\overline{i\lambda \varphi}\) given in (39) into \(I_6\), we get

Using the hypotheses on \(g\) given in (14) and taking into account the Poincaré and Young inequalities, we have \begin{eqnarray*} \frac{b_0}{2}\int_{\Omega}|\varphi|^2\;dx \leq \epsilon \int_{\Omega}(|\nabla u|^2+|\nabla v|^2)\;dx+K_{\epsilon}||\eta||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}}. \end{eqnarray*} Using the Lemma 1, follows the first inequality. To show the second inequality, we substitute the equation (38) into (42). This gives, Now, we substitute \(u\) given in (41) into (52). Then, we obtain, Multiplying the equation (53) by \(\overline{\int^{\infty}g(s)\psi}\), integrating by parts on \(\Omega\) and proceeding as to obtain the first estimate, we have \begin{eqnarray*} \frac{1}{2}\int_{\Omega}|\psi|^2\;dx\leq \frac{K}{|\lambda|^2}\int_{\Omega}|\psi|^2\;dx+K|\lambda|^2||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K||F||^2_{\mathcal{H}}. \end{eqnarray*} From where follows the second inequality. The proof is now complete.

Now, we are in the position the main result of this paper.

Theorem 4. The semigroup associated to the system (7)-(13) is polynomially stable and \[ ||S(t)U_0||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||U_0||_{D(\mathcal{A})}. \] Moreover, this result is optimal.

Proof. From Lemmas 1, 2 and 3, choosing \(\epsilon >0\) small enough and for \(|\lambda|> 1\) large enough, we have \[ ||U||^2_{\mathcal{H}}\leq K|\lambda|^2||U||_{\mathcal{H}} ||F||_{\mathcal{H}}+K ||F||^2_{\mathcal{H}}. \] From where it follows that \[ ||U||^2_{\mathcal{H}}\leq K|\lambda|^4||F||^2_{\mathcal{H}} \] that can be written as \begin{eqnarray*} ||(\lambda I-\mathcal{A})^{-1}||\leq K|\lambda|^2, \end{eqnarray*} that is

Then using the Theorem of A. Borichev and Y. Tomilov, (see [15], Theorem 2.4), the condition (54) is equivalent to \begin{eqnarray*} ||S(t){\mathcal{A}}^{-1}||=K\mathcal{O}(t^{-\frac{1}{2}})\Rightarrow ||S(t){\mathcal{A}}^{-1}F||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||F||_{\mathcal{H}}, \,\,t \rightarrow \infty. \end{eqnarray*} Then taking \(\mathcal{A}U_0=F\), we get \[ ||S(t)U_0||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||U_0||_{D(\mathcal{A})}. \] Therefore the solution decays polynomially. To prove that the rate of decay is optimal, we will argue by contradiction. Suppose that the rate \(t^{-\frac{1}{2}}\) can be improved; for example that the rate is \(t^{-\frac{1}{2-\epsilon}}\) for some \(0< \epsilon < 2\). From Theorem 5.3 in [20], the operator \[ |\lambda|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||(\lambda I-{\mathcal{A}})^{-1}|| \] should be limited, but this does not happen. For this, let us suppose that there exist a sequence \((\lambda_{\mu})\subset {\mathbb{R}}\) with \(\lim_{\mu \rightarrow \infty}|\lambda_{\mu}|=\infty\) and \((U_{\mu})\subset D(\mathcal{A})\) for \((F_{\mu})\subset {\mathcal{H}}\) such that \[ (i\lambda_{\mu}I-{\mathcal{A}})U_{\mu}=F_{\mu} \] is bounded in \({\mathcal{H}}\) and \[ \lim_{\mu \rightarrow \infty}|\lambda_{\mu}|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||U_{\mu}||_{\mathcal{H}}=\infty. \] So, following the same steps as in the proof of Theorem 3 we can conclude that \[ |\lambda_{\mu}|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||U_{\mu}||_{\mathcal{H}}\geq \mathcal{O}\left(\mu^{\displaystyle{\frac{\epsilon}{2}}}\right)\rightarrow \infty,\quad \mbox{as}\quad \mu \rightarrow \infty. \] Therefore the rate cannot be improved. The proof is now complete.

Acknowledgments

This research is partially supported by PNPD/UFBA/CAPES(Brazil).

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

Consider the linear differential equation

where \(k\geq 1\) is an integer, \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic functions in the unit disc \(\mathbb{D}=\{z\in \mathbb{C}:|z|< 1\}\) and \(a_{0}(z)\not\equiv 0\). The theory of complex differential equations in the unit disc has been developed since 1980's, see [1]. In the year 2000, Heittokangas in [2] firstly investigated the growth and oscillation theory of Equation (1) when the coefficients \( a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic functions in the unit disc \(\mathbb{D}\) by introducing the definition of the function spaces. His results also gave some important tools for further investigations on the theory of meromorphic solutions of Equation (1).

In this article, we investigate the growth of solutions of the Equation (1) when the coefficients \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\), and we deal with the case that the coefficients are fast growing in \(\mathbb{D}\). To define the order of fast-growth of analytic functions, we define inductively for \(r\in \lbrack 0,+\infty ),\) \( \exp _{0}r=r\), \(\exp _{1}r=e^{r}\) and \(\exp _{n+1}r=\exp \left( \exp _{n}r\right) ,\) \(n\in \mathbb{N}\). For all \(r\) sufficiently large, we define \(\log _{0}r=r,\) \(\log _{1}r=\log r\) and \(\log _{n+1}r=\log \left( \log _{n}r\right) ,\) \(n\in \mathbb{N}\). Also, we need to be familiar with the fundamental results and the standard notations of the Nevanlinna's theory on the complex plane \(\mathbb{C}\) and in the unit disc \(\mathbb{D}\), for more details on Nevanlinna theory and its applications in complex differential equations in complex plane and in unit disc, we refer to [2,3, 4, 5, 6, 7].

Before stating our main results, we recall definitions and preliminary remarks concerning meromorphic and analytic functions in \(\mathbb{D}\). For the definitions and more discussions, we refer the reader to [7, 8, 9, 10].

Let \(p\geq 1\) be an integer and \(f\) be a meromorphic function in \(\mathbb{D}\) . Then, the iterated \(p\)-order of \(f\) is defined by \begin{equation*} \rho _{p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p}^{+}T(r,f)}{\log \frac{1}{1-r}}, \end{equation*} where \(\log _{1}^{+}r=\log ^{+}r=\max \{\log r;0\}\), \(\log _{p+1}^{+}r=\log ^{+}\left( \log _{p}^{+}r\right) \) and \(T(r,f)\) is the Nevanlinna characteristic function. If \(f\) is analytic in \(\mathbb{D}\), then the iterated \(p\)-order of \(f\) is defined by \begin{equation*} \rho _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p+1}^{+}M(r,f)}{\log \frac{1}{1-r}}, \end{equation*} where \(M(r,f)=\max \left\{ |f(z)|:|z|=r\right\} .\)

Remark 1. For \(p=1\), \(\rho _{1}(f)\) is called order, see [2]. And for \(p=2,\) \(\rho _{2}(f)\) is called hyper-order, see [11].

Remark 2. It follows by [7, page 205] that if \(f\) is an analytic function in \(\mathbb{D}\), then we have the inequalities \begin{equation*} \rho _{1}(f)\leq \rho _{M,1}(f)\leq \rho _{1}(f)+1 \end{equation*} which are the best possible in the sense that there are analytic functions \( g \) and \(h\) such that \(\rho _{1}(g)=\rho _{M,1}(g)\) and \(\rho _{M,1}(h)=\rho _{1}(h)+1\), see [12]. However, it follows by [4,Proposition 2.2.2] {laine} that \(\rho _{p}(f)=\rho _{M,p}(f)\) for \(p\geq 2\).

The iterated \(p\)-type of a meromorphic function \(f\) \ in \(\mathbb{D}\) with \( 0< \rho _{p}(f)< +\infty \) is defined by \begin{equation*} \tau _{p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho _{p}(f)}\log _{p-1}^{+}T(r,f), \end{equation*} and if \(f\) is analytic in \(\mathbb{D}\) with \(0< \rho _{M,p}(f)< +\infty \), then the iterated \(p\)-type is defined by \begin{equation*} \tau _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho _{M,p}(f)}\log _{p}^{+}M(r,f). \end{equation*}

Remark 3. It follows by [4,Proposition 2.2.2] that \(\tau _{p}(f)=\tau _{M,p}(f)\) for \(p\geq 2\).

2. Basic results

Heittokangas et al. in [10] proved the following results.

Theorem 1. ([10]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})< \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\), then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Theorem 2.([10]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \sum_{\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})}\tau _{M,p}(a_{j})< \tau _{M,p}(a_{0}), \end{equation*} then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Hamouda in [13], gave an improvement of Theorem 2 as follows.

Theorem 3.([13]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})\right\} < \tau _{M,p}(a_{0}), \end{equation*} then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Our proofs depend mainly upon the following lemmas. Before starting these lemmas, we recall the concept of logarithmic measure. The logarithmic measure of a set \(S\subset (0,1)\) is given by \begin{equation*} lm(S):=\int_{S}\frac{dt}{1-t}. \end{equation*} The set \(F\subset \lbrack 0,1)\) in all this paper is not necessarily the same at each occurrence, but it is always of finite logarithmic measure, that is \(lm(F)< +\infty \). To avoid some problems of the exceptional sets, we need the following lemma.

Lemma 1. ([2, 14) Let \(g:[0,1)\mapsto \mathbb{R}\) and \(h:[0,1)\mapsto \mathbb{R}\) be monotone non-decreasing functions such that \(g(r)\leq h(r)\) holds outside of an exceptional set \(F\subset \lbrack 0,1)\) of finite logarithmic measure. Then there exists a \(d\in (0,1)\) such that if \(s(r)=1-d(1-r)\), then \( g(r)\leq h(s(r))\) for all \(r\in \lbrack 0,1)\).

Lemma 2.[12, Theorem 3.1] Let \(k\) and \(j\) be integers satisfying \(k>j\geq 0\), and let \( \varepsilon >0\) and \(d\in (0,1)\). If \(f\) is a meromorphic in \(\mathbb{D}\) such that \(f^{(j)}\not\equiv 0\), then \begin{equation*} \left\vert \frac{f^{(k)}(z)}{f^{(j)}(z)}\right\vert \leq \left[ \left( \frac{ 1}{1-|z|}\right) ^{2+\varepsilon }\max \left\{ \log \frac{1}{1-|z|} ;T(s(|z|),f)\right\} \right] ^{k-j} \end{equation*} for \(|z|\not\in F,\) where \(F\subset \lbrack 0,1)\) is a set of finite logarithmic measure, and where \(s(|z|)=1-d(1-r)\).

Lemma 3. ([10]) \label{lem2}Let \(f\) be a meromorphic function in the unit disc with \(\rho _{p}(f):=\rho < +\infty \) for some \(p\in \mathbb{N}\), and let \(\varepsilon >0\) be a given constant. Then, there exists a set \(F\subset (0,1)\) of finite logarithmic measure such that for all \(z\) with \(|z|=r\not\in F\) and for all integer \(j\geq 1\), we have:

• (i) If \(p=1\), then
  \begin{equation} \left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \frac{1}{(1-r)^{j(\rho +2+\varepsilon )}}. \label{conlem3} \end{equation}

  (2)
• (ii) If \(p\geq 2\), then
  \begin{equation} \left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \exp _{p-1}\left\{ \frac{1 }{(1-r)^{\rho +\varepsilon }}\right\} . \label{conlem4} \end{equation}

  (3)

Lemma 4. ([10]) Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\). Then, every solution \(f\not\equiv 0\) of the Equation \((1)\) satisfies \begin{equation*} \rho _{p+1}(f)=\rho _{M,p+1}(f)\leq \max \left\{ \rho _{M,p}(a_{j}):j=0,\dots ,k-1\right\} . \end{equation*}

Remark 4. If \(p\geq 2\), then by Remark 2 and Lemma 4, we obtain that very solution \(f\not\equiv 0\) of the Equation \((1)\) satisfies \begin{equation*} \rho _{p+1}(f)\leq \max \left\{ \rho _{p}(a_{j}):j=0,\dots ,k-1\right\} . \end{equation*}

Lemma 5. ([2, 3, 7]) Let \(f\) be a meromorphic function in the unit disc \(\mathbb{D}\) and let \(k\in \mathbb{N}\). Then \begin{equation*} m\left( r,\frac{f^{(k)}}{f}\right) =S(r,f), \end{equation*} where \(S(r,f)=O\left( \log ^{+}T(r,f)+\log \left( \frac{1}{1-r}\right) \right) \), possibly outside a set \(F\subset \lbrack 0,1)\) with finite logarithmic measure.

