1. Introduction

The study of fourth-order three-point boundary value problems (BVP) for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. Various authors studied the existence of positive solutions for \(n\)th-order \(m\)-point boundary value problems using different methods, for example, fixed point theorems in cones, nonlinear alternative of Leray-Schauder, and Krasnoselskii's fixed point theorem, see [1, 2, 3, 4, 5] and the references therein.

In 2003, by using the Leray-Schauder degree theory, Liu and Ge [6] proved the existence of positive solutions for \((n-1, 1)\) three-point boundary value problems with coefficient that changes sign given as follows: \begin{gather*} u^{(n)}(t)+\lambda a(t)f(u(t))=0,\quad\text t\in(0,1),\\ u(0)=\alpha u(\eta),\quad u(1)=\beta u(\eta),\quad u^{(i)}(0)=0~~for~~i=1,2,...,n-2,\\ and~~~u^{(n-2)}(0)=\alpha u^{(n-2)}(\eta),~~~u^{(n-2)}(1)=\beta u^{(n-2)}(\eta),~~~u^{(i)}(0)=0~~for~~i=1,2,..,n-3, \end{gather*} where \(\eta\in(0,1)\), \(\alpha\geq0\), \(\beta\geq0\), and \(a: (0,1)\rightarrow \mathbb{R}\) may change sign and \(\mathbb{R}=(-\infty,\infty)\), \(f(0)>0\), \(\lambda>0\) is a parameter.

In 2005, Eloea and Ahmad [7] studied the existence of positive solutions of a nonlinear \(n\)th-order boundary value problem with nonlocal conditions as follows: \begin{gather*} u^{(n)}(t)+a(t)f(u(t))=0,\quad\text t\in(0,1),\\ u(0)=0,\quad u^{'}(0)=0,...,u^{(n-2)}(0)=0,\quad \alpha u(\eta)=u(1), \end{gather*} where \(0< \eta< 1\), \(0< \alpha\eta^{n-1}< 1\), \(f\) is either superlinear or sublinear.

In 2009, Bai et al. [8] used fixed-point theory to study the existence of positive solutions of a singular nth-order three-point boundary value problem on time scales represented as: \begin{gather*} u^{n}(t)+a(t)f(u(t))=0,\quad\text t\in(0,1),\\ u(a)=\alpha u(\eta),\quad u^{'}(a)=0,...,u^{(n-2)}(a)=0,\quad u(b)=\beta u(\eta), \end{gather*} where \(a< \eta< b\), \(0\leq a< 1\), \(0< \beta(\eta-a)^{n-1}< (1-\alpha)(b-a)^{n-1}+\alpha(\eta-a)^{n-1}\), \(f\in C([a,b]\times[0,\infty), [0,\infty))\) and \(h\in C([a,b], [0,\infty))\) may be singular at \(t=a\) and \(t=b\).

In 2004, Sun [9] studied the existence of nontrivial solution for the three-point boundary value problem: \begin{gather*} u^{''}(t)+f(t,u(t))=0,\quad\text 0\leq t\leq 1, \\ u^{'}(0)=0,\quad u(1)=\alpha u^{'}(\eta), \end{gather*} where \(\eta\in(0,1)\), \(\alpha\in\mathbb{R}\), \(f\in C([0,1]\times\mathbb{R},\mathbb{R})\). The same author in [10], studied solvability of a nonlinear second-order three-point boundary value problem: \begin{gather*} u^{''}(t)+f(t,u(t))=0,\quad\text 0\leq t\leq 1, \\ u^{'}(0)=0,\quad u(1)=\alpha u(\eta), \end{gather*} where \(\eta\in(0,1)\), \(\alpha\in\mathbb{R}\), \(\alpha\neq0\), \(f\in C([0,1]\times\mathbb{R},\mathbb{R})\).

Li and Sun [11], also used the same method to study nontrivial solution of a nonlinear second-order three-point boundary value problem: \begin{gather*} u^{''}(t)+f(t,u(t))=0,\quad\text 0\leq t\leq 1, \\ au(0)-bu^{'}(0)=0,\quad u(1)-\alpha u(\eta)=0, \end{gather*} where \(\eta\in(0,1)\), \(a,b,\alpha\in\mathbb{R}\), with \(a^{2}+b^{2}>0.\)

Motivated by the above work, we extend the results obtained for second-order boundary value problem to fourth-order boundary value problem using a different method from [7]. We prove the existence of nontrivial solution for the fourth-order three-point boundary value problem (BVP):

where \(0< \eta< 1\), \(\alpha\in\mathbb{R}\) , \(\alpha\eta^{4}\neq1\), \(f\in C([0,1]\times\mathbb{R},\mathbb{R})\), \(\mathbb{R}=(\mathbb{-\infty,\infty})\).

This paper is organized as follows: in Section 2, we present two lemmas that will be helpful in proving our main results, in Section 3, we present our main results and finally, in Section 4, we illustrated our results with examples.

2. Preliminaries

Let \(E=C[0,1]\) with the norm \(\|y\|=\sup_{t\in[0,1]}|y(t)|\) for any \(u\in E\). A solution \(u(t)\) of the BVP (1)-(2) is called nontrivial solution if \(u(t)\neq0\). To get our results, we need to the following lemma.

Lemma 1. Let \(y\in C([0,1])\), \(\alpha\eta^{4}\neq1\), then the boundary value problem \begin{gather*} u^{(5)}(t)+ y(t)=0,\quad 0< t< 1, \\ u(0)=0,\quad u^{'}(0)=u^{''}(0)=u^{'''}(0)=0,\quad u(1)=\alpha u(\eta), \end{gather*} has a unique solution $$u(t)=-\frac{1}{24}\int_{0}^{t}(t-s)^{4}y(s)ds+\frac{t^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}y(s)ds- \frac{\alpha t^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}y(s)ds.$$

Proof. Rewriting the differential equation as \(u^{(5)}(t)=-y(t)\), and integrating five times from \(0\) to \(1\), we obtain

By the boundary conditions (2), we have $$u(0)=0,\quad u'(0)=u''(0)=u^{'''}(0)=0,~~i.e.~~c_{1}=c_{2}=c_{3}=c_{4}=0,$$ and \(u(1)=\alpha u(\eta)\), we get Compensate Equation (3) in the Equation (4), we obtain $$u(t)=-\frac{1}{24}\int_{0}^{t}(t-s)^{4}y(s)ds+\frac{t^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}y(s)ds- \frac{\alpha t^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}y(s)ds.$$ This completes the proof.

Define the integral operator \(T: E\longrightarrow E\), by By Lemma 1, the BVP (1)-(2) has a solution if and only if the operator \(T\) has a fixed point in \(E\). So we only need to seek a fixed point of \(T\) in \(E\). By Ascoli-Arzela theorem, we can prove that \(T\) is a completely continuous operator. Now we cite the Leray-Schauder nonlinear alternative.

Lemma 2. [1]. Let \(E\) be a Banach space and \(\Omega\) be a bounded open subset of \(E\), \(0\in\Omega\). \(T:\overline{\Omega}\rightarrow E\) be a completely continuous operator. Then, either
\((i)\) there exists \(u\in \partial \Omega\) and \(\lambda>1\) such that \(T(u)=\lambda u\), or
\((ii)\) there exists a fixed point \(u^{\ast}\in \overline {\Omega}\) of \(T\).

3. Existence of nontrivial solution

In this section, we prove the existence of a nontrivial solution for the BVP (1)-(2). Suppose that \(f\in C([0,1]\times\mathbb{R},\mathbb{R}).\)

Theorem 3. \label{thm1} Suppose that \(f(t,0)\neq 0\), \(\alpha\eta^{4}\neq1\), and there exist nonnegative functions \(k,h \in L^{1}[0,1]\), such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R},$$ $$\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds< 1.$$ Then the BVP (1)-(2) has at least one nontrivial solution \(u^{\ast}\in C[0,1].\)

Proof. Let $$M=\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds,$$ $$N=\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}h(s)ds+ \frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}h(s)ds.$$ Then \(M< 1\). Since \(f(t,0)\neq 0\), there exists an interval \([a,b]\subset [0,1]\) such that \(\min_{a\leq t\leq b}|f(t,0)|>0\), and as \(h(t)\geq |f(t,0)|\), a.e. \(t\in [0,1]\), we have \(N>0\). Let \(A=N(1-M)^{-1}\) and \(\Omega=\{u\in E: \|u\|< A\}\). Assume that \(u\in \partial \Omega\) and \(\lambda>1\) such that \(Tu=\lambda u\), then \begin{eqnarray*} \lambda A&=&\lambda \|u\|=\|Tu\|=\max_{0\leq t\leq 1}|(Tu)(t)|\\ &\leq& \frac{1}{24}\max_{0\leq t\leq1}\int_{0}^{t}(t-s)^{4}|f(s,u(s))|ds+\max_{0\leq t\leq1}\left|\frac{t^{4}}{24(1-\alpha\eta^{4})}\right|\int_{0}^{1}(1-s)^{4}|f(s,u(s))|ds\\ &&+\max_{0\leq t\leq1}\left|\frac{\alpha t^{4}}{24(1-\alpha\eta^{4})}\right|\int_{0}^{\eta}(\eta-s)^{4}|f(s,u(s))|ds\\ &\leq&\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}|f(s,u(s))|ds+\frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}|f(s,u(s))|ds\\ &\leq&\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}k(s)|u(s)|ds+\frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}k(s)|u(s)|ds\\ &&+\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}h(s)ds+\frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}h(s)ds\\ &=& M \|u\|+N. \end{eqnarray*} Therefore, $$\lambda \leq M+\frac{N}{A}=M+\frac{N}{N(1-M)^{-1}}=M+(1-M)=1.$$ This contradicts \(\lambda>1\). By Lemma 2, \(T\) has a fixed point \(u^{\ast}\in\overline{\Omega}\). In view of \(f(t,0)\neq0\), the BVP (1)-(2) has a nontrivial solution \(u^{\ast}\in E\). This completes the proof.

Theorem Suppose that \(f(t,0)\neq0\), \(\alpha\eta^{4}< 1\), and there exist nonnegative functions \(k,h\in L^{1}[0,1]\), such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times\mathbb{R}.$$ If one of the following conditions holds:

1. there exists a constant \(p>1\) such that $$\int_{0}^{1}k(s)^{p}ds< \left[\frac{24(1-\alpha\eta^{4})(1+4q)^{1/q}}{2-\alpha\eta^{4}+|\alpha|\eta^{(1+4q)/q}}\right]^{p}, \quad\frac{1}{p}+\frac{1}{q}=1;$$
2. there exists a constant \(\mu>-1\) such that $$k(s)\leq \frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}s^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\{s\in[0,1] : k(s)< \frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}s^{\mu}\}>0;$$
3. there exists a constant \(\mu>-5\) such that $$k(s)\leq \frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}(1-s)^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}(1-s)^{\mu}\right\}>0;$$
4. \(k(s)\) satisfies $$k(s)\leq\frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}},\quad a.e.~~~s\in [0,1], meas\left\{s\in[0,1] : k(s)< \frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}}\right\}>0,$$

then the BVP (1)-(2) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. Let \(M\) be defined as in the proof of Theorem 3. To prove Theorem 4, we only need to prove that \(M< 1\). Since \(\alpha\eta^{4}< 1\), we have $$M=\left(\frac{1}{24}+\frac{1}{24(1-\alpha\eta^{4})}\right)\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds$$ $$=\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds.~~~~\quad$$