Lemma 6. ([15]) Let \(f\) be a meromorphic function in the unit disc \(\mathbb{D}\) for which \(i\left( f\right) =p\geq 1\) and \(\rho _{p}\left( f\right) =\rho < +\infty \), and let \(k\geq 1\) be an integer. Then for any \(\varepsilon >0,\) \begin{equation*} m\left( r,\frac{f^{\left( k\right) }}{f}\right) =O\left( \exp _{p-2}\left\{ \frac{1}{1-r}\right\} ^{\rho +\varepsilon }\right) \end{equation*} holds for all \(r\) outside a set \(F\) \(\subset \lbrack 0,1)\) with \(\int_{F}\frac{dr}{1-r}< +\infty .\)

Lemma 7. For an integer \(p\geq 2\), let \(f\) be a meromorphic function in \( \mathbb{D}\) such that \(0< \rho _{p}(f)=\rho < +\infty \) (see Definition 7), \(0< \tau _{p}(f)=\tau < +\infty \) and \(0< \tau _{p}^{\ast }(f)=\tau ^{\ast }< +\infty \). Then for any given \(\eta < \tau ^{\ast }\), there exists a subset \(E\subset \lbrack 0,1)\) that has an infinite logarithmic measure \(\int_{E}\frac{dr}{1-r}=+\infty \) such that for all \(r\in E,\) we have \begin{equation*} \log _{p-2}T(r,f)>\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} . \end{equation*}

Proof. By the definition of \(\tau _{p}^{\ast }(f)\), there exists an increasing sequence \(\{r_{m}\}_{m=1}^{+\infty }\subset \lbrack 0,1)\) satisfying \(\frac{1 }{m}+\left( 1-\frac{1}{m}\right) r_{m}< r_{m+1},\;(r_{m}\longrightarrow 1^{-}, \) \(m\longrightarrow +\infty )\) and \begin{equation*} \lim_{m\rightarrow +\infty }\frac{\log _{p-2}T(r_{m},f)}{\exp \left\{ \frac{ \tau }{\left( 1-r_{m}\right) ^{\rho }}\right\} }=\tau ^{\ast }. \end{equation*} Then, for any given \(0< \varepsilon < \tau ^{\ast }\), there exists a positive integer \(m_{0}\) such that for all \(m\geq m_{0}\), we have

For \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we get \begin{equation*} \lim_{m\rightarrow +\infty }\frac{\exp \left\{ \tau \left[ \left( 1-\frac{1}{ m}\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} }{\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }=1. \end{equation*} Then for any given \(0< \eta < \tau ^{\ast }-\varepsilon \), there exists a positive integer \(m_{1}\) such that for all \(m\geq m_{1}\), and for all \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we have By (4) and (5), for all \(m\geq m_{2}=\max \left\{ { m_{0};m_{1}}\right\} \) and for all \(r\in \left[ r_{m},\frac{1}{m}+\left( 1- \frac{1}{m}\right) r_{m}\right] \), we have \begin{eqnarray*} \log _{p-2}T(r,f)&\geq& \log _{p-2}^{+}T(r_{m},f)>(\tau ^{\ast }-\varepsilon )\exp \left\{ \frac{\tau }{\left( 1-r_{m}\right) ^{\rho }}\right\}\\ &\geq& (\tau ^{\ast }-\varepsilon )\exp \left\{ \tau \left[ \left( 1-\frac{1}{m }\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} >\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} . \end{eqnarray*} Set \begin{equation*} E=\bigcup_{m=m_{2}}^{+\infty }\left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m} \right) r_{m}\right] . \end{equation*} Then \begin{equation*} \int_{E}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty }\int_{r_{m}}^{\frac{1}{m} +\left( 1-\frac{1}{m}\right) r_{m}}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty }\log \frac{m}{m-1}=+\infty . \end{equation*}

By using similar reasoning as in the proof of Lemma 7, we easily obtain the following lemma.

Lemma 8. For an integer \(p\geq 2\), let \(f\) be a meromorphic function in \(\mathbb{D}\) such that \(0< \rho _{M,p}(f)=\rho < +\infty \), \(0< \tau _{M,p}(f)=\tau < +\infty \) and \(0< \tau _{M,p}^{\ast }(f)=\tau ^{\ast }< +\infty \). Then for any given \(\eta < \tau ^{\ast }\), there exists a subset \(E\subset \lbrack 0,1)\) that has an infinite logarithmic measure \(\int_{E} \frac{dr}{1-r}=+\infty \) such that for all \(r\in E,\) we have \begin{equation*} \log _{p-1}M(r,f)>\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} . \end{equation*}

Lemma 9. ([16]) Let \(f\) be a solution of Equation \(\left( 1\right) ,\)where the coefficients \(a_{j}\left( z\right) \) \(\left( j=0,...,k-1\right) \) are analytic functions in the disc \(\Delta _{R}=\left\{ z\in\mathbb{C}:\left\vert z\right\vert < R\right\} ,\) \(0< R\leq \infty .\) Let \(n_{c}\in \left\{ 1,...,k\right\} \) be the number of nonzero coefficients \(a_{j}\left( z\right) \) \(\left( j=0,...,k-1\right) ,\) and let \(\theta \in \left[ 0,2\pi \right] \) and \( \varepsilon >0.\) If \(z_{\theta }=\nu e^{i\theta }\in \Delta _{R}\) is such that \(a_{j}\left( z_{\theta }\right) \neq 0\) for some \(j=0,...,k-1,\) then for all \(\nu < r< R,\) \begin{equation*} \left\vert f\left( re^{i\theta }\right) \right\vert \leq C\exp \left( n_{c} \overset{r}{\underset{\nu }{\int }}\underset{j=0,...,k-1}{\max }\left\vert a_{j}\left( te^{i\theta }\right) \right\vert ^{\frac{1}{k-j}}dt\right) , \end{equation*} where \(C>0\) is a constant satisfying \begin{equation*} C\leq \left( 1+\varepsilon \right) \underset{j=0,...,k-1}{\max }\left( \frac{ \left\vert f^{\left( j\right) }\left( z_{\theta }\right) \right\vert }{ \left( n_{c}\right) ^{j}\underset{n=0,...,k-1}{\max }\left\vert a_{n}\left( z_{\theta }\right) \right\vert ^{\frac{j}{k-n}}}\right) . \end{equation*}

Lemma 10. Let \(\{a_{j}(z)\}_{0\leq j\leq k-1}\) be analytic functions in the disc \(\mathbb{D}\) such that \(0< p< \infty \) and \(0< \max \{\rho _{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \rho _{M,p}\left( a_{0}\right) =\rho < \infty \) and \(\max \{\tau _{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \tau _{M,p}\left( a_{0}\right) =\tau < \infty \). Then, every solution \(f\not\equiv 0\) \textit{\ }of the Equation \(\left( 1\right) \) with \(\rho _{p+1}(f)=\rho \) satisfies \(\tau _{p+1}(f)\leq \tau \).

Proof. Let \(f\not\equiv 0\) be a solution of \(\left( 1\right) \) with \(\rho _{p+1}(f)=\rho .\) Let \(\theta _{0}\in \left[ 0,2\pi \right) \) be such that \( \left\vert f\left( re^{i\theta _{0}}\right) \right\vert =M\left( r,f\right) . \) By Lemma 9, we have

We have \(\max \left\{ \rho _{M,p}(a_{j}):j=0,1,...,k-1\right\} =\rho _{M,p}\left( a_{0}\right) =\rho \). By the definition of \(\tau _{M,p}(a_{j})\) , for any given \(\varepsilon >0\) and \(r\rightarrow 1^{-},\) we obtain By (6) and (7), we have for \(r\rightarrow 1^{-}\) Then, it follows from (8), arbitrariness of \(\varepsilon >0\) that \(\tau _{p+1}(f)=\tau _{M,p+1}(f)\leq \tau .\)

3. Main results

In this article, we aim to answer the following questions:

1. What can be said about the growth of solutions of the Equation \((1)\) in the case when \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})\right\} \leq \tau _{M,p}(a_{0})? \end{equation*}
2. What happened when we replace \(\rho _{M,p}\) and \(\tau _{M,p}\) by \(\rho _{p}\) and \(\tau _{p}\)?

As the first result, we give an improvement to Theorems 1 and 2 of [13].

Theorem 4. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be meromorphic functions in the unit disc \(\mathbb{D}\), and \(a_{0}(z)\not\equiv 0\). Suppose that there exist a point \(\omega \in \partial \mathbb{D}\), a curve \(\gamma \) tending to \(\omega \) and a set \(F_{1}\subset (0,1)\) of finite logarithmic measure such that for \(z\in \gamma \) and \(|z|=r\not\in F_{1},\) we have for the largest integer \(p\geq 1\)

for all \(\lambda >0\) and \(\alpha >0\). Then, every nontrivial meromorphic solution \(f\) of the Equation \((1)\) satisfies \(\rho _{p+1}(f)=+\infty \).

Proof. Suppose that \(f\not\equiv 0\) is a solution of the Equation (1) with \(\rho _{p+1}(f)=\rho < +\infty \). By (1), \(f\) satisfies

For \(p\geq 1\), by (3), for all \(z\) satisfying \(|z|=r\not\in F\) (\( F\) has finite logarithmic measure), we obtain where \(\alpha >0\) is a constant which depends on \(\rho \), \(\varepsilon \) and \(j=1,\dots ,k\). By substituting (10) into (11), it yields By (9), for \(z\in \gamma \) such that \(|z|=r\not\in F_{1}\) (\( F_{1} \) has finite logarithmic measure), we know that as \(z\rightarrow \omega \) Thus, for all \(z\in \gamma \) with \(|z|=r\not\in F_{1}\cup F\), by (12) and (13), we get a contradiction. Hence, every meromorphic solution \( f\not\equiv 0\) of (1) has an infinite \((p+1)\)-order.

Remark 5. In [13], under the same hypotheses of Theorem 4, Hamouda obtained that \(\rho _{p+1}(f)\geq \alpha .\)

In all the next, we consider \(p\in \mathbb{N}\backslash \{1\}\). In trying to give an answer on the above questions, we prove the following results.

Theorem 5. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau _{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose that there exist two positive real numbers \(\alpha \) and \(\beta \) with \( 0\leq \beta < \alpha \), such that

and as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{1}\) (\(E_{1}\) is of infinite logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \( \left( 1\right) \) satisfies \(\rho _{p}\left( f\right) =+\infty ,\) \( \rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1), \(f\) satisfies

By hypotheses of Theorem 5 and Lemma 4, we know that \( \rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \). Then, by (3) for all \(0< \varepsilon < \rho -\rho _{1}\), we have where \(|z|=r\not\in F\). By substituting (14), (15) and (17) into (16) we obtain for all \(r\in E_{1}\backslash F\). Hence, we get for some constant \(C_{1}>0,\) which is a contradiction as \(|z|=r\rightarrow 1^{-}\), \(r\in E_{1}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho >\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \). Now, by Lemma 2, we have for \(j=1,\dots ,k\) for all \(r\not\in F_{2}\cup \lbrack 0,1]\). From, (14), (15), (16) and (20), we obtain for all \(r\in E_{1}\backslash (F_{2}\cup \lbrack 0,1])\). Hence for some constant \(C_{2}>0\) and for all \(r\in E_{1}\backslash (F_{2}\cup \lbrack 0,1])\). Setting \(R=s(r)=1-d(1-r)\), \(d\in (0,1)\). We have \(1-r=\frac{ 1-R}{d},\) \(d\in (0,1)\). Then by Lemma 1, we obtain we obtain for \( R\longrightarrow 1^{-}\) Since \(0< \rho _{p+1}(f)=\rho < \infty ,\) from (23) we deduce that \begin{equation*} \tau _{p+1}(f)=\underset{R\rightarrow 1^{-}}{\lim \sup }\frac{\log _{p}^{+}T(R,f)}{\frac{1}{\left( 1-R\right) ^{\rho }}}\geq d^{\rho }\tau . \end{equation*} By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau .\)

Theorem 6. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho _{p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{p}(a_{j})\leq \tau _{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose that there exist two positive real numbers \(\alpha \) and \(\beta \) with \( 0\leq \beta < \alpha \), such that

and as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{2}\) (\(E_{2}\) is of infinite logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \( \left( 1\right) \) satisfies \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1) we can write

By hypotheses of Theorem 6 and Lemma 4, we know that \( \rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \). Then by Lemma 6, for all \(0< \varepsilon < \rho -\rho _{1}\) and for all \(|z|=r\notin F\), we have for \(j=1,\dots ,k\) Now, let \(p\geq 2\), it follows by (24), (25), (26) and (27) that holds for all \(z\) satisfying \(|z|=r\in E_{2}\backslash F\) as \(r\rightarrow +\infty ,\) and \(M>0\) is some constant. Hence, from (\ref{24}) we obtain \begin{equation*} (\alpha -\beta )\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} \leq \exp \left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon }}\right\} +C_{3} \end{equation*} for some constant \(C_{3}>0,\) which is a contradiction as \(|z|=r\rightarrow 1^{-}\), \(r\in E_{2}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho >\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \). Now, it follows by (24), (25), (26) and Lemma 5 that for all sufficiently large \(|z|=r\in E_{2}\backslash F\). Then, for all sufficiently large \(|z|=r\in E_{2}\backslash F\) for some constant \(C_{4}>0.\) Then by Lemma 1, we obtain for \( |z|=r\in E_{2},\) \(s\left( r\right) \rightarrow 1^{-}\) where \(s(r)=1-d(1-r)\), \(d\in (0,1)\). Hence, by (31) we obtain \begin{equation*} \tau _{p+1}(f)=\underset{s\left( r\right) \rightarrow 1^{-}}{\lim \sup } \frac{\log _{p}^{+}T(s\left( r\right) ,f)}{\frac{1}{\left( 1-s\left( r\right) \right) ^{\rho }}}\geq d^{\rho }\tau . \end{equation*} By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau .\)

Hamouda in [17], to study the growth of meromorphic solutions of differential equations with finite iterated \(p\)-order in complex plane, introduced new type of growth (see[17, p. 46]). According to the definition of this new type of growth, we introduce a new definition of type of growth that we note \(\tau _{p}^{\ast }(f)\) related to iterated \(p\)-order of meromorphic function \(f\) in the unit disc, as follows.

Definition 7. For \(p\geq 2,\) let \(f\) be a meromorphic function of finite iterated \(p\) -order in \(\mathbb{D}\) such that \(0< \rho _{p}\left( f\right) =\rho < +\infty \) and \(0< \tau _{p}\left( f\right) =\tau < +\infty \), we define \(\tau _{p}^{\ast }(f)\) by \begin{equation*} \tau _{p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p-2}^{+}T(r,f) }{\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }. \end{equation*} If \(f\)\ is an analytic function in \(\mathbb{D}\) with \(0< \tau _{M,p}\left( f\right) =\tau _{M}< +\infty \), we also define \begin{equation*} \tau _{M,p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p-1}^{+}M(r,f)}{\exp \left\{ \frac{\tau _{M}}{\left( 1-r\right) ^{\rho }} \right\} }. \end{equation*}

The following theorems improves and extends Theorems 2 and 3.

Theorem 8. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau _{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \ \begin{equation*} \max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau _{M,p}^{\ast }(a_{0}). \end{equation*} Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \( d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Suppose that all coefficients of the Equation (1) satisfy the hypotheses of Theorem 8. Now, let \(\alpha \) and \(\beta \) be two real numbers such that \begin{equation*} \max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta < \alpha < \tau _{M,p}^{\ast }(a_{0}). \end{equation*} Because all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)

and by Lemma 8, we have for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). \ From (32) and (33), and by Theorem 5, we obtain the result.

Theorem 9. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho _{p}(a_{0})=\rho \) \((0< \rho < +\infty )\), \(\tau _{p}(a_{j})\leq \tau _{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \ \begin{equation*} \max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau _{p}^{\ast }(a_{0}). \end{equation*} Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \( d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Suppose that all coefficients of the Equation (1) satisfy the hypotheses of Theorem 9. Now, let \(\alpha \) and \(\beta \) be two real numbers such that \begin{equation*} \max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta < \alpha < \tau _{p}^{\ast }(a_{0}). \end{equation*} Since all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)

and by Lemma 7, we have for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). From (34) and (35), and by Theorem 6, we obtain the result.

For some related results in the whole complex plane, see [18].

Acknowledgments

The authors would like to thank the referee for his/her valuables remarks, which led to an improvement of the presentation of this paper. This paper is supported by University of Mostaganem (UMAB) (PRFU Project Code C00L03UN270120180005).

Author Contributions

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

Competing Interests

The authors do not have any competing interests in the manuscript.

1. Introduction

The numerical study of the piezoelectric [1, 2] contact problems [3, 4, 5, 6, 7] presents a great challenges, because the non coercivity and non differentiability of some terms. The linear term coupling the mechanical field and electric potential is non coercive and non symmetric. The term corresponding to the friction is convex and non differentiable, in the variational formulation in one hand. In the other hand, the non-linear coupling mechanical field and frictional contact, it can be shown in the norm of the tangential component of the mechanical field present in the frictional function.

To overcome these difficulties, the authors develop some methods like finite elements method [8, 9], penalty method and fixed point method [4]. The most and sufficient method for this type of problem is the ones based on convex duality [10] and the introduction of Lagrange multipliers [11, 12, 13]. Primal-dual active sets strategy, which is an equivalent to infinite dimensional semismooth Newton method, is applied in [14] while in [15] the author propose a numerical approximation and based Uzawa block relaxation method. Alternating directions method of multipliers (ADMM) is based in [16].

In mechanic of structures one is always interested in the determination of the stress tensor \(\sigma\) more than the mechanical displacement itself. Methods have been developed for calculate an approximation of \(\sigma\) from \(u\) and the drawback is not easy to build an approximation space of tensors satisfying the equilibrium relations and required regularity. A mixed variational formulation have been developed to handle this difficulty. Concerning the piezoelectric contact problems, mixed formulation were developed in [17, 18, 19].

In this paper, we introduce a mixed variational approach based Lagrange multipliers which describes the static frictional contact between a piezoelectric body and non-conductive foundation. The standard mixed variational formulation of contact problems is formally in the following form [17, 18, 20]:

where \(a(\cdot,\cdot)\) is symmetric, coercive and the term \(b(\cdot,\cdot)\) coupling the normal and tangential Lagrange multipliers. This coupling is difficult to handle numerically and for model describing the contact problem with friction it is important to use the decoupling form to identify the slid and slip on the contact zone. Moreover, when the model (1) describe problem with electroelastic body the bi-linear form \(a(\cdot,\cdot)\) become not symmetric.

The idea consists in decoupling the contact from the friction by introducing two convex sets, one is reserved to the contact multiplier and the second is stated for the friction. This approach leads to decoupled inequalities in mixed variational problem. Since \(a(\cdot,\cdot)\) is not symmetric we follow the standards techniques and steps based on [17,18 ] but with different fixed point map (\(Step\, 3\) in the proof) and hence more analysis is needed to get the existence and uniqueness result. The resulting problem is an system by blocks of two unknowns (displacement and potential), regarding its structure and the form of blocks we use Gauss elimination technique and then we obtain a Schur complement in the matrix corresponding to displacement subproblem. It is well known (see [16, 21]) that this technique allows a suitable preconditioner for conjugate gradient method employed to solve the resulting symmetric and positive definite system.

To prove the efficiency of this approach, we state suitable numerical fixed point scheme. The convergence is proved basing abstract perturbed problem and fixed point process, Banach fixed point is hence applied. For details concerning the mathematical tools we refer to [22, 23, 24, 25].

The paper is structured as: In Section 2, we present the model of equilibrium process of the elastic piezoelectric body in frictional contact with a non-conductive foundation,we introduce the functional spaces for various quantities, list the assumptions on given data and derive the weak formulation of the problem. In Section, 2.2 we state and prove our main existence and uniqueness result, Theorem 1. The proof of these theorem are carried out in several steps and are based on an abstract result in the study of elliptic variational inequalities and Banach fixed point technique. The successive iterative method is detailed followed by the convergence result in Section 4. In Section 5, we conclude with finite element discretization and we give some numerical experiments by simple example .

2. Problem setting and main results

2.1. Problem setting

The piezoelectric body occupies in its reference configuration (initial configuration) the domain \(\Omega\subset\mathbb{R}^{d}\), \(d=2,3\). We suppose that \(\Omega\) is bounded with a smooth (enough) boundary \(\partial \Omega=\Gamma\). We denote by \(n\) be the outer normal to \(\Gamma\) and summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the variable. The indices take values in \(\{1,\,\cdots,\,d\}\) the summation convention over repeated indices is used.

Below we use \(\mathbb{S}^{d}\) to denote the space of second order symmetric tensors on \(\mathbb{R}^{d}\) while "\(.\)" and \(\| \cdot \|\) will denote the inner product and the Euclidean norm on \(\mathbb{S}^{d}\) and \(\mathbb{R}^{d}\), that is \begin{eqnarray*}u. v=u_{i}v_{i}, \hspace{1cm}\|v\|=(v. v)^{\frac{1}{2}},\hspace{0.5cm}\forall\;\; u,v\in \mathbb{R}^{d},\\ \sigma. \tau=\sigma_{ij}\tau_{ij}, \hspace{1cm}\|\tau\|=(\tau.\tau)^{\frac{1}{2}},\hspace{0.5cm}\forall\;\; \sigma,\tau\in \mathbb{S}^{d}.\end{eqnarray*}

We also use the notations \(u_{n}\) and \(u_\tau\) for the normal and tangential displacement, that is \(u_{n}=u. n\) and \(u_\tau=u-u_{n}n\). Similarly we denote by \(\sigma_{n}\) and \(\sigma_\tau\) the normal and tangential stress tensor given by \(\sigma_{n}=\sigma n. n\), \(\sigma_\tau=\sigma n-\sigma_{n}n\).\\ We introduce the following functional spaces on \(\Omega\); \begin{eqnarray*} &&H= L^{2}(\Omega)^{d}=\left\lbrace u=(u_{i})\; | \;u_{i}\in L^{2}(\Omega)\right\rbrace ,\;\;\; \mathcal{H}= \left\lbrace \sigma = \sigma_{ij},\; \sigma_{ij}=\sigma_{ji}\in L^{2}(\Omega)\right\rbrace, \\ &&H_{1}=\left\lbrace u \in H\; | \;\epsilon(u) \in \mathcal{H}\right\rbrace ,\;\;\;\mathcal{H}_1= \left\lbrace \sigma \in \mathcal{H}\;|\; Div \;\sigma \in H \right\rbrace \end{eqnarray*} endowed with the inner products \begin{eqnarray*}&&(u,v)_{H}=\int_{\Omega} u_{i}v_{i}dx,\hspace{0.5cm} (\sigma,\tau)_{\mathcal{H}}=\int_{\Omega} \sigma_{ij}\tau_{ij}dx.\\ &&(u,v)_{H_{1}}= (u,v)_{H}+(\varepsilon(u),\varepsilon(v))_{\mathcal{H}},\hspace{0.5cm} (\sigma,\; \tau)_{\mathcal{H}_{1}}=(\sigma,\; \tau)_{\mathcal{H}}+(Div\; \sigma,\; Div\; \tau)_{H} .\end{eqnarray*}