1. Using the H\"{o}lder inequality, we have \begin{eqnarray*} M&\leq&\left[\int_{0}^{1}k(s)^{p}ds\right]^{1/ p}\left\{\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\left[\int_{0}^{1}(1-s)^{4q}ds\right]^{1/ q} +\frac{|\alpha|}{24(1-\alpha\eta^{4})}\left[\int_{0}^{\eta}(\eta-s)^{4q}ds\right]^{1/q}\right\}\\ &\leq&\left[\int_{0}^{1}k(s)^{p}ds\right]^{1/ p}\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}(\frac{1}{1+4q})^{1/q} +\frac{|\alpha|}{24(1-\alpha\eta^{4})}(\frac{\eta^{1+4q}}{1+4q})^{1/q}\right]\\ &<&\frac{24(1-\alpha\eta^{4})(1+4q)^{1/q}}{2-\alpha\eta^{4}+|\alpha|\eta^{(1+4q)/q}}\times \frac{2-\alpha\eta^{4}+|\alpha|\eta^{(1+4q)/q}}{24(1-\alpha\eta^{4})(1+4q)^{1/q}}\\ &=&1. \end{eqnarray*}
2. Here, we have \begin{eqnarray*}M&<&\frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}\\&&\times\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}s^{\mu}ds+\frac{|\alpha|}{24(1-\alpha\eta^{4})} \int_{0}^{\eta}(\eta-s)^{4}s^{\mu}ds\right]\\ &\leq&\frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}\left[\frac{2-\alpha\eta^{4}}{(1-\alpha\eta^{4})}\frac{1} {(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\right.\\&&\left.+ \frac{|\alpha|}{(1-\alpha\eta^{4})}\frac{\eta^{5+\mu}}{(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\right]\\&=&\frac{(1-\alpha\eta^{4})(1+\mu) (2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}. \frac{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\\&=&1.\end{eqnarray*}
3. Here, we have \begin{eqnarray*}M&<&\frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4+\mu}ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}(1-s)^{\mu}ds\right]\\ &\leq&\frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4+\mu}ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4+\mu}ds\right]\\ &=&\frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}.\frac{1}{5+\mu}+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}.\frac{1}{5+\mu}\right]\\ &=&\frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}.\frac{2-\alpha\eta^{4}+|\alpha|}{24(1-\alpha\eta^{4})(5+\mu)}=1.\end{eqnarray*}
4. Here, we have \begin{eqnarray*}M&<&\frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}}\left[\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}ds\right]\\ &=&\frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}}.\frac{2-\alpha\eta^{4}+|\alpha|\eta^{5}}{120(1-\alpha\eta^{4})}=1.\end{eqnarray*}

This completes the proof.

Theorem 5. Suppose that \(f(t,0)\neq0\), \(\alpha\eta^{4}>1\), and there exist nonnegative functions \(k,h\in L^{1}[0,1]\) such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ If one of the following conditions holds:

1. there exists a constant \(p>1\) such that $$\int_{0}^{1}k(s)^{p}ds< \left[\frac{24(\alpha\eta^{4}-1)(1+4q)^{1/q}}{\alpha(\eta^{4}+\eta^{(1+4q)/q})}\right]^{p}, \quad\left(\frac{1}{p}+\frac{1}{q}=1\right);$$
2. there exists a constant \(\mu>-1\) such that $$k(s)\leq \frac{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+\eta^{5+\mu})}s^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+\eta^{5+\mu})}s^{\mu}\right\}>0;$$
3. there exists a constant \(\mu>-5\) such that $$k(s)\leq \frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}(1-s)^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\{s\in[0,1] : k(s)< \frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}(1-s)^{\mu}\}>0;$$
4. \(k(s)\) satisfies $$k(s)\leq\frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+\eta^{5})},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+\eta^{5})}\right\}>0,$$

then the BVP (1)-(2) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. Let \(M\) be defined as in the proof of Theorem 3. To prove Theorem 5, we only need to prove that \(M< 1\). Since \(\alpha\eta^{4}>1\), we have \begin{eqnarray*}M&=&\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds\\ &=&\frac{\alpha}{24(\alpha\eta^{4}-1)}[\eta^{4}\int_{0}^{1}(1-s)^{4}k(s)ds+\int_{0}^{\eta}(\eta-s)^{4}k(s)ds].\end{eqnarray*}

1. Using the H\"{o}lder inequality, we have \begin{eqnarray*}M&\leq&\left[\int_{0}^{1}k(s)^{p}ds\right]^{1/ p}\left\{\frac{\alpha\eta^{4}}{24\left(\alpha\eta^{4}-1\right)}\left[\int_{0}^{1}(1-s)^{4q}ds\right]^{1/q} +\frac{\alpha}{24\left(\alpha\eta^{4}-1\right)}\left[\int_{0}^{\eta}(\eta-s)^{4q}ds\right]^{1/q}\right\}\\ &\leq&\left[\int_{0}^{1}k(s)^{p}ds\right]^{1/ p}\left[\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}(\frac{1}{1+4q})^{1/q} +\frac{\alpha}{24(\alpha\eta^{4}-1)}(\frac{\eta^{1+4q}}{1+4q})^{1/q}\right]\\ &<&\frac{24\left(\alpha\eta^{4}-1\right)(1+4q)^{1/q}}{\alpha(\eta^{4}+\eta^{(1+4q)/q})}\times \frac{\alpha(\eta^{4}+\eta^{(1+4q)/q})}{24(\alpha\eta^{4}-1)(1+4q)^{1/q}}=1.\end{eqnarray*}
2. Here, we have \begin{eqnarray*}M&<&\frac{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+\eta^{5+\mu})}\left[\frac{\alpha\eta^{4}}{24 (\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}s^{\mu}ds\right.\end{eqnarray*} \begin{eqnarray*} \\&&+\left.\frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{\eta}(\eta-s)^{4}s^{\mu}ds\right]\\ &\leq&\frac{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+\eta^{5+\mu})}\left[\frac{\alpha\eta^{4}} {(\alpha\eta^{4}-1)}\frac{1}{(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\right.\\&&\left.+\frac{\alpha}{(\alpha\eta^{4}-1)}\times\frac{\eta^{5+\mu}}{(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\right]\\ &=&\frac{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+\eta^{5+\mu})}. \frac{\alpha(\eta^{4}+\eta^{5+\mu})}{(\alpha\eta^{4}-1)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}\\&=&1.\end{eqnarray*}
3. Here, we have \begin{eqnarray*}M&<&\frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}\left[\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4+\mu}ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{\eta}(\eta-s)^{4}(1-s)^{\mu}ds\right]\\ &\leq&\frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}\left[\frac{\alpha\eta^{4}}{24(\alpha(\eta^{4}-1)}\int_{0}^{1}(1-s)^{4+\mu}ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4+\mu}ds\right]\\ &=&\frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}\left[\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}.\frac{1}{5+\mu}+ \frac{\alpha}{24(\alpha\eta^{4}-1)}.\frac{1}{5+\mu}\right]\\ &=&\frac{24(\alpha\eta^{4}-1)(5+\mu)}{\alpha(\eta^{4}+1)}.\frac{\alpha(\eta^{4}+1)}{24(\alpha\eta^{4}-1)(5+\mu)}=1.\end{eqnarray*}
4. Here, we have \begin{eqnarray*}M&<&\frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+\eta^{5})}\left[\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{\eta}(\eta-s)^{4}ds\right]\\ &=&\frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+\eta^{5})}.\frac{\alpha(\eta^{4}+\eta^{5})}{120(\alpha\eta^{4}-1)}=1.\end{eqnarray*}

This completes the proof.

Corollary 6. Suppose \(f(t,0)\neq0\), \(\alpha\eta^{4}< 1\), and there exist nonnegative functions \(k, h\in L^{1}[0,1]\) such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ If one of following conditions holds:

1. there exists a constant \(p>1\) such that $$\int_{0}^{1}k(s)^{p}ds< \left[\frac{24(1-\alpha\eta^{4})(1+4q)^{1/q}}{2-\alpha\eta^{4}+|\alpha|}\right]^{p}; \quad\frac{1}{p}+\frac{1}{q}=1;$$
2. there exists a constant \(\mu>-1\) such that $$k(s)\leq \frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}s^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\{s\in[0,1] : k(s)< \frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}s^{\mu}\}>0;$$
3. \(k(s)\) satisfies $$k(s)\leq\frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|}\right\}>0,$$

then the BVP (1)-(2) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. We have \begin{eqnarray*}M&=&\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds\\ &\leq&\frac{2-\alpha\eta^{4}}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}k(s)ds\\ &=&\frac{2-\alpha\eta^{4}+|\alpha|}{24(1-\alpha\eta^{4})}\int_{0}^{1}(1-s)^{4}k(s)ds.\end{eqnarray*} Proof of this corollary 6 is the same method in the proof Theorem 4. The proof is complete.

Corollary 7. Suppose that \(f(t,0)\neq0\), \(\alpha\eta^{4}>1\), and there exist nonnegative functions \(k,h\in L^{1}[0,1]\) such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ If one of the following conditions holds:

1. there exists a constant \(p>1\) such that $$\int_{0}^{1}k(s)^{p}ds< \left[\frac{24(\alpha\eta^{4}-1)(1+4q)^{1/q}}{\alpha(\eta^{4}+1)}\right]^{p}; \quad\frac{1}{p}+\frac{1}{q}=1;$$
2. there exists a constant \(\mu>-1\) such that $$k(s)\leq \frac{\left(\alpha\eta^{4}-1\right)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+1)}s^{\mu},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{\left(\alpha\eta^{4}-1\right)(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{\alpha(\eta^{4}+1)}s^{\mu}\right\}>0;$$
3. \(k(s)\) satisfies $$k(s)\leq\frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+1)},\quad a.e.~~~s\in [0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{120(\alpha\eta^{4}-1)}{\alpha(\eta^{4}+1)}\right\}>0,$$

then the BVP (1)-(2) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. We have \begin{eqnarray*}M&=&\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds\\ &\leq&\frac{\alpha\eta^{4}}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{\alpha}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}k(s)ds\\ &=&\frac{\alpha(\eta^{4}+1)}{24(\alpha\eta^{4}-1)}\int_{0}^{1}(1-s)^{4}k(s)ds.\end{eqnarray*} The rest procedure is the same as for Theorem 5. This completes the proof.

4. Examples

In order to illustrate the above results, we consider some examples.

Example 1. Consider the following problem

Set \(\eta=1/2\), \(\alpha=-3\), and $$f(t,x)=\frac{t}{3}x\sin x^{2}-\sqrt{t}+2,$$ $$k(t)=t,\quad h(t)=\sqrt{t}+2,$$ It is easy to prove that \(k, h\in L^{1}[0,1]\) are nonnegative functions, and $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R},$$ and $$\alpha\eta^{4}=-\frac{3}{16}\neq1.$$ Moreover, we have $$M=\left(\frac{1}{24}+\frac{1}{24|1-\alpha\eta^{4}|}\right)\int_{0}^{1}(1-s)^{4}k(s)ds+ \frac{|\alpha|}{24|1-\alpha\eta^{4}|}\int_{0}^{\eta}(\eta-s)^{4}k(s)ds\quad\quad\quad$$ $$M=\frac{105}{1368}\int_{0}^{1}(1-s)^{4} sds+\frac{6}{57}\int_{0}^{1/2}\left(\frac{1}{2}-s\right)^{4} sds=\frac{105}{41040}+\frac{6}{109440}< 1.\quad\quad\quad\quad$$ Hence, by Theorem 3, the BVP (6) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 2. Consider the following problem

Set \(\eta=1/4\), \(\alpha=-4\), and $$f(t,x)=\frac{2/3\sqrt{8+t}}{5+x^{2}}x\cos x^{3}-e^{t}+1,$$ $$k(t)=\frac{2}{3}\sqrt{8+t},\quad h(t)=e^{t}+1.$$ It is easy to prove that \(k, h\in L^{1}[0,1]\) are nonnegative functions, and $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ and $$\alpha\eta^{4}=-\frac{1}{64}< 1.$$ Let \(p=q=2\), such that \(\frac{1}{p}+\frac{1}{q}=1,\) then $$\int_{0}^{1}k(s)^{p}ds=\int_{0}^{1}\frac{4}{9}(8+s)ds=\frac{34}{9}.$$ Moreover, we have $$\left[\frac{24(1-\alpha\eta^{4})(1+4q)^{1/q}}{2-\alpha\eta^{4}+|\alpha|\eta^{(1+4q)/q}}\right]^{p}=1307.88.$$ Therefore, $$\int_{0}^{1}k(s)^{p}ds< \left[\frac{24(1-\alpha\eta^{4})(1+4q)^{1/q}}{2-\alpha\eta^{4}+|\alpha|\eta^{(1+4q)/q}}\right]^{p}.$$ Hence, by Theorem 4(1), the BVP (7) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 3. Consider the following problem