The associated norms on the spaces \(H\), \(\mathcal{H}\), \(H_{1}\) and \(\mathcal{H}_{1}\) are denoted by \(\|.\|_{H}\), \(\|.\|_{\mathcal{H}}\), \(\|.\|_{H_{1}}\) and \(\|.\|_{\mathcal{H}_{1}}\) respectively. We recall the well known Green's formula $$ \left( \sigma,\varepsilon(v) \right)_{\mathcal{H}} + \left( Div \; \sigma,v \right)_{H} = \int_{\Gamma} \sigma n. v \; da \;\;\forall\;\; v \in H_{1}, $$ where \(Div \; \sigma = (\sigma_{ij,j})\) and for more details of this formula see [24].
In addition we shall use the following notations. \(u\) is the displacement field , \(\varepsilon (u)=(\varepsilon_{ij}(u))\) to denote the strain tensor, given by \(\varepsilon_{ij}(u)=\frac{1}{2}(u_{i,j}+u_{j,i})\) and \(\sigma =(\sigma_{ij})\) being the stress tensor. Lets \(\varphi\) denote the electric potential, \(E(\varphi)=(E_{i}(\varphi))\) is the electric field, which is defined by \(E_{i}(\varphi)=-\varphi_{,i}\) and \(D=(D_{i})\) is the electric displacement field.
The equilibrium equations are given by where the constitutive relations for the piezoelectric material are: where \(\mathcal{A}=(a_{ijkl})\) is a (fourth-order) elasticity tensor, \(\mathcal{B}=(b_{ijk})\) is the (third-order) piezoelectric tensor, \(\mathcal{B}^{*}\) is the transpose of \(\mathcal{B}\) and \(\beta=(\beta_{ij})\) is the electric permitivity and \(div(D)=D_{i,i}\) (see [26]).
To give the mechanical and electrical boundary conditions, we subdivide \(\Gamma\) into three disjoints measurable parts \(\Gamma_{1},\Gamma_{2},\Gamma_{3}\) such that \(meas(\Gamma_{1})>0\). The body is assumed to be clamped on \(\Gamma_{1}\) and surfaces traction of density \(f_{2}\) act on \(\Gamma_{2}\), on \(\Gamma_{3}\) the body can reaches a frictional contact with the so called foundation (insulating foundation). A second partition of \(\Gamma\), that is \(\Gamma=\Gamma_{3}\cup\Gamma_{a}\cup \Gamma_{b}\). Surface electric charge of density \(q_{2}\) acts on \(\Gamma_{_{b}}\), and the electric potential vanishes on \(\Gamma_{b}\). We use the same symbol \(v\) for the trace of \(v\) on \(\Gamma\). The contact and the Coulomb friction conditions: where \(\mathcal{F}\) is the friction coefficient and \(g\) is the gap between the body and the rigid foundation. The electric contact condition is; To resume, we consider the following problem:

Problem 1. Find the displacement field \(u:\Omega\longrightarrow \mathbb{R}^{d}\) and the electric potential field \(\varphi:\Omega\longrightarrow \mathbb{R}\) such that (2)-(12) hold.

To study of Problem 1 we will assume, under Einstein summation convention, that:

Let us introduce the following Hilbert spaces: $$V=\left\lbrace v\in [H^{1}(\Omega)]^{d}/v=0\text{ on }\Gamma_{1}\right\rbrace, $$ $$W=\left\lbrace \varphi\in H^{1}(\Omega)/\varphi =0\text{ on }\Gamma_{a}\right\rbrace, $$ $$K=\left\lbrace v\in [H^{\frac{1}{2}}(\Gamma_{3})]^{d}/v_{n}\leq g\text{ on }\Gamma_{3}\right\rbrace.$$

If \(u\) and \(\varphi\) are regular functions which satisfy (2)-(10), then we find: $$\int_{\Omega}\mathcal{A}\varepsilon(u)\varepsilon(v)dx+\int_{\Omega}\mathcal{B}^*\nabla \varphi\varepsilon(v) dx=\int_{\Omega}f_{0}vdx+\int_{\Gamma_{2}}f_{2}vd\Gamma+\int_{\Gamma_{3}}(\sigma n).vd\Gamma$$

$$-\int_{\Omega}\mathcal{B}\varepsilon(u)\nabla \psi dx +\int_{\Omega}\beta \nabla \varphi\nabla \psi dx=\int_{\Omega}q_{0}\psi dx-\int_{\Gamma_{b}}q_{2}\psi d\Gamma .$$ Let us introduce the functional space \(\tilde{V}=V\times W\), which is the Hilbert space endowed with the inner product:
\((\tilde{u},\tilde{v})_{\tilde{V}}=(u,v)_{V}+(\varphi,\psi )_{W}\) where \(\tilde{u}=(u,\varphi ),\;\tilde{v}=(v,\psi )\in \tilde{V}\). Let \( a:\tilde{V}\times\tilde{V}\longrightarrow\mathbb{R}\) be the bi-linear form given by: \(\displaystyle{a(\tilde{u},\tilde{v})=\int_{\Omega}\mathcal{A}\varepsilon(u)\varepsilon(v)dx+\int_{\Omega}\mathcal{B}^*\nabla \varphi\varepsilon(v) dx -\int_{\Omega}\mathcal{B}\varepsilon(u)\nabla \psi dx +\int_{\Omega}\beta\nabla \varphi \nabla \psi dx}.\) Moreover, by Riesz's representation theorem, we define \(\tilde{f}\in\tilde{V}\) by: $$(\tilde{f},\tilde{v})_{\tilde{V}}:=\int_{\Omega}f_{0}vdx+\int_{\Gamma_{2}}f_{2}vd\Gamma +\int_{\Omega}q_{0}\psi dx-\int_{\Gamma_{b}}q_{2}\psi d\Gamma.$$ Using the previous tools, we find: $$a(\tilde{u},\tilde{v})=(\tilde{f},\tilde{v})_{\tilde{V}}+\int_{\Gamma_{3}}(\sigma n).vd\Gamma.$$ Since \((\sigma n).v=\sigma_\tau v_\tau+\sigma_{n}v_{n}\), then: $$a(\tilde{u},\tilde{v})=(\tilde{f},\tilde{v})_{\tilde{V}}+\int_{\Gamma_{3}}\sigma_\tau v_\tau+\sigma_{n}v_{n}d\Gamma.$$ Let \(H_\Gamma^{*}\) be the dual space of the space \(H_\Gamma=[H^{\frac{1}{2}}(\Gamma_{3})]^{d}\) and let us define where \(\left\langle \cdot , \cdot\right\rangle_{H_\Gamma^{*},H_\Gamma}\) denotes the duality product between \(H_\Gamma^{*}\) and \(H_\Gamma\), \(K_n\) the set of the normal component of admissible displacement, i.e. \(K_n=\left\lbrace v_n, v_n\leq g\right\rbrace .\)

It is straightforward that \(M_{T,N}\) are two closed convex sets of \(H_\Gamma^{*}\) and \(0_{H_\Gamma^{*}}\in M_{N,T}\). We introduce two dual Lagrange multipliers \(\lambda_N\) and \(\lambda_T \in M\) as follows: $$\left\langle \lambda_N, v\right\rangle_{\Gamma_{3}}:=-\int_{\Gamma_{3}}\sigma_{n}v_{n}d\Gamma ,\hspace{0.5cm}v\in V, \; \text{and}\;\left\langle \lambda_T, v\right\rangle_{\Gamma_{3}}:=-\int_{\Gamma_{3}}\sigma_\tau v_\tau d\Gamma , \hspace{0.5cm}v\in V.$$

We define two bi-linear and continuous forms \(b_{1}\) and \(b_{2}\) for all \(v \in V,\; \delta_{1}, \delta_{2} \in H_\Gamma^{*}\) as follows: $$b_{1}: \tilde{V} \times H_\Gamma^{*} \longrightarrow \mathbb{R},\hspace{1cm}b_{1}(\tilde{v},\delta_{1}):=\left\langle \delta_{1} ,v \right\rangle_{\Gamma_{3}},$$ $$b_{2}:\tilde{V} \times H_\Gamma^{*} \longrightarrow \mathbb{R},\hspace{1cm}b_{2}(\tilde{v},\delta_{2}):=\left\langle \delta_{2} ,v \right \rangle_{\Gamma_{3}}.$$

We see that \(\displaystyle{b_{1}(\tilde{u},\lambda_N)=-\int_{\Gamma_{3}}\sigma_{n}u_{n}d\Gamma}\), and by definition of \(M_{N}\) we have: $$ b_{1}(\tilde{u},\delta_{1}-\lambda_N)\leq 0,\hspace{1cm} \forall\;\; \delta_{1} \in M_{N}. $$ Also, taking into account the definition of \(M_T\), \(\lambda_T\), and the assumption (10), we have: $$b_{2}(\tilde{u},\lambda_T)=\left\langle \lambda_T, u\right\rangle_{\Gamma_{3}}=-\int_{\Gamma_{3}}\sigma_\tau u_\tau d\Gamma.$$ Keeping in mind that the Sobolev trace operator is linear and continuous, it is clear that there exists \(M_{b_{i}}>0\) such that: In addition, using the properties of the Sobolev trace operator it can be shown that there exists \(\alpha_{i}>0\) such that: The following weak formulation of Problem 1 is then obtained :

Problem 2.(Weak formulation of Problem Problem 1) Find \(\tilde{u}\in\tilde{V}\) and \(\lambda=(\lambda_N,\lambda_T)\in M_{N}\times M_{T}(\mathcal{F}\lambda_N)\) such that:

2.2. Main results

In this section we present our main results.

Theorem 1. Assume (13)-(17), then the Problem 2 has unique solution \((\tilde{u},\lambda)\in\tilde{V}\times M\). Moreover if \((\tilde{u}_{1},\lambda)\) and \((\tilde{u}_{2},\beta)\) are two solutions of Problem 2 for given data \(\tilde{f}_{1}\) and \(\tilde{f}_{2}\) respectively, then $$||\tilde{u}_{1}-\tilde{u}_{2}||_{\tilde{V}}+||\lambda -\beta||_{H_\Gamma^{*} \times H_\Gamma^{*}}\leq C(||\tilde{f}_{1}-\tilde{f}_{2}||).$$

We denote \(b_{1}(v,\delta_{1})+b_{2}(v,\delta_{2})=\left\langle \delta_{1}, v\right\rangle_{\Gamma_{3}}+\left\langle \delta_{2}, v\right\rangle_{\Gamma_{3}}=\left\langle \delta_{1}+\delta_{2}, v\right\rangle_{\Gamma_{3}}\), hence there exists \(\alpha >0\) such that where \(b(\cdot,\cdot)= b_{1}(\cdot,\cdot) + b_{2}(\cdot,\cdot):\tilde{V}\times M_{N} \times M_{T}\longrightarrow \mathbb{R}\). Now we introduce a numerical scheme to get numerically the solution of the Problem 2. The scheme is an fixed point iterative and is stated in the following Algorithm 1.

Proposition 1. Let \((u^{\ell},\lambda^\ell)\) be the solution generated by the Algorithm 1, then

The proof of the main results will be presented in the next section.

3. Proof of the main result

Let \(X\) and \(Y\) two Hilbert spaces endowed with the inner product \((\cdot,\cdot)_{X}\) and \((\cdot,\cdot)_{Y}\) respectively and let us consider two bi-linear forms as follows:
\(a(\cdot,\cdot):X\times X\longrightarrow\mathbb{R}\), generally non symmetric, such that and \(b(\cdot,\cdot):X\times Y\times Y\longrightarrow\mathbb{R}\), \(b(v,\lambda)=b_{1}(v,\lambda_N)+b_{2}(v,\lambda_T)\) such that \(\exists \;\;M_{b_{i}}>0 \) such that there exists \(\alpha >0\) such that Now, let \(M=M_N \times M_T \subset Y\times Y\) be closed and convex set that contain \(0_{Y\times Y}\), we consider the following problem:

Problem 3. For given \(f\in X\), find \(u\in X\) and \(\lambda=(\lambda_N,\lambda_T)\in M\) such that:

We have the following result;

Theorem 2. Let \(f \in X\) and assume that (30)-(34) hold. Then, there exists a unique solution \((u,\lambda)\) of \textbf{Problem \(AWF\)} Moreover, if \((u_{1},\lambda)\) and \((u_{2},\gamma)\) are two solutions of the \textbf{Problem \(AWF\)} for given data functions \(f_{1}\in X\) and \(f_{2}\in X\) respectively, then, there exists \(SC>0\) such that:

Proof. We consider the symmetric \(a_{0}(.,.)\) and anti-symmetric \(c(.,.)\) part of \(a(.,.)\) respectively, defined by $$a_{0}:X\times X\longrightarrow \mathbb{R} ,\;\;a_{0}(u,v):=(a(u,v)+a(v,u))/2,\hspace{1cm}u, \forall\;\; v\in X,$$ $$c:X\times X\longrightarrow \mathbb{R} , \;\;c(u,v):=(a(u,v)-a(v,u))/2,\hspace{1cm}u,\forall\;\; v\in X.$$ For given \(0

where $$\mathbf{U}=\left[\begin{array}{c} \textbf{u} \\ \varphi \end{array} \right] ,\; \mathcal{A}=\left[ \begin{array}{cc} \mathbf A & -\mathbf E^\top \\ \mathbf E & \mathbf B \end{array} \right] ,$$ $$b^\ell=\left[ \begin{array}{c} \mathbf b_1 \\ \mathbf b_2\end{array}\right] =\left[ \begin{array}{c} \mathbf f +\mathbf M_{n}\textbf{N}\lambda_{N}^{\ell} +\mathbf M_\tau\textbf{T}\lambda_{T}^{\ell} \\ \mathbf q \end{array}\right] ,$$ \(B(0,-\mathcal{F}\lambda_N^{\ell+1})\) denote the ball of center \(0\) and radius \(-\mathcal{F}\lambda_N^{\ell}>0\) and \(x^+\) denote the non negative part of \(x\) i.e. \(x^+=\max(0,x).\)

Remark 1. In practice, the Algorithm 2 is solved using Tresca friction with slip bound \(S^\ell\) and an fixed point iteration is used to compute the problem with Coulomb friction, i.e. \(S^{\ell+1}=-\mathcal{F}\lambda_N^{\ell}\).

To compute the solution of the system (70), we proceed by the following elimination technique: where \(\mathbf S_c\) is the Schur complement given by \(\mathbf S_c=\mathbf A -\mathbf E^\top \mathbf B^{-1} \mathbf E\). Since the matrices \(\mathbf A\) and \(\mathbf B^{-1}\) are symmetric and positive definite, the suitable method for solving the subsystem (74) is the Conjugate Gradient method (CG). We take profit from the Schur complement \(\mathbf S_c\) to obtain a convenient preconditioner, as discussed in [16] and reference therein the (CG)-preconditioner is \(\mathbf P=\mathbf A\). The preconditioned Conjugate Gradient method for solving the system (74) is stated in the Algorithm 3. Once \(\textbf{u}\) is computed one can compute \(\varphi\) by the explicit formula (73).

4.2. Numerical example

The algorithms are implemented in MATLAB on computer equipped running Windows 10 core i7 of 2.4GHz clock frequency and 6 GB RAM. As example, the domain consists of two-dimensional rectangular domain \(\Omega=(0,\;2)\times (0,\;1)\) as in the Figure 1, with boundaries \(\Gamma_D =\{0\}\times \left[0,\;1\right]\cup \{2\}\times \left[0,\;1\right]\), \(\Gamma_3 = \left[0,\;2\right]\times \{0\}\) and \(\Gamma_N =\left[0,\;2\right]\times \{1\}\). External body force and charge are \(f=0\) and \(q=0\), respectively. On \(\Gamma_D\) the displacements and the electric potential are prescribed, i.e., \(\textbf{u}=0\) and \(\varphi=0\) on \(\Gamma_D\). On \(\Gamma_N\), non-homogeneous Neumann boundary conditions are prescribed \(f_0=\sigma (u).n=-2\). On \(\{1\}\times(0,\;1)\) the homogeneous Neumann boundary condition is applied (\(\sigma n=0\) and \(Dn=0\)). For seek of simplicity, the normalized gap between \(\Gamma_3\) and the foundation is \(g(\mathbf x)=0\) and the friction coefficient is \(\mathcal{F}=0.6\). The mesh is generated by using Matlab function "kmg.m" built by the author in [29].
The deformed configuration is showed in the Figure 2 and the the contour plot of the electric potential distribution are showed in the Figure 3. The Figure 4 show the Lagrange multipliers \(\lambda_{T,N}\) in the contact zone \(\Gamma_3\) corresponding to the problem with Coulomb friction condition. The Figures 5 and 6 show the Lagrange multipliers \(\lambda_{T,N}\) with different choices of loads acting on \(\Gamma_2\), it is clear that the slide occurs when the load is important (large enough). The performance of the algorithm is presented in the Table 1, as showed the number of iterations is independent of mesh refinement and the time of execution is significant.

Table 1. Performance of the algorithms.

Mesh size \(h\)	1/32	1/64	1/128	1/256
Number of iterations of FP	3	4	4	4
Number of iterations of UA	39	39	39	39
Number of iterations of PCG	2	2	2	2
CPU time (seconds)	0.1341	2.0863	21.1882	238.4585

5. Conclusion

We have investigated numerical analysis of a model describing the process of contact between a piezoelectric body and non-conductive foundation. The behavior of the material is modeled with a electro-elastic constitutive law. The contact is formulated by Signorini conditions and Coulomb friction. In coming works, more general problem with non linear constitutive equation and non monotone friction will be treated. This is may be handled by fixed point iteration and hemivariational inequalities [30]migorski2012nonlinear}.

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

Certainly, the Lyapunov direct method has been, for more than 100 years, the efficient tool for the study of stability properties of ordinary, functional, partial differential and difference equations. Nevertheless, the application of this method to problems of stability in differential and difference equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms ([1, 2, 3, 4, 5, 6, 7, 8 9, 10, 11, 12, 13, 14, 15, 16]). Recently, Burton, Furumochi, Zhang, Raffoul, Islam, Yankson and others have noticed that some of these difficulties vanish or might be overcome by means of fixed point theory (see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]). The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov's direct method. The conditions of the former are often averages but those of the latter are usually pointwise (see [1]). In this paper, we consider the following mixed type neutral difference equation with an assumed initial condition where \(\psi :[m\left( t_{0}\right) ,t_{0}]\cap \mathbb{Z}\rightarrow \mathbb{\mathbb{R}}\) is a bounded sequence and for \(t_{0}\geq 0\) \begin{equation*} m\left( t_{0}\right) =\inf \{\sigma _{i}(s):s\geq t_{0},\ i=1,...k\}. \end{equation*} Here \(\Delta \) denotes the forward difference operator \(\Delta x(t)=x(t+1)-x(t)\) for any sequence \(\left\{ x\left( t\right) ,\ t\in \mathbb{ Z}^{+}\right\} \). For more details on the calculus of difference equations, we refer the reader to [11] and [24]. Throughout this paper, we assume that \(a\), \(b_{i}\) and \(c_{j}\) are bounded sequences, and \(\tau \), \( \sigma _{i}\) and \(\tau _{j}\) are non-negative sequences such that \begin{eqnarray*} \tau \left( t\right) &\rightarrow &\infty \text{ as }t\rightarrow \infty ,\ \tau \left( t\right) \geq t,\ t\geq t_{0}, \\ \sigma _{i}\left( t\right) &\rightarrow &\infty \text{ as }t\rightarrow \infty ,\ i=1,...,k,\ \sigma _{i}\left( t\right) \leq t,\ t\geq t_{0}, \\ \tau _{j}\left( t\right) &\rightarrow &\infty \text{ as }t\rightarrow \infty ,\ j=1,...,l,\ \tau _{j}\left( t\right) \geq t,\ t\geq t_{0}. \end{eqnarray*} Equation (1) can be viewed as a discrete analogue of the mixed type neutral differential equation; In [25], Bicer investigated (3) and obtained the asymptotic behavior of solutions. Our purpose here is to show the asymptotic behavior of solutions for (1). An asymptotic stability theorem with a necessary and sufficient condition is proved by using the contraction mapping theorem. For details on contraction mapping principle we refer the reader to [33] . An example is given to illustrate our main results.

2. Main results

Theorem 1. Let \(a\), \(b_{i}\) and \(c_{j}\) non positive sequences. Assume that the following inequality has a nonnegative solution \begin{equation*} -a\left( t\right) \lambda \left( \tau \left( t\right) \right) \prod_{u=t}^{\tau \left( t\right) -1}\left( 1-\lambda \left( u\right) \right) -\sum_{i=1}^{k}b_{i}\left( t\right) \prod_{u=t}^{\sigma _{i}\left( t\right) -1}\left( 1-\lambda \left( u\right) \right) \notag -\sum_{j=1}^{l}c_{j}\left( t\right) \prod_{u=t}^{\tau _{j}\left( t\right) -1}\left( 1-\lambda \left( u\right) \right) \leq\lambda \left( t\right) ,\ t\geq t_{0}, \label{3} \end{equation*} with \(\lambda \left( t\right) < 1\). Then, (1) has a positive solution.

Proof. Let \(\lambda _{0}\) be a nonnegative solution of (\ref{3}). Set \begin{equation*} \lambda _{n}\left( t\right) =\left\{ \begin{array}{l} \lambda _{n-1}\left( t\right) ,\text{ \ \ if }m\left( t_{0}\right) \leq t\leq t_{0}, \\ -a\left( t\right) \lambda _{n-1}\left( \tau \left( t\right) \right) \prod\limits_{u=t}^{\tau \left( t\right) -1}\left( 1-\lambda _{n-1}\left( u\right) \right) -\sum\limits_{i=1}^{k}b_{i}\left( t\right) \prod\limits_{u=t}^{\sigma _{i}\left( t\right) -1}\left( 1-\lambda _{n-1}\left( u\right) \right) \\ -\sum\limits_{j=1}^{l}c_{j}\left( t\right) \prod\limits_{u=t}^{\tau _{j}\left( t\right) -1}\left( 1-\lambda _{n-1}\left( u\right) \right) ,\text{ }t\geq t_{0}, \end{array} \right. \end{equation*} for \(n=1,2,...\). Then, by (\ref{3}), we get \begin{equation*} \lambda _{0}\left( t\right) \geq -a\left( t\right) \lambda _{0}\left( \tau \left( t\right) \right) \prod_{u=t}^{\tau \left( t\right) -1}\left( 1-\lambda _{0}\left( u\right) \right) -\sum_{i=1}^{k}b_{i}\left( t\right) \prod_{u=t}^{\sigma _{i}\left( t\right) -1}\left( 1-\lambda _{0}\left( u\right) \right) -\sum_{j=1}^{l}c_{j}\left( t\right) \prod_{u=t}^{\tau _{j}\left( t\right) -1}\left( 1-\lambda _{0}\left( u\right) \right) =\lambda _{1}\left( t\right) . \end{equation*} Then, we obtain \(\lambda _{0}(t)\geq \lambda _{1}(t)\geq ...\geq \lambda _{n}(t)\geq 0\). So, there exists a pointwise limit \(\lambda (t)=\lim\limits_{n\rightarrow \infty }\lambda _{n}(t)\). So, from the Lebesgue convergence theorem, we obtain \begin{equation*} \lambda \left( t\right) =-a\left( t\right) \lambda \left( \tau \left( t\right) \right) \prod_{u=t}^{\tau \left( t\right) -1}\left( 1-\lambda \left( u\right) \right) -\sum_{i=1}^{k}b_{i}\left( t\right) \prod_{u=t}^{\sigma _{i}\left( t\right) -1}\left( 1-\lambda \left( u\right) \right) -\sum_{j=1}^{l}c_{j}\left( t\right) \prod_{u=t}^{\tau _{j}\left( t\right) -1}\left( 1-\lambda \left( u\right) \right) . \end{equation*} Hence, \begin{equation*} x\left( t\right) =\left\{ \begin{array}{l} \lambda \left( t\right) ,\text{ \ \ if }m\left( t_{0}\right) \leq t\leq t_{0}, \\ \lambda \left( t_{0}\right) \prod\limits_{u=t_{0}}^{t-1}\left( 1-\lambda \left( u\right) \right) ,\ t\geq t_{0}, \end{array} \right. \end{equation*} is a positive solution of (1).

Theorem 2. Let \(a\), \(b_{i}\) and \(c_{j}\) be non positive sequences and let \( \Delta a(t)>0\), \(a\left( t_{0}\right) \neq -\infty \). If \begin{equation*} \sum_{u=t_{0}}^{\infty }\sum_{j=1}^{l}c_{j}\left( u\right) =-\infty , \end{equation*} and \(x\) is a eventually positive solution of (1), then \( x(t)\rightarrow \infty \) as \(t\rightarrow \infty \).