Set \(\eta=1/3\), \(\alpha=-3\), and $$f(t,x)=\frac{x}{7(4+x^{2})\sqrt[3]{t}}e^{-\cos x}-2t-1,$$ $$k(t)=\frac{1}{7\sqrt[3]{t}},\quad h(t)=2t+1.$$ It is easy to prove that \(k, h\in L^{1}[0,1]\) are nonnegative functions, and $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ and $$\alpha\eta^{4}=-\frac{1}{27}< 1.$$ Let \(\mu=-\frac{1}{3}>-1\), then $$\frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}=16.873.$$ Therefore, $$k(s)=\frac{1}{7\sqrt[3]{s}}=\frac{1}{7}s^{-\frac{1}{3}}< 16.873.s^{-\frac{1}{3}},$$ $$meas\left\{s\in[0,1]: k(s)< \frac{(1-\alpha\eta^{4})(1+\mu)(2+\mu)(3+\mu)(4+\mu)(5+\mu)}{2-\alpha\eta^{4}+|\alpha|\eta^{5+\mu}}s^{\mu}\right\}>0.$$ Hence, by Theorem 4(2), the BVP (8) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 4. Consider the following problem

Set \(\eta=1/2\), \(\alpha=-2\), and $$f(t,x)=\frac{3x^{3}}{7(1+x^{2})\sqrt[3]{(1-t)^{2}}}\sin x+t^{5}-1,$$ $$k(t)=\frac{3}{7\sqrt[3]{(1-t)^{2}}},\quad h(t)=t^{5}+1.$$ It is easy to prove that \(k, h\in L^{1}[0,1]\) are nonnegative functions, and $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ and $$\alpha\eta^{4}=-\frac{1}{8}< 1.$$ Let \(\mu=- \frac{2}{3} >-4\), then $$\frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}=\frac{2808}{33}.$$ Therefore, $$k(s)=\frac{3}{7\sqrt[3]{(1-s)^{2}}}=\frac{3}{7}(1-s)^{-\frac{2}{3}}< \frac{2808}{33}(1-s)^{-\frac{2}{3}},$$ $$meas\left\{s\in[0,1] : k(s)< \frac{24(1-\alpha\eta^{4})(5+\mu)}{2-\alpha\eta^{4}+|\alpha|}(1-s)^{\mu}\right\}>0.$$ Hence, by Theorem 4(3), the BVP (9) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 5. Consider the following problem

Set \(\eta=1/5\), \(\alpha=-5\), and $$f(t,x)=\frac{tx}{2(3+x^{2})}+e^{t}-3,$$ $$k(t)=\frac{t}{2},\quad h(t)=e^{t}+3.$$ It is easy to prove that \(k, h\in L^{1}[0,1]\) are nonnegative functions, and $$|f(t,x)|\leq k(t)|x|+h(t),\quad a.e.~~(t,x)\in [0,1]\times \mathbb{R}.$$ and $$\alpha\eta^{4}=-\frac{1}{125}< 1.$$ Moreover, we have $$\frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}}=\frac{9450}{157}.$$ Therefore, $$k(s)=\frac{s}{2}< \frac{9450}{157},\quad s\in[0,1],$$ $$meas\left\{s\in[0,1] : k(s)< \frac{120(1-\alpha\eta^{4})}{2-\alpha\eta^{4}+|\alpha|\eta^{5}}\right\}>0.$$ Hence, by Theorem 4(4), the BVP (10) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

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 concept of fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non integer order. Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,15, 16, 17] and the references therein.

Recently, Ahmad and Ntouyas [3] discussed the existence of solutions for the hybrid Hadamard differential equation \begin{equation*} \left\{ \begin{array}{l} ^{H}D^{\alpha }\left( \frac{x(t)}{g\left( t,x\left( t\right) \right) } \right) =f\left( t,x\left( t\right) \right) ,\text{ }t\in \left[ 1,T\right] , \\ \left. ^{H}I^{\alpha }x(t)\right\vert _{t=1}=\eta , \end{array} \right. \end{equation*} where \(^{H}D^{\alpha }\) is the Hadamard fractional derivative of order \( 0< \alpha \leq 1\). By employing the Dhage fixed point theorem, the authors obtained existence results.

In [4], Ardjouni and Djoudi studied the existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions \begin{equation*} \left\{ \begin{array}{l} \mathfrak{D}_{1}^{\alpha }\left( x\left( t\right) \right) =f\left( t,x\left( t\right) ,\mathfrak{D}_{1}^{\alpha }\left( x\left( t\right) \right) \right) ,\ t\in \left[ 1,T\right] , \\ x\left( 1\right) +g\left( x\right) =x_{0}, \end{array} \right. \end{equation*} where \(f:[1,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) and \(g:C\left( [1,T],\mathbb{R}\right) \rightarrow\mathbb{R}\) are nonlinear continuous functions and \(\mathfrak{D}_{1}^{\alpha }\) denotes the Caputo-Hadamard fractional derivative of order \(0< \alpha < 1\).

Motivated by these works, we study the existence, interval of existence and uniqueness of solution for the following nonlinear hybrid implicit Caputo-Hadamard fractional differential equation

where \(f:[1,T]\times\mathbb{R}\rightarrow\mathbb{R}\), \(g:[1,T]\times\mathbb{R}\rightarrow\mathbb{R}\backslash \{0\}\) and \(h:[1,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) are nonlinear continuous functions and \(\mathfrak{D}_{1}s^{\alpha }\) denotes the Caputo-Hadamard fractional derivative of order \(0< \alpha < 1\). To show the existence, interval of existence and uniqueness of solutions, we transform (1) into an integral equation and then use the Banach fixed point theorem. Further, by the generalization of Gronwall's inequality we obtain the estimate of solutions of (1).

2. Preliminaries

In this section, we present some basic definitions, notations and results of fractional calculus [2, 7, 12, 15] which are used throughout this paper.

Definition 1. [12] The Hadamard fractional integral of order \(\alpha >0\) for a continuous function \(x:\left[ 1,+\infty \right) \rightarrow\mathbb{R}\) is defined as \begin{equation*} \mathfrak{I}_{1}^{\alpha }x\left( t\right) =\frac{1}{\Gamma \left( \alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}x\left( s\right) \frac{ds}{s},\text{ }\alpha >0. \end{equation*}

Definition 2. [12] The Caputo-Hadamard fractional derivative\ of order \(\alpha \) for a continuous function \(x:\left[ 1,+\infty \right) \rightarrow\mathbb{R}\) is defined as \begin{equation*} \mathfrak{D}_{1}^{\alpha }x\left( t\right) =\frac{1}{\Gamma \left( n-\alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{n-\alpha -1}\delta ^{n}\left( x\right) \left( s\right) \frac{ds}{s},\text{ }n-1< \alpha < n, \end{equation*} where \(\delta ^{n}=\left( t\frac{d}{dt}\right) ^{n}\), \(n=\left[ \alpha \right] +1\).s

Lemma 3. [12] Let \(\alpha >0\), \(n\in\mathbb{N}\). Suppose \(x\in C^{n-1}\left( \left[ 1,+\infty \right) \right) \) and \( \delta ^{\left( n\right) }x\) exists almost everywhere on any bounded interval of \(\left[ 1,+\infty \right) \). Then \begin{equation*} \mathfrak{I}_{1}^{\alpha }\left[ \mathfrak{D}_{1}^{\alpha }x\right] \left( t\right) =x(t)-\sum\limits_{k=0}^{n-1}\frac{\delta ^{\left( k\right) }x\left( 1\right) }{\Gamma \left( k+1\right) }\left( \log t\right) ^{k}. \end{equation*} In particular, when \(0< \alpha < 1,\ \mathfrak{I}_{1}^{\alpha }\left[ \mathfrak{D}_{1}^{\alpha }x\right] \left( t\right) =x(t)-x(1)\).

Lemma 4.[12] For all \(\mu >0\) and \(v>-1\), then \begin{equation*} \frac{1}{\Gamma \left( \mu \right) }\int_{1}^{t}\left( \log \frac{t}{s} \right) ^{\mu -1}\left( \log s\right) ^{v}ds=\frac{\Gamma \left( v+1\right) }{\Gamma \left( \mu +v+1\right) }\left( \log t\right) ^{\mu +v}. \end{equation*}

The following generalization of Gronwall's lemma for singular kernels plays an important rsole in obtaining our main results.

Lemma 5. [15] Let \(x:\left[ 1,T\right] \rightarrow \left[ 0,\infty \right) \) be a real function and \(w\) is a nonnegative locally integrable function on \( \left[ 1,T\right] \). Assume that there is a constant \(a>0\) such that for \( 0< \alpha < 1\) \begin{equation*} x(t)\leq w(t)+a\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}x(s) \frac{ds}{s}. \end{equation*} Then, there exists a constant \(k=k(\alpha )\) such that \begin{equation*} x(t)\leq w(t)+ka\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}w(s) \frac{ds}{s}, \end{equation*} for every \(t\in \left[ 1,T\right] \).

3. Main results

In this section, we give the equivalence of the initial value problem (1) and prove the existence,interval of existence, uniqueness and estimate of solution of (1).
The proof of the following lemma is close to the proof Lemma 6.2 given in [7].

Lemma 6. If the functions \(f:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\), \(g:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\) are continuous, then the initial value problem (1) is equivalent to the nonlinear fractional Volterra integro-differential equation \begin{eqnarray*} x(t) &=&f\left( t,x(t)\right) +\theta g(t,x\left( t\right) ) +\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) } \int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}h\left( s,x\left( s\right) ,\mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( s\right) -f\left( s,x(s)\right) }{g(s,x\left( s\right) )}\right) \right) \frac{ds}{s}, \end{eqnarray*} for \(t\in \left[ 1,T\right] \).

Theorem 7. Let \(T>0\). Assume that the continuous functions \(f:\left[ 1,T \right] \times\mathbb{R}\rightarrow\mathbb{R}\), \(g:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\) satisfy the following conditions
(H1) There exists \(M_{g}\in\mathbb{R}^{+}\) such that \begin{equation*} \left\vert g\left( t,u\right) \right\vert \leq M_{g}, \end{equation*} for all \(u\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \).
(H2) There exists \(M_{h}\in\mathbb{R}^{+}\) such that \begin{equation*} \left\vert h\left( t,u,v\right) \right\vert \leq M_{h}, \end{equation*} for all \(u,v\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \).
(H3) There exist \(K_{1},K_{2},K_{3}\in\mathbb{R}^{+},K_{4}\in \left( 0,1\right) \) with \(K_{1}+K_{2}\left\vert \theta \right\vert \in \left( 0,1\right) \) such that \begin{eqnarray*} \left\vert f\left( t,u\right) -f\left( t,u^{\ast }\right) \right\vert &\leq &K_{1}\left\vert u-v\right\vert , \\ \left\vert g\left( t,u\right) -g\left( t,u^{\ast }\right) \right\vert &\leq &K_{2}\left\vert u-v\right\vert , \end{eqnarray*} and \begin{equation*} \left\vert h\left( t,u,v\right) -h\left( t,u^{\ast },v^{\ast }\right) \right\vert \leq K_{3}\left\vert u-u^{\ast }\right\vert +K_{4}\left\vert v-v^{\ast }\right\vert , \end{equation*} for all \(u,v,u^{\ast },v^{\ast }\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \). Let

Then (1) has a unique solution \(x\in C\left( \left[ 1,b\right],\mathbb{R}\right) \).

Proof. Let \begin{equation*} \mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( t\right) -f\left( t,x(t)\right) }{g\left( t,x(t)\right) }\right) =z_{x}\left( t\right) ,\text{ }x\left( 1\right) =\theta g\left( 1,x(1)\right) +f\left( 1,x(1)\right) , \end{equation*} then by Lemma 6, we have \begin{equation*} x(t)=f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) } \int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}z_{x}\left( s\right) \frac{ds}{s}, \end{equation*} where \begin{equation*} z_{x}\left( t\right) =h\left( t,f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) ,z_{x}\left( t\right) \right) . \end{equation*} That is \(x\left( t\right) =f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \). Define the mapping \(P:C\left( \left[ 1,b\right] ,\mathbb{R}\right) \rightarrow C\left( \left[ 1,b\right] ,\mathbb{R}\right) \) as follows \begin{equation*} \left( Px\right) \left( t\right) =f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}z_{x}\left( s\right) \frac{ds}{s}. \end{equation*} It is clear that the fixed points of \(P\) are solutions of (1). Let \( x,y\in C\left( \left[ 1,b\right] ,\mathbb{R}\right) \), then we have

and By replacing (4) in the inequality (3), we get \begin{eqnarray*} \left\vert \left( Px\right) \left( t\right) -\left( Py\right) \left(t\right) \right\vert &\leq& K_{1}\left\vert x(t)-y(t)\right\vert +K_{2}\left\vert \theta \right\vert \left\vert x(t)-y(t)\right\vert +K_{2}\left\vert x(t)-y(t)\right\vert \frac{M_{h}}{\Gamma \left( \alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}\frac{ds}{s} \\ &&+\frac{M_{g}}{\Gamma \left( \alpha \right) }\frac{K_{3}}{1-K_{4}} \int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}\left\vert x(s)-y(s)\right\vert \frac{ds}{s} \\ &\leq& K_{1}\left\Vert x-y\right\Vert +K_{2}\left( \left\vert \theta \right\vert +\frac{M_{h}\left( \log t\right) ^{\alpha }}{\Gamma \left( \alpha +1\right) }\right) \left\Vert x-y\right\Vert +\frac{K_{3}}{1-K_{4}}\left( \frac{M_{g}\left( \log t\right) ^{\alpha }}{ \Gamma \left( \alpha +1\right) }\right) \left\Vert x-y\right\Vert \\ &\leq& (K_{1}+K_{2}\left\vert \theta \right\vert +\left( M_{h}K_{2}+\frac{ M_{g}K_{3}}{1-K_{4}}\right) \frac{\left( \log t\right) ^{\alpha }}{\Gamma \left( \alpha +1\right) })\left\Vert x-y\right\Vert . \end{eqnarray*} Since \(t\in \left[ 1,b\right] \), Then \begin{equation*} \left\Vert Px-Py\right\Vert \leq \beta \left\Vert x-y\right\Vert , \end{equation*} where \begin{equation*} \beta =K_{1}+K_{2}\left\vert \theta \right\vert +\left( M_{h}K_{2}+\frac{ M_{g}K_{3}}{1-K_{4}}\right) \frac{\left( \log b\right) ^{\alpha }}{\Gamma \left( \alpha +1\right) }. \end{equation*} That is to say the mapping \(P\) is a contraction in \(C\left( \left[ 1,b\right],\mathbb{R}\right) \). Hence, by the Banach fixed point theorem, \(P\) has a unique fixed point \(x\in C\left( \left[ 1,b\right] ,\mathbb{R}\right) \). Therefore, (1) has a unique solution.

Theorem 8. Assume that \(f:\left[ 1,T\right] \times \mathbb{R}\rightarrow \mathbb{R}\), \(g:\left[ 1,T\right] \times \mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\) satisfy (H1), (H2) and (H3). If \(x\) is a solution of (1), then \begin{eqnarray*} \left\vert x\left( t\right) \right\vert &\leq &\left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) , \end{eqnarray*} where \(Q_{1}=\underset{t\in \left[ 1,T\right] }{\sup }\left\vert f\left( t,0\right) \right\vert ,\) \(Q_{2}=\underset{t\in \left[ 1,T\right] }{\sup } \left\vert g\left( t,0\right) \right\vert ,\ Q_{3}=\underset{t\in \left[ 1,T \right] }{\sup }\left\vert h\left( t,0,0\right) \right\vert \) and \(K\in\mathbb{R}^{+}\) is a constant.

Proof. Let \begin{equation*} \mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( t\right) -f\left( t,x(t)\right) }{g\left( t,x(t)\right) }\right) =z_{x}\left( t\right) ,\text{ }x\left( 1\right) =\theta g\left( 1,x(1)\right) +f\left( 1,x(1)\right) , \end{equation*} then by Lemma 6, \(x(t)=f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \). Then by (H1), (H2) and (H3), for any \(t\in \left[ 1,T\right] \) we have \begin{align*} \left\vert x\left( t\right) \right\vert & \leq \left\vert f\left( t,x(t)\right) \right\vert +\left\vert \theta \right\vert \left\vert g(t,x\left( t\right) )\right\vert +\left\vert g\left( t,x\left( t\right) \right) \right\vert \left\vert \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \right\vert \\ & \leq \left\vert f\left( t,x(t)\right) -f\left( t,0\right) \right\vert +\left\vert f\left( t,0\right) \right\vert +\left\vert \theta \right\vert \left( \left\vert g(t,x\left( t\right) )-g(t,0)\right\vert +\left\vert g(t,0)\right\vert \right) +M_{g}\left\vert \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \right\vert \\ & \leq K_{1}\left\vert x\left( t\right) \right\vert +Q_{1}+\left\vert \theta \right\vert \left( K_{2}\left\vert x\left( t\right) \right\vert +Q_{2}\right) +M_{g}\mathfrak{I}_{1}^{\alpha }\left\vert z_{x}\left( t\right) \right\vert . \end{align*} On the other hand, for any \(t\in \left[ 1,T\right] \) we get \begin{eqnarray*} \left\vert z_{x}\left( t\right) \right\vert =\left\vert h\left( t,x\left( t\right) ,z_{x}\left( t\right) \right) \right\vert &\leq& \left\vert h\left( t,x\left( t\right) ,z_{x}\left( t\right) \right) -h\left( t,0,0\right) \right\vert +\left\vert h\left( t,0,0\right) \right\vert \\ &\leq& K_{3}\left\vert x\left( t\right) \right\vert +K_{4}\left\vert z_{x}\left( t\right) \right\vert +\left\vert h\left( t,0,0\right) \right\vert \\ &\leq& \frac{K_{3}}{1-K_{4}}\left\vert x\left( t\right) \right\vert +\frac{ Q_{3}}{1-K_{4}}. \end{eqnarray*} Therefore \begin{equation*} \left\vert x\left( t\right) \right\vert \leq K_{1}\left\vert x\left( t\right) \right\vert +Q_{1}+\left\vert \theta \right\vert \left( K_{2}\left\vert x\left( t\right) \right\vert +Q_{2}\right) +M_{g}\mathfrak{I} _{1}^{\alpha }\left( \frac{K_{3}}{1-K_{4}}\left\vert x\left( t\right) \right\vert +\frac{Q_{3}}{1-K_{4}}\right) . \end{equation*} Thus \begin{eqnarray*}\left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert &\leq& Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) } +\left( \frac{M_{g}K_{3}}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) }\right)\\&&\times \left( \mathfrak{I}_{1}^{\alpha }\left\{ \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert \right\} \right) . \end{eqnarray*} By Lemma 5, there is a constant \(K=K\left( \alpha \right) \) such that \begin{eqnarray*} \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert &\leq& Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }+\left( \frac{M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \\ && \times\left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) \\&\leq& \left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \\ && \times \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) . \end{eqnarray*} Hence \begin{eqnarray*} \left\vert x\left( t\right) \right\vert &\leq &\left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) . \end{eqnarray*} 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

During the investigation of convexity, many researchers founded new classes of functions which are not convex in general. Some of them are the so called harmonic convex functions [1], harmonic \((\alpha, m)\)-convex functions [2], harmonic \((s,m)\)-convex functions [4, 5] and harmonic \((p,(s,m))\)-convex functions [3]. For a quick glance on importance of these classes and applications, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and references therein.

Definition1. A function \(f:I \subseteq \mathbb{R}\backslash \{0\} \rightarrow \mathbb{R}\) is said to be harmonic convex function on \(I\) if

holds for all \(x,y\in I\) and \(t\in \left[ 0,1\right] \). If the inequality is reversed, then \(f\) is said to be harmonic concave.

In [5, 20], Baloch et al. and Noor et al. also gave the definition of harmonic log-convex functions as follow:

Definition 2. A function \(f:I \subseteq \mathbb{R}\backslash \{0\} \rightarrow (0,\infty)\) is said to be harmonic log-convex function on \(I\) if

% holds for all \(x,y\in I\) and \(t\in \left[ 0,1\right] \). If the inequality is reversed, then \(f\) is said to be harmonic log-concave.

In [20], Noor et al. proved the following result for harmonic log-convex functions:

Theorem 3. Let \(I \subseteq \mathbb{R}\backslash \{0\}\) be an interval. If \(f: I \rightarrow (0,\infty)\) is harmonic convex function, then

for all \(a, b \in I\) and \( a < b\).

Here, motivated by the above result we study the class of harmonic log-convex functions and present some new inequalities for this class of functions.

2. Main Results

The following result holds.

Theorem 4. Let \(f:I \subseteq \mathbb{R} \backslash \{0\} \rightarrow (0,\infty)\) be harmonic log-convex function. Then, for every \(t \in [0,1]\), we have

Proof. The cases \(t = 0, \frac{1}{2}, 1\) are obvious. Assume that \(t \in (0,1) \backslash \left\{\frac{1}{2}\right\}.\) By the harmonic log-convexity of \(f\) we have

for any \(x \in [a,b]\). This allows that Integrating the inequality (6) over \(x\) on \([a,b]\), we have \begin{equation*} \int_{a}^{b}[f(x)]^{1 - t} \left[\frac{a^{2}b^{2}}{[(a + b)x - ab]^{2}}f\left(\frac{abx}{(a + b)x - ab}\right)\right]^{t}dx \geq \int_{a}^{b}\frac{a^{2t}b^{2t}}{[(a + b)x - ab]^{2t}}f\left( \frac{abx}{(a + b)tx - (2t - 1)ab} \right)dx. \end{equation*} Since \(t \neq \frac{1}{2},\) then \(u = \frac{abx}{(a + b)tx - (2t - 1)ab}\) is the change of variable with \( dx = \frac{(1 - 2t) a^{2}b^{2}}{[(a + b)tu - ab]^{2}}du\). For \(x = a\), we get \(u = \frac{ab}{ta + (1 - t)b}\) and for \(x = b\), we get \(u = \frac{ab}{(1 - t)a + tb}.\) Therefore, $$ \int_{a}^{b}\frac{a^{2t}b^{2t}}{[(a + b)x - ab]^{2t}}f\left( \frac{abx}{(a + b)tx - (2t - 1)ab} \right)dx= (1 - 2t) a^{2}b^{2}\int_{\frac{ab}{ta + (1 - t)b}}^{\frac{ab}{(1 - t)a + tb}} \frac{[(a + b)tu - ab]^{2(t - 1)}}{[ab - (1 - t)(a + b)u]^{2t}}f(u)du, $$ and hence the second inequality (4) is proved. By the Holder integral inequality for \(p = \frac{1}{1 - t}\), \(q = \frac{1}{t}\), we have \begin{eqnarray*} &&\int_{a}^{b}[f(x)]^{1 - t} \left[\frac{a^{2}b^{2}}{[(a + b)x - ab]^{2}}f\left(\frac{abx}{(a + b)x - ab}\right)\right]^{t}dx\\ &&\leq \left(\int_{a}^{b}\left([f(x)]^{1 - t}\right)^{\frac{1}{1 - t}}dx\right)^{1 - t} \left(\int_{a}^{b}\left(\left[\frac{a^{2}b^{2}}{[(a + b)x - ab]^{2}}f\left(\frac{abx}{(a + b)x - ab}\right)\right]^{t}\right)^{\frac{1}{t}}dx\right)^{t}\\ &&=\left(\int_{a}^{b}f(x)dx\right)^{1 - t} \left(\int_{a}^{b}\frac{a^{2}b^{2}}{[(a + b)x - ab]^{2}}f\left(\frac{abx}{(a + b)x - ab}\right)dx\right)^{t}\\ &&=\left(\int_{a}^{b}f(x)dx\right)^{1 - t} \left(\int_{a}^{b}f(x)dx\right)^{t}=\int_{a}^{b}f(x)dx.\end{eqnarray*} This proves the first part of inequality (4).

Author Contributions

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

Competing Interests

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

1. Introduction

The use of integral transforms (Laplace, Sumudu, Natural, Elzaki, Aboodh, Shehu and other transforms) in solving linear differential equations as well as integral equations has developed significantly as a result of the advantages of these transformations. Through these transforms, many problems of engineering and sciences have been solved. However, it was found that these transforms remain limited in solving equations that contain a nonlinear part.