Proof. Assume that \(x(t)>0\) for \(t\geq T_{1}\). Choose \(T\geq T_{1}\) such that \( T_{1}\leq \inf \{\sigma _{i}(s):s\geq T,\ i=1,...,k\}\). Then \(\Delta x(t)+a(t)\Delta x(\tau (t))\geq 0\), for \(t\geq T\), \begin{equation*} \Delta x\left( t\right) +a\left( t\right) \Delta x\left( \tau \left( t\right) \right) =-\sum_{i=1}^{k}b_{i}\left( t\right) x\left( \sigma _{i}\left( t\right) \right) -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) , \end{equation*} and \begin{equation*} \Delta \left[ a(t)x\left( \tau \left( t\right) \right) \right] =a\left( t\right) \Delta x\left( \tau \left( t\right) \right) +\Delta a\left( t\right) x\left( \tau \left( t+1\right) \right) , \end{equation*} that is \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] -\Delta a\left( t\right) x\left( \tau \left( t+1\right) \right) \geq -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) . \end{equation*} From this, we can write \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] \geq -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) , \end{equation*} so \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] \geq -x\left( T\right) \sum_{j=1}^{l}c_{j}\left( t\right) , \end{equation*} which implies \begin{equation*} x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \geq a(t_{0})x\left( \tau \left( t_{0}\right) \right) -x\left( T\right) \sum_{u=t_{0}}^{t-1}\sum_{j=1}^{l}c_{j}\left( u\right) . \end{equation*} So, we get \begin{equation*} x\left( t\right) \geq a(t_{0})x\left( \tau \left( t_{0}\right) \right) -x\left( T\right) \sum_{u=t_{0}}^{t-1}\sum_{j=1}^{l}c_{j}\left( u\right) . \end{equation*} Then \(x(t)\rightarrow \infty \) as \(t\rightarrow \infty \).

Theorem 3. Let \(a(t)>0\), \(b_{i}\) and \(c_{j}\) be nonnegative sequences and let \(\Delta a(t)< 0\), \(a\left( t_{0}\right) \neq \infty \). If \begin{equation*} \sum_{u=t_{0}}^{\infty }\sum_{j=1}^{l}c_{j}\left( u\right) =\infty , \end{equation*} and \(x\) is a eventually positive solution of (1), then \( x(t)\rightarrow 0\) as \(t\rightarrow \infty \).

Proof. For \(t\geq T_{1}\), since \(x(t)>0\) we Choose \(T\geq T_{1}\) such that \( T_{1}\leq \inf \{\sigma _{i}(s):s\geq T,\ i=1,...,k\}\). Then \(\Delta x(t)+a(t)\Delta x(\tau (t))\leq 0\), for \(t\geq T\), and \begin{equation*} \Delta x\left( t\right) +a\left( t\right) \Delta x\left( \tau \left( t\right) \right) \leq -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) , \end{equation*} that is \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] -\Delta a\left( t\right) x\left( \tau \left( t+1\right) \right) \leq -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) . \end{equation*} From this, we can write \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] \leq -\sum_{j=1}^{l}c_{j}\left( t\right) x\left( \tau _{j}\left( t\right) \right) , \end{equation*} so \begin{equation*} \Delta \left[ x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \right] \leq -x\left( T\right) \sum_{j=1}^{l}c_{j}\left( t\right) , \end{equation*} which implies \begin{equation*} x\left( t\right) +a(t)x\left( \tau \left( t\right) \right) \leq a(t_{0})x\left( \tau \left( t_{0}\right) \right) -x\left( T\right) \sum_{u=t_{0}}^{t-1}\sum_{j=1}^{l}c_{j}\left( u\right) . \end{equation*} So, we get \begin{equation*} x\left( t\right) \leq a(t_{0})x\left( \tau \left( t_{0}\right) \right) -x\left( T\right) \sum_{u=t_{0}}^{t-1}\sum_{j=1}^{l}c_{j}\left( u\right) . \end{equation*} Since \(x(t)>0\), we get a contradiction.Then \(x(t)\rightarrow 0\) as \( t\rightarrow \infty \).

Now, we investigate the asymptotic behavior of solutions of (1), free of the sign of the coefficients. During the process of inverting (1), an summation by parts will have to performed on the term involving \(\Delta x(\tau (t))\).

Lemma 1. A sequence \(x\) is a solution of (1)--(2) if and only if

for \(t\geq t_{0}\), where \begin{equation*} B\left( t\right) =\sum_{i=1}^{k}b_{i}\left( t\right) +\sum_{j=1}^{l}c_{j}\left( t\right) ,\ 0 < B\left( t\right) < 1, \end{equation*} and

Proof. Since \begin{eqnarray*} x\left( \tau _{j}\left( t\right) \right) =x\left( t+1\right) +\sum_{u=t+1}^{\tau _{j}\left( t\right) -1}\Delta x\left( u\right) , \text{ and } x\left( \sigma _{i}\left( t\right) \right) =x\left( t+1\right) +\sum_{u=t+1}^{\sigma _{i}(t)-1}\Delta x\left( u\right). \end{eqnarray*} We can rewrite (1) as

Multiplying both sides of (6) with \(\prod\limits_{s=t_{0}}^{t}\left( 1-B(s\right) )^{-1}\), by summing from \(t_{0}\) to \(t-1\), we obtain \begin{align*} & \underset{r=t_{0}}{\overset{t-1}{\sum }}\Delta \left[ \prod \limits_{s=t_{0}}^{r-1}\left( 1-B(s\right) )^{-1}x\left( r\right) \right] =-\sum_{r=t_{0}}^{t-1}\prod_{s=t_{0}}^{r}\left( 1-B(s\right) )^{-1}a\left( r\right) \Delta x\left( \tau \left( r\right) \right) -\sum_{r=t_{0}}^{t-1}\prod_{s=t_{0}}^{r}\left( 1-B(s\right) )^{-1}B\left( r\right) x\left( r+1\right) \\ &\,\,\, -\sum_{r=t_{0}}^{t-1}\prod_{s=t_{0}}^{r}\left( 1-B(s\right) )^{-1}\sum_{i=1}^{k}b_{i}\left( t\right) \sum_{u=t+1}^{\sigma _{i}(r)-1}\Delta x\left( u\right) -\sum_{r=t_{0}}^{t-1}\prod_{s=t_{0}}^{r}\left( 1-B(s\right) )^{-1}\sum_{j=1}^{l}c_{j}\left( t\right) \sum_{u=t+1}^{\tau _{j}(r)}\Delta x\left( u\right) . \end{align*} By dividing both sides of the above expression by \(\prod \limits_{s=t_{0}}^{t-1}\left( 1-B(s\right) )^{-1}\) we get By performing an summation by parts, we get But, \begin{align*} \sum_{r=t_{0}}^{t-1}\Delta \left[ \prod_{s=r}^{t-1}\left( 1-B(s\right) )a\left( r-1\right) \right] x\left( \tau \left( r\right) \right) =\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s\right) )\left[ a(r)-a\left( r-1\right) \left( 1-B(r)\right) \right] x\left( \tau \left( r\right) \right) \\ & =\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s\right) )h\left( r\right) x\left( \tau \left( r\right) \right) , \end{align*} where \(h\) is given by (5). We obtain (4) by replacing (8) into (7). Since each step is reversible, the converse follows easily. This completes the proof.

Theorem 4. Assume that \(0< B\left( t\right) < 1\) and the following conditions hold

and Then for each initial condition (2), every solution of (1) converges to zero.

Proof. Let \(x\in C([m\left( t_{0}\right) ,\infty )\cap \mathbb{Z})\) is the space of all bounded sequences and \( M=\{x\in C([m\left( t_{0}\right) ,\infty )\cap \mathbb{Z}):x(t)\rightarrow 0 \text{ as }t\rightarrow \infty \}, \) be a closed subspace. Then \((M,\left\Vert .\right\Vert )\) is a Banach space with the norm \( \left\Vert x\right\Vert =\sup_{t\geq m\left( t_{0}\right) }\left\vert x\left( t\right) \right\vert . \) Define the operator \(\phi :M\rightarrow M\) by

It is clear that for \(x\in M\), \(\phi x\) is bounded. Now, we will show that \( \phi \) is a contraction. Let \(x\) and \(y\) be two bounded sequences on \( [m\left( t_{0}\right) ,\infty )\cap \mathbb{Z}\) and satisfying same initial condition (2). Then for \(t\geq t_{0}\), we get \begin{align*} & \left\vert \left( \phi x\right) \left( t\right) -\left( \phi y\right) \left( t\right) \right\vert \leq \left\vert a(t-1)\right\vert \left\vert x\left( \tau \left( t\right) \right) -y\left( \tau \left( t\right) \right) \right\vert +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \left\vert x\left( \tau \left( r\right) \right) -y\left( \tau \left( r\right) \right) \right\vert \\ & +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert x\left( r+1\right) -y\left( r+1\right) \right\vert \left\vert B\left( r\right) \right\vert +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \sum_{i=1}^{k}\left\vert b_{i}\left( r\right) \right\vert \left\vert x\left( \sigma _{i}\left( r\right) \right) -y\left( \sigma _{i}\left( r\right) \right) \right\vert \\ & +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \sum_{j=1}^{l}\left\vert c_{j}\left( r\right) \right\vert \left\vert x\left( \tau _{j}\left( r\right) \right) -y\left( \tau _{j}\left( r\right) \right) \right\vert \leq \beta \left\Vert x-y\right\Vert . \end{align*} Thus, the operator \(\phi \) has a unique fixed point in \(M\), which solves ( 1). Now, we will show that, \(\left( \phi x\right) \left( t\right) \rightarrow 0\) as \(t\rightarrow \infty \). Actually, for \(x\in M\), we have Note that by (9), \begin{equation*} \left\vert \left( x\left( t_{0}\right) +a\left( t_{0}-1\right) x\left( \tau \left( t_{0}\right) \right) \right) \right\vert \prod_{s=t_{0}}^{t-1}\left( 1-B(s)\right) \rightarrow 0\text{ as }t\rightarrow \infty . \end{equation*} Moreover, since \(x(t)\rightarrow 0\) as \(t\rightarrow \infty \), for each \( \varepsilon >0\), there exists \(T_{1}>t_{0}\) such that \(u\geq T_{1}\) implies that \(\left\vert x(\tau (u))\right\vert < \frac{\varepsilon }{2}\). Thus, for \( t\geq T_{1}\), the third term \(I_{3}\) in (12) satisfies \begin{align*} I_{3}& =\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \left\vert x\left( \tau \left( r\right) \right) \right\vert \\&\leq \sum_{r=t_{0}}^{T_{1}-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \left\vert x\left( \tau \left( r\right) \right) \right\vert +\sum_{r=T_{1}}^{t-1}\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \left\vert x\left( \tau \left( r\right) \right) \right\vert\\& \leq \sum_{r=t_{0}}^{T_{1}-1}\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \left\vert x\left( \tau \left( r\right) \right) \right\vert +\frac{\varepsilon }{2}\sum_{r=T_{1}}^{t-1}\sum_{r=t_{0}}^{t-1} \prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert \\& \leq \frac{\varepsilon }{2}+\beta \frac{\varepsilon }{2}\\&\leq \varepsilon . \end{align*} Thus \(I_{3}\rightarrow 0\) as \(t\rightarrow \infty \). By a similar technique, we can prove that the rest of terms in (12) tend zero as \(t\rightarrow \infty \). Therefore \((\phi x)(t)\rightarrow 0\) as \(t\rightarrow \infty \). This completes the proof.

Theorem 5. Suppose that \(0< B\left( t\right) < 1\). If all solutions of (1) converge to zero, then (9) holds.

Proof. Suppose that (9) does not holds. That is,

So, from (13), we can write \(\delta \neq 0\). Then, there exists a sequence \(\left\{ t_{n}\right\} \) approaching \(\infty \), such that \begin{equation*} \prod\limits_{s=t_{0}}^{t_{n}-1}\left( 1-B(s)\right) \rightarrow \delta \text{ as }n\rightarrow \infty . \end{equation*} For \(x(t_{0})\neq 0\), let \(x\) be a solution. Then, From Lemma 1, \(x(t_{n})\) satisfies (4). On the other hand, we know that Since all solutions tend zero, from (4), (14) and (15), we get \begin{equation*} \lim_{n\rightarrow \infty }x\left( t_{n}\right) =\left\vert \left( x\left( t_{0}\right) +a\left( t_{0}-1\right) x\left( \tau \left( t_{0}\right) \right) \right) \right\vert \delta \neq 0, \end{equation*} which contradicts all solutions of (1) converge to zero. The proof is completed.