To take advantage of these transformations and to use them to solve nonlinear differential equations, researchers in the field of mathematics were guided to the idea of their composition with some methods such as: Adomian decomposition method (ADM) [1, 2, 3, 4], homotopy analysis method [5, 6, 7, 8], variational iteration method (VIM) [9, 10, 11,12], homotopy perturbation method (HPM) [13, 14, 15,16] and DJ iteration method [17, 18, 19, 20].

The objective of the present study is to combine two powerful methods, Adomian decomposition method and Shehu transform method to get a better method to solve nonlinear partial differential equations. The modified method is called Shehu transform decomposition method (STDM). We apply our modified method to solve some examples of nonlinear partial differential equations.

2. Basics of Shehu transform

In this section we define Shehu transform and gave its important properties [21].

Definition

The Shehu transform of the function \(v(t)\) of exponential order is defined over the set of functions:

by the following integral It converges if the limit of the integral exists, and diverges if not.

The inverse Shehu transform is given as:

Equivalently where \(s\) and \(u\) are the Shehu transform variables, and \(\alpha \) is a real constant and the integral in Equation (4) is taken along \(s=\alpha \) in the complex plane \(s=x+iy.\)

Theorem 2. (The sufficient condition for the existence of Shehu transform [21]. If the function \(v(t)\) is piecewise continues in every finite interval \(0\leqslant t\leqslant \beta \) and of exponential order \(\alpha \) for \(t>\beta \). Then its Shehu transform \(V(s;u)\) exists.

Theorem 3. (Derivative of Shehu transform [21]. If the function \(v^{\left( n\right) }(t)\) is the \(n\)th derivative of the function \(v(t)\in A\) with respect to \( ^{\prime }t^{\prime }\) then its Shehu transform is defined as:

Taking \(n=1,2\) and \(3\) in Equation (5), we obtain the following derivatives with respect to \(t\): Now, we summarize some important properties of this transform [21].

1. Linearity: \(\hat{S}\left[ (\alpha f(t)+\beta g(t)\right] =\alpha \hat{S}\left[ f(t)\right] +\alpha \hat{S}\left[ g(t)\right] .\)
2. Change of scale: \(\hat{S}\left[ f(\beta t)\right] =\frac{u}{ \beta }V\left( \frac{s}{\beta },u\right) .\)

Other properties are given in the Table 1.

Table 1. Some important properties of Shehu transform.

\(v(t)\)	\(\hat{S}\left[ v(t)\right]\)	\(v(t)\)	\(\hat{S}\left[ v(t)\right]\)
\(1\)	\(\frac{u}{s}\)	\(\sin at\)	\(\frac{au^{2}}{s^{2}+a^{2}u^{2}}\)
\(t\)	\(\frac{u^{2}}{s^{2}}\)	\(\cos at\)	\(\frac{us}{s^{2}+a^{2}u^{2}}\)
\(\frac{t^{n}}{n!},\) \(n=0,1,2,..\)	\(\left( \frac{u}{s}\right)^{n+1}\)	\(\sinh at\)	\(\frac{au^{2}}{s^{2}-a^{2}u^{2}}\)
\(\frac{t^{n}\exp (at)}{n!}\)	\(\frac{u^{n+1}}{\left(s-au\right) ^{n+1}}\)	\(\cosh at\)	\(\frac{us}{s^{2}-a^{2}u^{2}}\)

Proposition 4. If \(\frac{\partial v(x,t)}{\partial t}\) and \(\frac{\partial ^{2}v(x,t)}{ \partial t^{2}}\) exist, then

Proof. By means of integration by parts, we get \begin{eqnarray*} \hat{S}\left[ \frac{\partial v(x,t)}{\partial t}\right] &=&\int\limits_{0}^{ \infty }e^{\frac{-st}{u}}\frac{\partial v(x,t)}{\partial t}dt=\lim_{\tau \longrightarrow \infty }\int\limits_{0}^{\tau }e^{\frac{-st}{u}}\frac{ \partial v(x,t)}{\partial t}dt \\ &=&\lim_{\tau \longrightarrow \infty }\left( \left[ v(x,t)e^{\frac{-st}{u}} \right] _{0}^{\tau }+\frac{s}{u}\int\limits_{0}^{\tau }e^{\frac{-st}{u} }v(x,t)dt\right) \\ &=&\frac{s}{u}V(x,s,u)-v(x,0). \end{eqnarray*} Let \(\frac{\partial v(x,t)}{\partial t}=w(x,t)\), then, by using Equation (2) and Equation (9), we get: \ \ \begin{eqnarray*} \hat{S}\left[ \frac{\partial ^{2}v(x,t)}{\partial t^{2}}\right] &=&\hat{S} \left[ \frac{\partial w(x,t)}{\partial t}\right] =\frac{s}{u}\hat{S}\left[ w(x,t)\right] -w(x,0) \\ &=&\frac{s}{u}\hat{S}\left[ \frac{\partial v(x,t)}{\partial t}\right] -\frac{ \partial v(x,0)}{\partial t} \\ &=&\frac{s^{2}}{u^{2}}V(x,s,u)-\frac{s}{u}v(x,0)-\frac{\partial v(x,0)}{ \partial t}. \end{eqnarray*}

Proposition 5. Let \(V(x,s,u)\) is the Shehu transform of \(v(x,t),\) then

Proof. We use use mathematical induction to prove (11). By means of Equation (9), the formula (11) is true for \(n=1\) and suppose

Let \(\frac{\partial ^{n}v(x,t)}{ \partial t^{n}}=w(x,t)\) and using (9) and (12), we have: \begin{eqnarray*} \ \ \hat{S}\left[ \frac{\partial ^{n+1}v(x,t)}{\partial t^{n+1}}\right] &=& \hat{S}\left[ \frac{\partial w(x,t)}{\partial t}\right] =\frac{s}{u}\hat{S} \left( w(x,t)\right) -w(x,0) \\ &=&\frac{s}{u}\left[ \frac{s^{n}}{u^{n}}V(x,s,u)-\sum_{k=0}^{n-1} \left( \frac{s}{u}\right) ^{n-(k+1)}\frac{\partial ^{k}v(x,0)}{\partial t^{k} }\right] -\frac{\partial ^{n}v(x,0)}{\partial t^{n}} \\ &=&\frac{s^{n+1}}{u^{n+1}}V(x,s,u)-\sum_{k=0}^{n-1}\left( \frac{s}{u} \right) ^{n+1-(k+1)}\frac{\partial ^{k}v(x,0)}{\partial t^{k}}-\frac{ \partial ^{n}v(x,0)}{\partial t^{n}} \\ &=&\frac{s^{n+1}}{u^{n+1}}V(x,s,u)-\sum_{k=0}^{n}\left( \frac{s}{u} \right) ^{n+1-(k+1)}\frac{\partial ^{k}v(x,0)}{\partial t^{k}}. \end{eqnarray*}

3. Main results

3.1. Shehu transform decomposition method

To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous partial differential equation where \(\frac{\partial ^{m}U(x,t)}{\partial t^{m}}\) is the partial derivative of the function \(U(x,t)\) of order \(m\) \((m=1,2,3)\), \(R\) is the linear differential operator, \(N\) represents the general nonlinear differential operator, and \(g(x,t)\) is the source term. Applying the Shehu transform (denoted in this paper by \(\hat{S}\)) on both sides of Equation (13), we get Using the properties of Shehu transform, we obtain where \(m=1,2,3.\) Thus, we have Operating the inverse transform on both sides of Equation (16), we get where \(G(x,t)\), represents the term arising from the source term and the prescribed initial conditions. The second step in Shehu transform decomposition method, is that we represent the solution as an infinite series given below and the nonlinear term can be decomposed as: where \(A_{n}\) are Adomian polynomials [22] of \( U_{0},U_{1},U_{2},...,U_{n}\) and it can be calculated by the formula:\(\ \ \) Substituting (18) and (19) in (17), we have On comparing both sides of the Equation (21), we get In general, the recursive relation is given as: where \(m=1,2,3,\) and \(n\geqslant 0.\) Finally, we approximate the analytical solution \(U(x,t)\) by: The above series solutions generally converge very rapidly [23].

3.2. Application

Here, we apply Shehu transform decomposition method to solve some nonlinear partial differential equations.

Example 1. Consider the nonlinear KdV equation [24]:

with the initial condition: Applying the Shehu transform on both sides of Equation (25), we get By means of the properties of Shehu transform, we get Taking the inverse Shehu transform on both sides of Equation (28), we obtain By applying the aforesaid decomposition method, we have On comparing both sides of Equation (30), we get The first few components of \(A_{n}(U)\) polynomials [22], for example, are given by: Using the iteration formulas (31) and the Adomian polynomials (32), we obtain Based on the formula (24), we get which gives which is an exact solution to the KdV equation as presented in [25]. The graphs of exact solution and approximate solutions of Equation (25) for 3 terms and 4 terms is given in Figure 1.

Example 2. Consider the nonlinear gas dynamics equation:

with the initial condition:\(\ \) Applying the Shehu transform and its inverse on both sides of Equation (36), we get By applying the aforesaid Decomposition Method, we have On comparing both sides of Equation (39), we obtain The first few components of \(A_{n}(U)\) polynomials [2] is given by (32), and for \(B_{n}(U)\) for example, given as follows: Using the iteration formulas (40) and the Adomian polynomials (32), (41), we get the first terms of the solution series that is given by: So, the approximate series solution of Equation (36) is given as: And in the closed form, is given by: This result is the same as that obtained in [26] using homotopy analysis method. In Figure 2, \((a)\) represents the graph of exact solution, \((b)\) represents the graph of approximate solutions in 5 terms and \((c)\) represents the graph of approximate solutions in 4 terms.

Example 3. Consider the nonlinear wave-like equation with variable coefficients:

with\ the initial conditions: Applying the Shehu transform and its inverse on both sides of Equation (45), we get By applying the aforesaid Decomposition Method, we have Comparing both sides of Equation (48), we obtain The first few components of \(C_{n}(U)\) and \(D_{n}(U)\) Adomian polynomials [22], for example, are given by: and Using the iteration formulas (49) and the Adomian polynomials (50) and (51), we obtain The first terms of the approximate solution of Equation (45), is given by And in the closed form: This result represents the exact solution of the Equation (45) as presented in [27] The graphs of exact solution and approximate solutions of Equation (45) for 3 terms and 4 terms are shown in Figure 3.

4. Conclusion

The coupling of Adomian decomposition method (ADM) and Shehu transform method proved very effective to solve nonlinear partial differential equations. We can say that this method is easy to implement and is very effective, as it allows us to know the exact solution after calculate the first three terms only. As a result, the conclusion that comes through this work is that (STDM) can be applied to other nonlinear partial differential equations of higher order, due to the efficiency and flexibility.

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 investigate the generalized porous medium equation in the whole space \(\mathbb{R}^{3}\),

where \(u=u(t,x)\) is a real-valued function, which denotes a density or concentration. The dissipative coefficient \(\mu>0\) corresponds to the viscous case, while \(\mu=0\) corresponds to the inviscid case. The fractional Laplacian operator \(\Lambda^{\alpha}\) is defined by Fourier transform as \(\widehat{\Lambda^{\alpha}u}=|\xi|^{\alpha}\hat{u}\), and \(P\) is an abstract operator.

The equation (1) was introduced in the first by Zhou et al. [1]. In fact, Equation (1) is obtained by adding the fractional dissipative term \(\mu \Lambda^{\alpha}u\) to the continuity equation (PME) \(u_{t}+\nabla\cdot(u V)=0\) given by Caffarelli and Vázquez [2], where the velocity \(V\) derives from a potential, \(V=-\nabla p\) and the velocity potential or pressure \(p\) is related to \(u\) by an abstract operator \(p=Pu\) [3].

For \(\mu=0\) and \( Pu=(-\Delta)^{-s}u=\Lambda^{-2s}u,\, 0< s< 1\); X. Zhou et al. [4] were interested in finding the strong solutions of the equation (1) which becomes the fractional porous medium equation in the Besov spaces \(B_{p,\infty}^{\alpha}\) and they obtained the local solution for any initial data in \(B_{1,\infty}^{\alpha}\). Moreover, in the critical case \(s=1\), the Equation (1) leads to a mean field equation [4, 5]. Let's take this opportunity to briefly quote some works on the well-posedness and regularity on those equations such as [4, 6] and the references therein.

On the other hand, an another similar model occurs in the aggregation equation, and plays a fundamental role in applied sciences such as physics, biology, chemistry, population dynamics. It describes a collective motion and aggregation phenomena in biology and in mechanics of continuous media [7,8]. In the aggregation equation, the abstract form pressure term \(Pu\) can also be represented by convolution with a kernel \(K\) as \(Pu=K*u\). The typical kernels are the Newton potential \(|x|^{\gamma}\) [9, 10], and the exponent potential \(-e^{-|x|}\) [11, 12]. For more results on this equation, we refer to [13, 14] and the references therein.

Recently, Zhou et al. [1] obtained the local well-posedness in Besov spaces for large initial data, and proved that the solution becomes global if the initial data is small, also, they studied a blowup criterion for the solution.

In addition, we can represent the Equation (1) with the same initial data by

As a consequence, this equation must be compared to the geostrophic model. So, the convective velocity is not absolutely divergence-free for the generalized porous medium equation. Additionally, if we assume that \(v\) is divergence-free vector function (\(\nabla\cdot v=0\)), the form (2) can contain the quasi-geostrophic (Q-G) equation [15, 16].

Inspired by the works [1, 17]; the aim of this paper is to prove the well-posedness results of Equation (1) and to give the Gevrey class regularity of the solution in homogeneous Fourier Besov-Morrey spaces under the condition that the abstract operator \(P\) is commutative with the operator \(e^{-\mu\sqrt{t}|D|^{\frac{\alpha}{2}}}\) and

Clearly, for the fractional porous medium equation, i.e. \(Pu=\Lambda^{-2s}u\), we get \(\sigma=1-2s\). If \(Pu=K*u\) in the aggregation equation, Wu and Zhang [18] proved a similar result under the condition \(\nabla K\in W^{1,1}\), \(\alpha\in (0,1)\). Corresponding to their case we give a same result for \(\sigma=0\) when \(\nabla K\in L^1\), and also a similar result for \(\sigma=1\) when \(K\in L^1\).

Throughout this paper, we use \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) to denote the homogenous Fourier Besov-Morrey spaces, \(C\) will denote constants which can be different at different places, \({\mathsf U}\lesssim{\mathsf V}\) means that there exists a constant \(C>0\) such that \({\mathsf U}\leq C{\mathsf V}\), and \(p'\) is the conjugate of \(p\) satisfying \(\frac{1}{p}+\frac{1}{p'}= 1\) for \(1\leq p\leq\infty\).

2. Preliminaries and main results

We start with a dyadic decomposition of \(\mathcal {\mathbb{R}}^n\). Suppose \(\chi \in C_0^\infty(\mathcal {\mathbb{R}}^n),\;\varphi\in C_0^\infty(\mathcal {\mathbb{R}}^n\setminus \{0\})\) satisfying \begin{gather*} \operatorname{supp}\chi \subset \left\{\xi\in {\mathbb{R}}^n:|\xi|\leq \frac 43\right\},\\ \operatorname{supp}\varphi \subset \left\{\xi\in {\mathbb{R}}^n:\frac 34\leq|\xi|\leq \frac 83\right\},\\ \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n,\\ \sum_{j\in \mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n\backslash\{0\}, \end{gather*} and denote \(\varphi_{j}(\xi)=\varphi(2^{-j}\xi)\) and \(\mathcal{P}\) the set of all polynomials.

First, we recall the definition of Morrey spaces which are a complement of \(L^{p}\) spaces.

Definition 1. [19] For \(1 \leq p < \infty\), \(0\leq\lambda < n\), the Morrey spaces \(\mathrm{M}_{p}^{\lambda}=\mathrm{M}_{p}^{\lambda}(\mathbb{R}^{n})\) is defined as the set of functions \(f\in L_{loc}^{p}(\mathbb{R}^{n})\) such that

where \(B(x_{0},r)\) denotes the ball in \(\mathbb{R}^{n}\) with center \(x_{0}\) and radius \(r\).

It is easy to see that the injection \(\mathrm{M}_{p_{1}}^{\lambda}\hookrightarrow \mathrm{M}_{p_{2}}^{\mu}\) provided \(\frac{n-\mu}{p_{2}}\geq\frac{n-\lambda}{p_{1}}\) and \( p_{2}\leq p_{1}\), and \(\mathrm{M}_{p}^{0}=L^{p}\).

If \(1\leq p_{1},p_{2},p_{3}< \infty\) and \( 0\leq\lambda_{1},\;\lambda_{2},\,\lambda_{3}< n\) with \( \frac{1}{p_{3}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\) and \( \frac{\lambda_{3}}{p_{3}}=\frac{\lambda_{1}}{p_{1}}+\frac{\lambda_{2}}{p_{2}}\), then we have the H\"{o}lder type inequality \begin{equation*} \|fg\|_{\mathrm{M}_{p_{3}}^{\lambda_{3}}}\leq\|f\|_{\mathrm{M}_{p_{1}}^{\lambda_{1}}} \|g\|_{\mathrm{M}_{p_{2}}^{\lambda_{2}}}\,. \end{equation*}

Also, for \(1\leq p< \infty\) and \(0\leq\lambda< n,\)

for all \(\varphi\in L^{1}\) and \(g\in\mathrm{M}_{p}^{\lambda}\).

Definition 2.(homogeneous Fourier-Besov-Morrey spaces ) Let \(s\in\mathbb{R}, \;0\leq\lambda< n\), \(1\leq p< +\infty \) and \(1\leq q\leq+\infty\). The space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) denotes the set of all \(u\in \mathcal{S'}(\mathbb{R}^{n})/\mathcal{P}\) such that

with suitable modification made when \(q = \infty\).

Note that the space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) equipped with the norm (6) is a Banach space. Since \(\mathrm{M}_{p}^{0}=L^{p}\), we have \(\mathcal{F} \dot{\mathcal{N}}_{p, 0, q}^{s}=F \dot{B}_{p, q}^{s}, \, \mathcal{F} \dot{\mathcal{N}}_{1, 0, q}^{s}=F \dot{B}_{1, q}^{s}=\dot{\mathcal{B}}_{q}^{s}\) and \(\mathcal{F} \dot{\mathcal{N}}_{1, 0, 1}^{-1}=\chi^{-1}\) where \(\dot{\mathcal{B}}_{q}^{s}\) is the Fourier-Herz space and \(\chi^{-1}\) is the Lei-Lin space [20].

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

Definition 3. Let \(s\in\mathbb{ R},\;1\leq p< \infty,\; 1\leq q,\rho\leq\infty, \;0\leq\lambda< n\), and \(I=[0,T),\;T\in(0,\infty]\). The space-time norm is defined on \(u(t,x)\) by \begin{eqnarray*} \|u(t,x)\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}= \Big\{\sum_{j\in \mathbb{Z}}2^{jqs}\| \varphi_{j}\widehat{u}\| _{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} ^q \Big\}^{1/q}, \end{eqnarray*} and denote by \(\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\) the set of distributions in \(S'(\mathbb{R}\times\mathbb{R}^{n})/\mathcal{P}\) with finite \(\|.\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}\) norm. According to Minkowski inequality, we have \begin{equation*} \begin{gathered} L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\hookrightarrow \mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}),\quad \text{if } \rho\leq q, \\ \mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}) \hookrightarrow L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}),\quad \text{if } \rho\geq q\,, \end{gathered} \end{equation*} where \(\|u(t,x)\|_{L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})} :=\Big(\int_I\|u(\tau,\cdot)\|^\rho_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}}d\tau\Big)^{1/\rho}\,.\)

Our first main result is the following theorem.

Theorem 4. Assume that the abstract operator \(P\) satisfies the condition (3). If \(0\leq\lambda< 3,\, 1\leq q\leq \infty,\, 1\leq p< \infty\) and \(\max\{1+\sigma,0\}< \alpha< 2+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma\) then there exists a constant \(C_{0}\) such that for any \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\) satisfies \( \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}< C_{0}\mu\), the equation(1) admits a unique global solution \(u\), \begin{equation*} \|u\|_{\mathcal{L}^{\infty}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|u\|_{\mathcal{L}^{1}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq2C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma}} \end{equation*} where \(C\) is a positive constant.

Now, we give some remarks about this result.

Remark 1. The result stated in Theorem 4 is based on the works [3]. In particular, this result remains true if we replace the Fourier-Besov-Morrey space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) by other functional spaces such as Fourier-Herz space \(\mathcal{\dot{B}}_{q}^{ s}\), Fourier-Besov space \(\mathrm{F\dot{B}}_{p,q}^{s}\) and Lei-Lin space \(\chi^{-1}\).

The analyticity of the solution is also an important subject developed by several researchers, particularly with regard to the Navier-Stokes equations, see [17] and its references. In this paper, we will prove the Gevrey class regularity for (1) in the Fourier-Besov-Morrey space. Inspired by this, we have obtained the following specific results.

Theorem 5. Let \(0\leq\lambda< 3,\,1\leq q\leq \infty,\,1\leq p< \infty\) and \(\max\{1+\sigma,0\}< \alpha< \min \{2,2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\}\). There exists a constant \(C_{0}\) such that, if \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\) satisfies \(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}< C_{0}\mu\), then the Cauchy problem (1) admits a unique analytic solution \(u\), in the sense that $$ \|e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u\|_{\mathcal{L}^{\infty}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u\|_{\mathcal{L}^{1}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq2C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,.$$

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

Lemma 6.[21] Let \(1\leq q\leq p< \infty,\, 0\leq\lambda_{1},\lambda_{2}< n,\;\frac{n-\lambda_{1}}{p}\leq\frac{n-\lambda_{2}}{q}\), and let \(\gamma\) be a multiindex. If \(supp(\widehat{f})\subset\{|\xi|\leq A2^{j}\}\) then there is a constant \(C>0\) independent of \(f\) and \(j\) such that

3. The well-posedness

First, we consider the linear nonhomogeneous dissipative equation for which we recall the following result.

Lemma 7. [22] Let \(I=[0,T),\;0< T\leq \infty,\,s\in\mathbb{R},\,0\leq\lambda< 3, 1\leq p< \infty\), and \(1\leq q,\rho\leq \infty.\) Assume that \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) and \(f\in \mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s-\alpha+\frac{\alpha}{\rho}}\right)\). Then the Cauchy problem (8) has a unique solution \(u(t,x)\) such that for all \(\rho_{1}\in[\rho,+\infty]\) \begin{eqnarray*} \mu^{\frac{1}{\rho_{1}}}\|u\|_{\mathcal{L}^{\rho_{1}}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\frac{\alpha}{\rho_{1}} }\right)}\leq \Big(\frac{4}{3}\Big)^{\alpha}\Big(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}} +\mu^{\frac{1}{\rho}-1}\|f\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\frac{\alpha}{\rho}-\alpha})}\Big) \end{eqnarray*} and \begin{eqnarray*} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\right)} +\mu\|u\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\alpha }\right)} \leq(1+\left(\frac{4}{3}\right)^{\alpha})\left(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s }}+\|f\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s }\right)}\right). \end{eqnarray*} If in addition \(q\) is finite, then u belongs to \(\mathcal{C}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\).