We end the paper with the following example.

Example 1. consider the mixed type neutral difference equation

with an assumed initial condition \begin{equation*} x(t)=\psi (t)\text{ for }t\in \lbrack m\left( t_{0}\right) ,t_{0}]\cap \mathbb{Z}, \end{equation*} where \(t_{0}=0\), \(m\left( t_{0}\right) =-2\), \(\psi (t)=t/3\), \(a\left( t\right) =\dfrac{1}{3^{t+2}}\), \(b_{1}\left( t\right) =1-\dfrac{1}{2^{t}}\), \( c_{1}\left( t\right) =\dfrac{1}{2^{t+1}}\), \(\tau \left( t\right) =3t/2\), \( \sigma _{1}\left( t\right) =t/2-2\), \(\tau _{1}\left( t\right) =5t/2\). We have \begin{equation*} B\left( t\right) =1-\dfrac{1}{2^{t+1}},\ \prod\limits_{s=0}^{t-1}\left( 1-B(s)\right) =\prod\limits_{s=0}^{t-1}\dfrac{1}{2^{s+1}}\rightarrow 0\text{ as }t\rightarrow \infty , \end{equation*} and \begin{align*} & \left\vert a(t-1)\right\vert +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert h\left( r\right) \right\vert +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert B\left( r\right) \right\vert \\ & +\sum_{r=t_{0}}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert b_{1}\left( r\right) \right\vert +\sum_{r=0}^{t-1}\prod_{s=r+1}^{t-1}\left( 1-B(s)\right) \left\vert c_{1}\left( r\right) \right\vert \\ & =\dfrac{1}{3^{t+1}}+\sum_{r=0}^{t-1}\prod_{s=r+1}^{t-1}\dfrac{1}{2^{s+1}} \left\vert \frac{1}{3^{r+2}}-\frac{1}{3^{r+1}\times 2^{r+1}}\right\vert +\sum_{r=0}^{t-1}\prod_{s=r+1}^{t-1}\dfrac{1}{2^{s+1}}\left( 1-\dfrac{1}{ 2^{r+1}}\right) \\ & +\sum_{r=0}^{t-1}\prod_{s=r+1}^{t-1}\dfrac{1}{2^{s+1}}\left( 1-\dfrac{1}{ 2^{r}}\right) +\sum_{r=0}^{t-1}\prod_{s=r+1}^{t-1}\dfrac{1}{2^{s+1}}\times \dfrac{1}{2^{r+1}} \simeq 0.722< 1. \end{align*} Thus all the conditions of Theorem 4 are satisfied and every solution of (16) converges to zero.

3. Concluding remarks

In this article, a neutral mixed type difference equation is considered. The asymptotic behavior of solutions is obtained with a necessary and sufficient condition by using fixed point theorems. The results are supported with a suitable illustrative example.

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 and formulation of the result

Let where \(v_0\) is a smooth functions rapidly decaying with all its derivatives as \(t\to \infty\), \(v_0(t)=0\) if \(t< 0\). The integral in (1) diverges in the classical sense. Our result can be formulated as follows.

Theorem 1. The solution to equation (1) for \(\lambda=-\frac1 4\) does exist, is unique and less singular than \(t^{-\frac 5 4}\) as \(t\to 0\).

Proof. We define the integral in (1) as a convolution of the distribution \(t^{\lambda-1}\) and \(v\). The space of the test functions for this distribution is the space \(\mathcal{K}:=C^\infty_0(R_+)\) of compactly supported on \(R_+:=[0, \infty)\) infinitely differentiable functions \(\phi(t)\) defined on \(R_+:=[0,\infty)\). The topology on this space is defined by countably many norms \(\sup_{t\ge 0}t^m|D^p\phi(t)|\). A sequence \(\phi_n(t)\) converges to \(\phi(t)\) in \(\mathcal{K}\) if and only if all the functions \(\phi_n(t)\) have compact support on an interval \([a,b]\), \(a>0\), \(b< \infty\) and \(\phi_n\) converges on this interval to \(\phi\) in every of the above norms. Let us check that \(t^{\lambda-1}:=t_+^{\lambda-1}\) is a distribution on \(\mathcal{K}\) for \(\lambda< 0\), i.e., a linear bounded functional on \(\mathcal{K}\). Let \(\phi_n\in \mathcal{K}\) and \(\phi_n \to \phi\) in \(\mathcal{K}\). If \(\lambda< 0\) then \(\max_{t\in [a,b]}t^{\lambda -1}\le a^{\lambda-1}+b^{\lambda-1}\). Thus, $$|\int_0^\infty t^{\lambda-1}\phi_n(t)dt|\le [a^{\lambda-1}+b^{\lambda-1}] \int_0^\infty |\phi_n(t)|dt,$$ where \(a>0\) and \(b< \infty\). Since \(\phi_n \to \phi\) in \(\mathcal{K}\), we have $$\int_0^\infty |\phi_n(t)|dt\to \int_a^b | \phi|dt.$$ So, the integral \(\int_0^\infty t^{\lambda-1}\phi(t)dt\) is a bounded linear functional on \(\mathcal{K}\) and \(t^{\lambda-1}\) is a distribution on the set of the test functions \(\mathcal{K}\) for \(\lambda\neq 0,-1,-2,...\). The integral in (1) is the convolution \(t^{\lambda-1}\star v\). This convolution is defined for any distributions on the dual to \(\mathcal{K}\) space \(\mathcal{K}'\). This is done in [3], p.57. For another space of the test functions \(K=C^\infty_0(R)\) this is done in [2], p.135. It is known, see e.g. [1], p.39, that

where \(L\) is the Laplace transform, and \(f,h\) are distributions on \(\mathcal{K}\). Let us calculate \(L(t^{\lambda-1})\) using the new variable \(s=pt\): This formula is valid classically for \(Re \lambda>0\). By analytic continuation with respect to \(\lambda\) it is valid for all complex \(\lambda\neq 0,-1,-2,....\). Applying the Laplace transform to (1) and using formulas (2) and (3), one gets Let us assume that \(\lambda=-\frac 1 4\) so that \(\lambda-1=-\frac 5 4\). This value appears in the solution to the Navier-Stokes problem in \(R^3\), see [3], p.53. If \(\lambda=-\frac 1 4\), then equation (4) yields where we have used the relation \(\Gamma(-\frac 1 4)=-4\Gamma(3/4)\), which follows from the known formula \(\Gamma(z+1)=z\Gamma(z)\) with \(z=-\frac 1 4\). Thus, So, the solution \(v\) does exist and is unique. Moreover, \(v\) is not a distribution if \(v_0\) is smooth and rapidly decaying when \(t\to \infty\). This follows from the known results concerning the relation of asymptotic of \(L(f)(p)\) and \(f(t)\) for \(p\to \infty\) and \(t\to 0\) and for \(p\to 0\) and \(t\to \infty\), see [1], p.41. Namely, if \(f(t)\sim At^{\nu}\) as \(t\to 0\), then \(L(f)(p)\sim A\Gamma(\nu +1)p^{-\nu -1}\) as \(p\to \infty\), \(\nu\neq -1.-2,....\). If \(f(t)\sim At^{\nu}\) as \(t\to \infty\) then \(L(f)(p)\sim A\Gamma(\nu +1)p^{-\nu -1}\) as \(p\to 0\). Since \(p^{1/4}\to 0\) as \(p\to 0\), the asymptotic of \(v(t)\) as \(t\to \infty\) is of the same order as that of \(v_0\). As \(p\to \infty\) the singularity of \(v(t)\) as \(t\to 0\) is of the order less than that of \(t^{-\frac 5 4}\). For example, assume that \(v_0\) is continuous as \(t\to 0\). Then we can take \(\nu\ge 0\). Consider the worst case \(\nu=0\). In this case \(L(v)(p)\) is of the order \(p^{-1-\frac 1 4}=p^{-\frac 5 4}\). Therefore \(v\sim t^{-\frac 1 4}\) as \(t\to 0\). This is an integrable singularity. Thus, \(v\) is less singular as \(t\to 0\) than the distribution \(t^{-\frac 5 4}\). Theorem 1 is proved.

In [4] another result, similar to the one in this paper, is proved. In Zbl 07026037 in a review of paper [5] there is an erroneous claim that the proof in [5] is incorrect. The reviewer erroneously claims that the integral (1) diverges and therefore it is equal to infinity. While this is true classically it is not true in the sense of distributions. Therefore, the claim of the reviewer that the proof in [5] is not correct is false. The reviewer claims that \(\Phi_{-\frac 1 4}\) is not equal to \(\frac {t_+^{-\frac 5 4}}{\Gamma(-\frac 1 4)}\). This is not true if the space of the test functions is \(\mathcal{K}\) (although it is true if the space of the test functions is \(K\)).

2. Concluding remark.

Historically it is well known that equation (1) can be solved explicitly by the Laplace transform if \(\lambda>0\) and the function \(1-L(t^{\lambda -1})\neq 0\). To our knowledge, for \(\lambda< 0\) there were no results concerning the solvability of equation (1). The author got interested in (1) in the case \(\lambda=-\frac 1 4\) in connection with the millennium problem about unique global solvability of the Navier-Stokes problem (NSP) in \(R^3\) which was solved in [5], see also [3] Chapter 5.

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

We start with the following compact form of fractional integrals:

Definition 1.[1] Let \(f:[a,b]\rightarrow\mathbb{R}\) be an integrable function. Also let \(g\) be an increasing and positive function on \((a, b]\), having a continuous derivative \(g^{\prime}\) on \((a,b)\). The left-sided and right-sided fractional integrals of a function \(f\) with respect to another function \(g\) on \([a, b]\) of order \(\mu\in\mathbb{C}\,\,(\mathcal{R}(\mu) > 0)\) are defined as:

and where \(\Gamma(.)\) is the gamma function.

A \(k\)-fractional analogue of above definition is given as follows:

Definition 2.[2] Let \(f:[a,b]\rightarrow\mathbb{R}\) be an integrable function. Also let \(g\) be an increasing and positive function on \((a, b]\), having a continuous derivative \(g^{\prime}\) on \((a,b)\). The left-sided and right-sided \(k\)-fractional integral operators, \(k>0\) of a function \(f\) with respect to another function \(g\) on \([a, b]\) of order \(\mu\in\mathbb{C}\,\,(\mathcal{R}(\mu) > 0)\) are defined as:

and where \begin{equation*} \Gamma_{k}(\mu)=\int_{0}^{\infty}t^{\mu-1}e^{\frac{-t^{k}}{k}}dt \end{equation*} is the \(k\)-gamma function.

Fractional integral operators which have been studied in recent decades are special cases of generalized classical integral operators of Riemann-Liouville type which are defined in (1) and (2). Instead of studying the derivations of recently defined fractional integral operators from (3) and (4), the authors of these decades have derived them independently, see [1, 3, 4, 5, 6, 7, 8]. Remark 1 provides derivations of these fractional integrals from (1) and (2) and along with their \(k\)-analogues (3) and (4). A detail study of fractional integrals associated with (3) and (4), which have been investigated by the authors of recent years, is summarized in the following remark:

Remark \label{rem1} Fractional integrals elaborated in (3) and (4) particularly produce several known fractional integrals corresponding to different settings of \(k\) and \(g\).

1. For \(k=1\) (3) and (4) fractional integrals coincide with (1) and (2) fractional integrals.
2. By taking \(g\) as identity function (3) and (4) fractional integrals coincide with \(k\)-fractional Riemann-Liouville integrals defined by Mubeen et al. in [7].
3. For \(k = 1\), along with \(g\) as identity function (3) and (4) fractional integrals coincide with Riemann-Liouville fractional integrals [1].
4. For \(k = 1\) and \(g(x)=\frac{x^\rho}{\rho}\), \(\rho>0\), (3) and (4) produce Katugampola fractional integrals defined by Chen et al. in [3].
5. For \(k = 1\) and \(g(x)=\frac{x^{\tau+s}}{\tau+s}\) , (3) and (4) produce generalized conformable fractional integrals defined by Khan et al. in [6].
6. If we take \(g(x)=\frac{(x-a)^s}{s}\), \(s>0\) in (3) and \(g(x)=-\frac{(b-x)^s}{s}\), \(s>0\) in (4), then conformable \((k,s)\)-fractional integrals are achieved as defined by Habib et al. in [4].
7. If we take \(g(x)=\frac{x^{1+s}}{1+s}\), then conformable fractional integrals are achieved as defined by Sarikaya et al. in [8].
8. If we take \(g(x)=\frac{(x-a)^s}{s}\), \(s>0\) in (3) and \(g(x)=-\frac{(b-x)^s}{s}\), \(s>0\) in (4) with \(k=1\), then conformable fractional integrals are achieved as defined by Jarad et al. in [5].