Proposition 8. Let \(1\leq p< \infty,\,1\leq \rho,\,q\leq \infty,\,1+\sigma< \alpha< \frac{2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}{2-\frac{1}{\rho}},\, 0\leq\lambda< 3,\,I=[0,T),\,T\in(0,\infty]\), and set \begin{equation*} X=\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\frac{\alpha}{\rho}+\sigma}\right), \end{equation*} with the norm \begin{equation*} \|u\|_{X}=\|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|u\|_{\mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\frac{\alpha}{\rho}+\sigma}\right) }\,. \end{equation*} There exists a constant \(C=C(p,q)>0\) depending on \(p,q\) such that

Proof. Let us introduce some notations about the standard localization operators. We set \begin{align*} u_{j}=\dot{\Delta}_{j}u=\left(\mathscr{F}^{-1} \varphi_{j}\right)* u,\;\;\;\dot{S}_{j}u=\sum_{k\leq j-1}\dot{\Delta}_{k}u,\;\;\; \widetilde{\dot{\Delta}}_{j}u=\sum_{|k-j|\leq 1}\dot{\Delta}_{k}u,\;\;\; \forall j\in \mathbb{Z}\,. \end{align*} Using the decomposition of Bony's paraproducts for the fixed \(j\), we have \begin{align*} \dot{\Delta}_{j}(u\partial_{i}Pv) &=\sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}u \dot{\Delta}_{k}(\partial_{i}Pv))+ \sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}(\partial_{i}Pv) \dot{\Delta}_{k}u)+\sum_{k\geq j-3}\dot{\Delta}_{j}(\dot{\Delta}_{k}u \widetilde{\dot{\Delta}}_{k}(\partial_{i}Pv))\\ &=I_{j}+II_{j}+III_{j}\,. \end{align*} To prove this proposition, we can write

We treat the above three terms differently. First, using Young's inequality (5) in Morrey spaces, and Lemma 6 with \(|\gamma|=0\), we get \begin{align*} \|\widehat{I_{j}}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}&\leq \sum_{|k-j|\leq 4}\|\widehat{\dot{S}_{k-1}u \dot{\Delta}_{k}(\partial_{i}P v)}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}\\ &\leq \sum_{|k-j|\leq 4}\|\varphi_{k}\mathcal{F}(\partial_{i}P v)\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}\| \varphi_{l}\hat{u}\|_{L^{\infty}\left(I ,L^{1}\right)}\\ &\leq\sum_{|k-j|\leq 4}\|\varphi_{k}\mathcal{F}(\partial_{i}P v)\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\| \widehat{u}_{l}\|_{L^{\infty}\left(I ,\mathrm{M}_{p}^{\lambda}\right)}\\ &\lesssim\sum_{|k-j|\leq 4}2^{k\sigma}\|\widehat{v}_{k}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\Big(\sum_{l\leq k-2}2^{l(\alpha-1-\sigma)q'}\Big)^{\frac{1}{q'}} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\\ &\lesssim\sum_{|k-j|\leq 4}2^{k(\alpha-1)}\|\widehat{v}_{k}\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda} )} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Multiplying by \(2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\), and taking \(l^{q}-\)norm of both sides in the above estimate, we obtain Likewise, we prove that To evaluate \(III_{j}\), we apply the Young inequality (5) in Morrey spaces and Lemma 6 with \(|\gamma|=0\), we obtain \begin{eqnarray*} {2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\|\widehat{III_{j}}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}}\\ &\leq& 2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} \big\|\mathcal{F}(\dot{\Delta}_{k}u\dot{\Delta}_{l}(\partial_{i}P v))\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ &\leq&2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} \big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\big\|\varphi_{l}\mathcal{F}(\partial_{i}P v)\big\|_{L^{\infty}\left(I,L^{1}\right)}\nonumber\\ &\leq& 2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} 2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} 2^{l\sigma}\big\|\widehat{v}_{l}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\nonumber\\ &\leq& \sum_{k\geq j-3}\sum_{l=-1}^{1}2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+ \frac{\lambda}{p}+\sigma)(j-k)}2^{(\alpha-1)l} \big(2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)k} \big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\big)\\ \quad&\times&\big(2^{(l+k)(-(\alpha-1)+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)} \big\|\widehat{v}_{l+k}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\big)\,. \end{eqnarray*} Taking the \(l^{q}-\)norm on both sides in the above estimate and using H\"{o}lder's inequalities for series with \(-2(\alpha-1)+\frac{\alpha}{\rho}+\frac{3}{p'}+\frac{\lambda}{p}+\sigma>0\), we get \begin{eqnarray*} {\Big(\sum_{j\in\mathbb{Z}}2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)q} \|\widehat{III_{j}}\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}^{q}\Big)^{\frac{1}{q}}}\\ &\leq&\Big(\sum_{j\in\mathbb{Z}}\Big(\sum_{m\leq 3}\sum_{l=-1}^{1} 2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)m}2^{(\alpha-1)l} 2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)(j-m)}\\ &&\times\big\|\widehat{u}_{j-m}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} 2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)(j-m+l)} \big\|\widehat{v}_{j-m+l}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\Big)^{q}\Big)^{\frac{1}{q}}\\ &\leq&\sum_{l=-1}^{1}\sum_{m\leq 3} 2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)m}2^{(\alpha-1)l} \|u\|_{\mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma}\right)}\\ &&\times\|v\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,\infty}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{eqnarray*} Since \(l^{q} \hookrightarrow l^{\infty}\), we obtain Estimates (10), (11), (12) and (13) yield (9).

Lemma 9. Let \(X\) be a Banach space with norm \(\|.\|_{X}\) and \(B:X\times X\longmapsto X\) be a bounded bilinear operator satisfying \begin{equation*} \|B(u,v)\|_{X}\leq \eta \|u\|_{X}\|v\|_{X} \end{equation*} for all \(u,v\in X \) and a constant \(\eta >0\). Then, if \(0< \varepsilon< \frac{1}{4\eta}\) and if \(y\in X\) such that \(\|y\|_{X}\leq\varepsilon\), the equation \(x:=y+B(x,x)\) has a solution \(\overline{x}\) in \(X\) such that \(\|\overline{x}\|_{X}\leq 2 \varepsilon\). This solution is the only one in the ball \(\overline{B}(0,2\varepsilon)\). Moreover, the solution depends continuously on \(y\) in the sense: if \(\|y'\|_{X}\leq \varepsilon ,\;x'=y'+B(x',x')\), and \(\|x'\|_{X}\leq2\varepsilon\), then \begin{equation*} \|\overline{x}-x'\|_{X}\leq \frac{1}{1-4\varepsilon \eta}\|y-y'\|_{X}\,. \end{equation*}

Proof of theorem 4

Proof. To ensure the existence of global solutions with small initial data, we will use Lemma 9. In the following, we consider the Banach space \begin{equation*} X=\mathcal{L}^{\infty}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{1}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\,. \end{equation*} First, we start with the integral equation

We notice that \(B(u,v)\) can be thought as the solution to the heat Equation (8) with \(u_{0}=0\) and force \(f=\nabla\cdot(u(\tau)\nabla Pv(\tau))\). According to Lemma 7 with \(s=1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\) and Proposition 8 with \(\rho=1 \), we obtain \begin{align*} \|B(u,v)\|_{X}&\leq \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big) \|\nabla\cdot(u\nabla Pv)\|_{\mathcal{L}^{1}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\\ &\leq \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big)C \mu^{-1} \|u\|_{X}\|v\|_{X}\,. \end{align*} By Lemma 9, we know that if \(\|e^{-\mu t\Lambda^{\alpha}}u_{0}\|_{X}< R\) with \(R=\frac{\mu}{4(1+(\frac{4}{3})^{\alpha})C}\)\\ then the Equation (14) has a unique solution in \(B(0,2R):=\{x\in X:\|x\|_{X}\leq 2R\}\). To prove \(\|e^{-\mu t\Lambda^{\alpha}}u_{0}\|_{X}< R\), notice that \(e^{-\mu t\Lambda^{\alpha}}u_{0}\) is the solution to the dissipative equation with \(u_{0}=u_{0}\) and \(f=0\). So, Lemma 7 yields If \(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\leq C_{0}\mu\) with \(C_{0}=\frac{1}{4(1+(\frac{4}{3})^{\alpha})^{2}C}\), then (14) has a unique global solution \(u\in X\) satisfying \begin{eqnarray*} \|u\|_{X} \leq 2 \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big) \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{eqnarray*}

Proof of theorem 5

Proof. To prove Theorem 5, we note \(a(t, x) :=e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u(t, x)\,.\) Using the integral Equation (14), we obtain \begin{align*} a(t, x)&=e^{\mu(\sqrt{t}|D|^{\frac{\alpha}{2}}-\frac{1}{2}t\Lambda^{\alpha})} e^{-\frac{1}{2} \mu t \Lambda^{\alpha}} u_{0}\\ &\quad+\int_{0}^{t}e^{\mu[(\sqrt{t}-\sqrt{\tau})|D|^{\frac{\alpha}{2}}-\frac{1}{2}(t-\tau)\Lambda^{\alpha}]} e^{-\frac{1}{2}\mu(t-\tau)\Lambda^{\alpha}} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} \nabla \cdot(u \nabla(Pu))d \tau\\ &:=L u_{0}+\widetilde{B}(u,u)\,. \end{align*} In order to obtain the Gevrey class regularity of the solution, we use Lemma 9. Firstly, we start by estimating the term \(L u_{0}=e^{-\frac{1}{2}\mu (\sqrt{t}|D|^{\frac{\alpha}{2}}-1)^{2}+\frac{\mu}{2}}e^{-\frac{1}{2} \mu t \Lambda^{\alpha}} u_{0}\,.\) Using the Fourier transform, multiplying by \(\varphi_{j}\) and taking the \(\mathrm{M}_{p}^{\lambda}\)-norm we obtain \begin{equation*} \|\varphi_{j}\widehat{L u_{0}}\|_{\mathrm{M}_{p}^{\lambda}} \leq C e^{-\frac{1}{2}\mu t 2^{j \alpha}(3 / 4)^{\alpha}}\left\|\varphi_{j}\widehat{u_{0 }}\right\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{equation*} Multiplying by \(2^{j(1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\) and taking \(l^{q}-\)norm we get \begin{equation*} \left\|L u_{0}\right\|_{\mathcal{L}^{\infty}\left([0,+\infty) ; \mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} Similarly \begin{equation*} 2^{j(1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\left\|\varphi_{j}\widehat{L u_{0}}\right\|_{L^{1}\left([0,+\infty) ; \mathrm{M}_{p}^{\lambda}\right)} \leq \left(\int_{0}^{\infty} e^{-\frac{1}{2}\mu t 2^{j\alpha}(3 / 4)^{\alpha}} 2^{j \alpha} d t \right)2^{j (1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\left\|\varphi_{j}\widehat{u_{0}}\right\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{equation*} We conclude by taking \(l^{q}-\)norm that \begin{equation*} \mu\left\|L u_{0}\right\|_{\mathcal{L}^{1}\left([0,+\infty) ; \mathcal{F\dot{N}}_{p,\lambda, q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} Finally, \begin{equation*} \left\|L u_{0}\right\|_{X} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} On the other hand, we notice that \(\widetilde{B}(u, v)\) as \(\widetilde{B}\left(e^{-\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} a, e^{-\mu\sqrt{\tau} |D|^{\frac{\alpha}{2}}} b\right)\) with \(b :=e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} v\). Since \(e^{\mu[(\sqrt{t}-\sqrt{\tau})|\xi|^{\frac{\alpha}{2}}-\frac{1}{2}(t-\tau)|\xi|^{\alpha}]}\) is uniformly bounded on \(t \in(0, \infty)\) and \(\tau \in[0, t]\), it sufficient to consider the estimate of \(\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\) for which we prove the flowing lemma.