Moreover, the authors also have constructed various fractional integral operators by using Mittag-Leffler function and its generalizations. Recently, Andri\'c et al. studied an extended generalized Mittag-Leffler function and the associated fractional integral operators in [9]. The extended generalized fractional integrals defined in [9], produce all fractional integrals containing Mittag-Leffler function defined in [10, 11, 12, 13].

Definition 3.[9] Let \(\omega,\mu,\alpha,l,\gamma,c\in {C}\), \(\Re(\mu),\Re(\alpha),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\delta>0\) and \(0< \nu\leq\delta+\Re(\mu)\). Let \(f\in L_{1}[a,b]\) and \(x\in[a,b].\) Then the generalized fractional integral operators \(\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,\nu,c}f \) and \(\epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,\nu,c}f\) are defined by:

and where is the extended generalized Mittag-Leffler function, \begin{equation*} \beta_{p}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}e^{-\frac{p}{t(1-t)}}dt \end{equation*} and \((c)_{n\nu}=\frac{\Gamma(c+n\nu)}{\Gamma(c)}\).

Remark 2, The settings of \(\omega,\nu,\delta, l,p, \gamma\) into generalized Mittag-Leffler function obtain the following consequences:

1. Setting \(p=0\), (5) and (6) reduce to the fractional integral operators defined by Salim-Faraj in [10].
2. Setting \(l=\delta=1\), (5) and (6) reduce to the fractional integral operators defined by Rahman et al. in [11].
3. Setting \(p=0\) and \(l=\delta=1\), (5) and (6) reduce to the fractional integral operators defined by Shukla-Prajpati in [12].
4. Setting \(p=0\) and \(l=\delta= \nu=1\), (5) and (6) reduce to the fractional integral operators defined by Prabhakar in [13].
5. Setting \(p=\omega=0\), (5) and (6) reduce to the left-sided and right-sided Riemann-Liouville fractional integrals.

The extended generalized Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,\nu,c}(t;p)\) is absolutely convergent for \(\nu< \delta+\Re(\mu)\), (see [9]). If \(S\) is the sum of series of absolute terms of the Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,\nu,c}(t;p)\), then one can obtain \(\left|E_{\mu,\alpha,l}^{\gamma,\delta,\nu,c}(t;p)\right|\leq S\). This property of absolutely convergence of the Mittag-Leffler function is used in establishing Theorems 6, 7, 8. The rest of the paper is organized as follows:
In Section 2, we establish the existence of integral operators in a unified form. The bounds of these unified integral operators are also obtained. It is important to note that the unified integral operators produce almost all Riemann-Liouville type fractional integral operators as well as fractional integral operators containing the Mittag-Leffler function in their kernels. Furthermore, the bounds of all these fractional integral operators are obtained in Section 2 and Section 3, from the bounds which have been established for unified integral operators.

2. Existence of new unified integral operators

The first result provides the existence of new integral operators with upper bounds in variable form.

Theorem 4. Let \(f:[a,b]\longrightarrow \mathbb{R}\), \(0< a< b\), be a positive and integrable function, \(g:[a,b]\longrightarrow \mathbb{R}\) be differentiable and strictly increasing. Also let \(\frac{\phi}{x}\) be an increasing function on \([a,b]\) and \(\omega,\alpha,l,\gamma,c\in \mathbb{C}\), \(\Re(\alpha),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\mu,\delta>0\) and \(0< \nu\leq\delta+\mu\). Then for \(x\in[a,b]\), we have

and \begin{equation}\label{e2} \int_{x}^{b}\frac{\phi(g(t)-g(x))}{g(t)-g(x)}E^{\gamma, \delta, \nu, c}_{\mu, \alpha, l,}(\omega(g(t)-g(x))^{\mu}; p)g'(t)f(t)dt\leq \phi(g(b)-g(x)) E^{\gamma, \delta, \nu, c}_{\mu, \alpha, l,}(w(g(b)-g(x))^{\mu}; p)\lVert f\rVert_{[x,b]}\nonumber \end{equation} where \(||f||_{[a,x]}=\sup\limits_{t\in[a,x]}|f(t)|\) and \(||f||_{[x,b]}=\sup\limits_{t\in[x,b]}|f(t)|\).

Proof. As \(g\) is increasing, therefore for \(t\in[a,x);\,x\in[a,b]\), \(g(x)-g(t)\leq g(x)-g(a)\). The function \(\frac{\phi}{x}\) is increasing, therefore one can obtain:

It is given that \(f\) is positive and \(g\) is differentiable and increasing. Therefore from (9), the following inequality is valid: Multiplying (10) by \( E^{\gamma, \delta, \nu, c}_{\mu, \alpha, l,}(\omega(g(x)-g(t))^{\mu}; p)\), then integrating over \([a,x]\), we obtain From which by solving integral and using the fact \(\frac{1}{\mu n+1} \leq 1 \), inequality (8) can be obtained. Now on the other hand for \(t\in(x,b]\) and \(x\in[a,b]\), the following inequality holds true: Multiplying (13) by \( E^{\gamma, \delta, \nu, c}_{\mu, \alpha, l}(\omega(g(t)-g(x))^{\mu}; p)\), then integrating over \((x,b]\), we obtain From which the inequality (9) can be obtained.

From above theorem, we are motivated to give the definition of a new unified two-sided integral operator as follows:

Definition 5. Let \(f,g:[a,b]\longrightarrow \mathbb{R}\), \(0< a< b\), be the functions such that \(f\) be positive and \(f\in L_{1}[a,b]\), and \(g\) be differentiable and strictly increasing. Also let \(\frac{\phi}{x}\) be an increasing function on \([a,\infty)\) and \(\omega,\alpha,l,\gamma,c\in \mathbb{C}\), \(\Re(\alpha),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\mu,\delta>0\) and \(0< \nu\leq\delta+\mu\). Then for \(x\in[a,b]\) the left and right integral operators are defined by

and The upcoming theorem provides the boundedness of the unified integral operators.

Theorem 6. Under the assumption of Theorem 4, the following bounds hold for integral operators (15) and (16):

and Hence where \(S\) is the sum of absolute terms of series in (7) and \(||f||_{[a,b]}=\sup\limits_{t\in[a,b]}|f(t)|\).

Proof. From (11), it can be obtained \begin{equation*} \left|\int_{a}^{x}\frac{\phi(g(x)-g(t))}{g(x)-g(t)}E^{\gamma, \delta, \nu, c}_{\mu, \alpha, l}(\omega(g(x)-g(t))^{\mu}; p)g'(t)f(t)dt\right|\leq S\left| \frac{\phi(g(x)-g(a))}{g(x)-g(a)}\right| \int_{a}^{x}\left|g'(t)f(t)\right|dt\nonumber \end{equation*} by simplifying the above inequality we get \begin{equation*} \left|(_gF_{\mu, \alpha, l, {a^+}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)\right|\leq S\left|\frac{\phi(g(x)-g(a))}{g(x)-g(a)}\right|\lVert f\lVert_{[a,x]}(g(x)-g(a)). \end{equation*} As \(g(x)-g(a)\leq g(b)-g(a)\), therefore

and hence (17) can be achieved. Similarly from (14), one can obtain \begin{equation*} \left|(_gF_{\mu, \alpha, l, {b^-}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)\right|\leq S\left|\frac{\phi(g(b)-g(x))}{g(b)-g(x)}\right|\lVert f\lVert_{(x,b]}(g(b)-g(x)) \end{equation*} and hence (18) can be achieved. By using (17) and (18), the inequality in (19) can be obtained.

Integral operators defined in (15) and (16) have connection with fractional integral operators given in Section \ref{sec1}. The upcoming section describes the connection of these integral operators with fractional integral operators (5) and (6) and their bounds are computed.

3. Bounds of fractional integral operators containing Mittag-Leffler functions

In this whole section, we set the function \(\phi(x)=x^\alpha,\alpha>0\) and the function \(g(x)=I(x)\), where \(I\) denotes the identity function. In this case bounds of fractional integral operators containing Mittag-Leffler functions defined in [1, 3, 4, 5, 6, 7, 8] can be obtained at once from bounds of unified integral operators (15) and (16). As an example the bounds of fractional integral operators defined by Andri\'c et al. in [9]and} are obtained. Computation of rest of the bounds of related fractional integrals described in Remark 2 are left for the reader.

Theorem 7. The fractional integral operators of function \(f\) defined in (5) and (6) are bounded for \(\alpha>1\), further the following inequality holds:

Proof. Let \(g(x)=x\) and \(\phi(x)=x^\alpha\). Then \(\frac{\phi}{x}\) is increasing for \(\alpha>1\). Therefore \begin{equation*} (_gF_{\mu, \alpha, l, {a^+}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)=(_IF_{\mu, \alpha, l, {a^+}}^{x^{\alpha}, \gamma, \delta, \nu, c}f)(x;p):=\left( \epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,\nu,c}f \right)(x;p) \end{equation*} and \begin{equation*}(_gF_{\mu, \alpha, l, {b^-}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)=(_IF_{\mu, \alpha, l, {b^-}}^{x^{\alpha}, \gamma, \delta, \nu, c}f)(x;p):=\left( \epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,\nu,c}f \right)(x;p). \end{equation*} Thus

and Hence boundedness of fractional operators (5) and (6) is followed and (21) can be achieved.

Remark 3. By using (22) and (23), the boundedness of all fractional integrals containing Mittag-Leffler functions compiled in Remark 2 can be obtained.

Bounds of fractional integral operators associated with generalized \(k\)-fractional integrals

In this whole section, we set the function \(\phi(x)=\frac{x^{\frac{\beta}{k}}}{k\Gamma_k({\beta})},\beta,k>0\). In this case bounds of fractional integral operators defined in [10, 11, 12, 14] can be obtained at once from bounds of unified integral operators (15) and (16). As an example the bounds of fractional integral operators (3) and (4) are obtained. Computation of the rest of the bounds of related fractional integrals described in Remark 1 are left for the reader.

Theorem 8.\label{th4} The generalized fractional integral operators of function \(f\) defined in (3) and (4) are bounded for \(\beta>k\), further the following inequality holds:

Proof. Let \(\phi(x)=\frac{x^{\frac{\beta}{k}}}{k\Gamma_k({\beta})},\beta,k>0\) and \(p=\omega=0\). Then \(\frac{\phi}{x}\) is increasing for \(\beta>k\). Therefore \begin{equation*} (_gF_{\mu, \alpha, l, {a^+}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)=(_gF_{\mu, \alpha, l, {a^+}}^{\frac{x^{\frac{\beta}{k}}}{k\Gamma_k({\beta})}, \gamma, \delta, \nu, c}f)(x;0):=\,^{\beta}_{g}I^{k}_{a^+}f(x) \end{equation*} and \begin{equation*} (_gF_{\mu, \alpha, l, {b^-}}^{\phi, \gamma, \delta, \nu, c}f)(x;p)=(_gF_{\mu, \alpha, l, {b^-}}^{\frac{x^{\frac{\beta}{k}}}{k\Gamma_k({\beta})}, \gamma, \delta, \nu, c}f)(x;0):=\,^{\beta}_{g}I^{k}_{b^-}f(x). \end{equation*} Thus

and Hence boundedness of fractional operators (3) and (4) is followed and (24) can be achieved.

Remark 4. By using (25) and (26), the boundedness of all fractional integrals compiled in Remark 1 can be obtained. Also by setting \(g(x)=I(x)\) in (22) and (23) integral operators defined in [14]se} can be obtained and using Theorem 6 bounds of these integral operators can be achieved.

5. Concluding remarks

The aim of this study is to develop unified integral operators which provide the fractional integral operators of Riemann-Liouville type as well as fractional integral operators containing Mittag-Leffler functions in their kernels. The bounds of these new integral operators are computed which simultaneously provide the bounds of all fractional integral operators defined in [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13]. The existence of new generalized integral operators may be useful in applied sciences along with the theory and applications of fractional calculus.

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.