Lemma 10. Let \(1\leq p< \infty,\,1\leq q \leq \infty,\,0\leq\lambda< 3,\,1 +\sigma< \alpha< \min \{2,2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\},\, I=[0,T),\,T\in(0,\infty]\), and set \begin{equation*} X=\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\,. \end{equation*} There exists a constant \(C=C(p,q)>0\) depending on \(p,q\) such that \begin{equation*} \|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\leq C \mu^{-1}\|a\|_{X}\|b\|_{X}\,. \end{equation*}

Proof. Based on the same procedure in the proof of Proposition 8, we evaluate the estimate of \(\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\), in fact, we have for fixed \(j\) \begin{align*} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}(u \partial_{i}(Pv)) &=\sum_{|k-j| \leq 4} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{S}_{k-1} u \dot{\Delta}_{k} \partial_{i}(Pv)\right)\\ &\quad+\sum_{|k-j| \leq 4} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{S}_{k-1} \partial_{i}(Pv) \dot{\Delta}_{k} u\right)\\ &\quad+\sum_{k \geq j-3} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{\Delta}_{k} u \widetilde{\Delta}_{k} \partial_{i}(Pv)\right)\\ &:=S_{1,j}+S_{2,j}+S_{3,j}\,. \end{align*} Since \(e^{\mu\sqrt{\tau}\left(|\xi|^{\frac{\alpha}{2}}-|\xi-\eta|^{\frac{\alpha}{2}}-|\eta|^{\frac{\alpha}{2}}\right)}\) is uniformly bounded on \(\tau\) when \(\alpha\in[0,2]\), we obtain \begin{align*} \|\widehat{S_{1,j}}\|_{\mathrm{M}_{p}^{\lambda}} &=\|\sum_{|k-j| \leq 4}\varphi_{j} e^{\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}}\mathscr{F}\big(\dot{S}_{k-1} u \dot{\Delta}_{k} \partial_{i}(Pv)\big)\|_{\mathrm{M}_{p}^{\lambda}}\\ &=\|\sum_{|k-j| \leq 4}\varphi_{j} e^{\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}}\big[(\sum_{l \leq k-2} e^{-\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}} \widehat{a}_{l})* \big(e^{-\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}} \mathscr{F}(\dot{\Delta}_{k} \partial_{i}(Pb))\big) \big]\|_{\mathrm{M}_{p}^{\lambda}}\\ &=\|\sum_{|k-j| \leq 4} \varphi_{j}\int_{\mathbb{R}^{3}} e^{\mu\sqrt{\tau}\big(|\xi|^{\frac{\alpha}{2}}-|\xi-\eta|^{\frac{\alpha}{2}}-|\eta|^{\frac{\alpha}{2}}\big)}\big(\sum_{l \leq k-2} \widehat{a_{l}}\big)(\xi-\eta)\mathscr{F}(\dot{\Delta}_{k} \partial_{i}(Pb))(\eta) d \eta\|_{\mathrm{M}_{p}^{\lambda}}\\ &\leq C\|\sum_{|k-j| \leq 4}\mathscr{F}\big(\dot{S}_{k-1}a \dot{\Delta}_{k} \partial_{i}(Pb)\big)\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{align*} The same calculus as in Proposition 8 gives \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}&2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\| \widehat{S_{1,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}&\lesssim \|a\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|b\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Similarly, we show that \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\|\widehat{S_{2,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}\lesssim\|b\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|a\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Similarly, \begin{equation*} \left\|\widehat{S_{3,j}}\right\|_{\mathrm{M}_{p}^{\lambda}}\leq \sum_{k \geq j-3} \sum_{|l-k| \leq 1}\left\|\mathcal{F}\left(\dot{\Delta}_{k} a \dot{\Delta}_{l}\left(\partial_{i}(Pb)\right)\right)\right\|_{M_{p}^{\lambda}}\,. \end{equation*} Using again the same procedure described in the proof of Proposition 8 we obtain \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\|\widehat{S_{3,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}\lesssim\|a\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|b\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Finally, \begin{equation*} \left\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\right\|_{\mathcal{L}^{1}\left(I; \mathcal{F} \dot{\mathcal{N}}_{p, \lambda, q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \mu^{-1}\|a\|_{X}\|b\|_{X}\,. \end{equation*}

To finish the proof of Theorem 5, it is easy to obtain the requested result by repeating the same step in the proof of Theorem 4 and Proposition 8.

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 article, we consider the following non-autonomous strongly damped wave equation with linear memory on an unbounded domain: with initial data

Let \(\varepsilon, \alpha,\beta>0\), c is a positive constant and \(\mu(s)\leq 0\) for every \(s\in~\mathbb{R}^{+}\), where \(\triangle\) is the Laplacian with respect to the variable \(x\in\mathbb{R}^{n}\) with \(n=3\), \(u=u(t,x)\) is a real function of \(x\in\mathbb{R}^{n},(n=3)\),and \(t\geq\tau~,~\tau\in\mathbb{R}\). The function \(g(x,t)\in L_{b}^{2}(\mathbb{R},L^{2}(\mathbb{R}^{n}))\) is time-dependent external force, and \(W(x,t)\) is an independent two sided real-valued wiener processes of probability space.

Following a well-established procedure first devised by [1], we introduce a Hilbert " history " space \(\mathfrak{R}_{\mu }=L_{\mu}^{2}(\mathbb{R}^{+},H^{1}(\mathbb{R}^{n}))\) with the inner product and new variants. Then the Equation (1) can be transformed into the following system with the initial-boundary conditions The following conditions are necessary to obtain our main results.

1. concerning the memory kernel \(\mu\), it is required to satisfy the following hypotheses:
   \begin{equation}\label{equ6} \left\{\begin{aligned} &\ \mu\in\mathbb{C}^{1}(\mathbb{R^{+}})\cap\mathbb{L}^{1}(\mathbb{R^{+}}),\mu(s)\geq0 ,\mu'(s)\leq0~, \forall s\in\mathbb{R^{+}},\\ &\ \mu'(s)+\delta\mu(s)\leq 0~, \forall s\in\mathbb{R^{+}} and ~\delta> 0,\\ \end{aligned}\right. \end{equation}

   (6)
   and denote
   \begin{equation}\label{equ7} k_{0}:=\int_{0}^{\infty}\mu(s)ds< \infty. \end{equation}

   (7)
2. for the nonlinear term \(f(u)\), we assume that \(f\in\mathbb{C}^{1}(\mathbb{R})\) with \(f(0)=0\) ,and it satisfies the following growth conditions. There exist constant \(C_{1}>0\) such that
   \begin{equation}\label{equ8} |f'(u)|\leq C_{1}(1+|u|^{p})~,\forall~u\in\mathbb{R},0\leq p\leq 4,~when ~n=3. \end{equation}

   (8)
   And there exists constants \(k>0\) and \(\nu_{1}>\) such that for any \( \nu\in(0,\nu_{1})\), there exist \(C_{\nu}>0\) satisfying
   \begin{equation}\label{equ9} kF(u)-\nu u^{2}+C_{\gamma}\leq uf(u),~\forall~u\in \mathbb{R}, \end{equation}

   (9)
   and
   \begin{equation}\label{equ10} \limsup_{|u|\rightarrow\infty}\frac{f(u)}{u}\leq 0~,\forall~u\in~\mathbb{R}, \end{equation}

   (10)
   where \(F(s)=\int_{0}^{s}f(r)dr\). About the time-dependent forcing \(g(x,t)\) term we assume that \(g(x,t)\in L_{b}^{2}(\mathbb{R},L^{2}(\mathbb{R}^{n}))\), where space of translation -bounded function \(L_{b}^{2}(\mathbb{R},L^{2}(\mathbb{R}^{n}))=\{g(x,t)\in L_{loc}^{2}(\mathbb{R},L^{2}(\mathbb{R}^{n})) :\sup_{t\in\mathbb{R}}\int_{t}^{t+1}(\int_{\mathbb{R}^{n}}|g(\cdot,r)|^{2}dx)dr< \infty\}\) with the norm
   \begin{equation}\label{equ11} \|g(x,t)\|^{2}=\sup_{t\in\mathbb{R}}\int_{t}^{t+1}\int_{\mathbb{R}^{n}}|g(x,r)|^{2}dxdr < \infty,~~\forall r\in\mathbb{R}. \end{equation}

   (11)
   Finally, we introduce the product Hilbert space $$E=H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\times\mathfrak{R}_{\mu }.$$

In recent years, there have many results on the dynamics of a variety of systems related to Equation (1). The deterministic hyperbolic equations with memory have been studied to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects, see[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. For instance, Borini and Pata [13] proved the existence of a Uniform attractor for a strong damping wave equation with linear memory on a bounded domain. Qiaozhen Ma,Chengkui Zhong[7] obtained the strong global attractors, and Ghidaglia, and Marzocchi [14] showed global attractors and their finite Dimension. Crauel and Flandoli [15,16,17, 18] studied the random attractors for a stochastic dynamical system. Recently, many authors have established the existence of random attractors for other equations (see[13, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]). For Equation (1), there are fewer results and most previous authors have concentrated to the deterministic case, but there are no results of random attractors for the Equation (1).

In general, to prove the existence of random attractors for (1) in E, we must establish the pullback asymptotic compactness of solutions. Since Sobolev embedding are not compact on \(\mathbb{R}^{n}\), we cannot get the desired asymptotic compactness directly from the regularity of solutions. We were overcome the difficulty by using uniform estimates on the tails of solutions outside a bounded ball in \(\mathbb{R}^{n}\) and decomposing the solutions in a bounded domain as in [2, 6, 21, 29].

The rest of the article is organized as follows. In Section 2 we recall some basic concepts related to RDS and a random attractor for the random dynamical system. In Section 3, we devote to uniform estimates and the existence of bounded absorbing sets for the solutions and pullback compactness. In Section 4, the compactness of the RDS is established by the decomposition of a solution of the random differential equation into two parts. In Section 5, we prove the asymptotic compactness of the solutions, finally existence and uniqueness of a random attractor in \(E\).

2. Preliminaries

In this Section, we recall some basic concepts related to RDS and a random attractor for RDS in [16, 17], which are important for getting our main results. Let \((\Omega,\mathcal{F},P )\) be a probability space and \((X,d)\) is a Polish space with the Borel \(\sigma\)-algebra \(B(X)\).The distance between \(x\in\ X\) and \(B{\subseteq X}\) is denoted by \(d(x,B)\). If \(B{\subseteq X}\) and \(C{\subseteq X}\), the Hausdorff semi-distance from B to C is denoted by \( d(B,C)=\sup_{x\in B}d(x,C)\).

Definition 1. (\(\Omega,\mathcal{F},P,(\theta_{t})_{t\in\mathbb{R}}\)) is called a metric dynamical system if \(\theta\) : \(\mathbb{R} \times \Omega \longrightarrow \Omega\) is \((\mathcal{B}(\mathbb{R}) \times \mathcal{F},\mathcal{F})\)-measurable, \(\theta_{0}\) is the identity on \(\Omega\), \(\theta_{s+t}\) = \(\theta_{t}\circ\theta_{s}\), for all s,t \(\in\mathbb{R}\) and \(\theta_{0}\)P = P for all t\(\in\mathbb{R}\).

Definition 2. A mapping \(\Phi(t,\tau,\omega,x):\mathbb{R}^{+}\times \mathbb{R} \times\Omega \times X\rightarrow X\) is called continuous cocycle on X over \(\mathbb{R}\) and \((\Omega,\mathcal{F},P,(\theta_{t})_{t\in \mathbb{R}})\), if for all \(\tau\in\mathbb{R},\omega\in\Omega\) and \(t,s \in \mathbb{R}^{+}\), the following conditions are satisfied:

1. \(\Phi(t,\tau,\omega,x):\mathbb{R}^{+}\times \mathbb{R}\times\Omega \times X\rightarrow X\) is a \((\mathcal{B}(\mathbb{R}^{+})\times \mathcal{F},\mathcal{B}(\mathbb{R}))\) measurable mapping
2. \(\Phi(0,\tau,\omega,x)\) is identity on X.
3. \( \Phi(t+s,\tau,\omega,x)=\Phi(t,\tau+s,\theta_{s}\omega,x)\circ\Phi(s,\tau,\omega,x)\)
4. \(\Phi(t,\tau,\omega,x):X \rightarrow X \) is continuous.

Definition 3. Let \(2^{X}\) be the collection of all subsets of X, set valued mapping \((\tau,\omega)\mapsto \mathcal{D}(t,\omega):\mathbb{R}\times\Omega\mapsto 2^{X}\) is called measurable with respect to \(\mathcal{F}\) in \(\Omega\) if \(\mathcal{D}(t,~\omega)\) is a(usually closed ) nonempty subset of X and the mapping \(\omega\in\Omega\mapsto d(X,B(\tau,\omega))\) is \((\mathcal{F},\mathcal{B}(\mathbb{R}))\) -measurable for every fixed \(x\in X\) and \(\tau\in \mathbb{R} \). Let \(B={B(t,\omega)\in \mathcal{D}(t,\omega):\tau\in \mathbb{R}, \omega\in\Omega}\) is called a random set.

Definition 4. A random bounded set \(B=\{B(\tau,\omega):\tau\in \mathbb{R},\omega\in\Omega\}\in \mathcal{D}\) of X is called tempered with respect to~\(\{\theta(t)\}_{t\in\Omega}\), if for p-a.e \(\omega\in\Omega~,\) $$\lim_{t~\mapsto\infty}~ e^{-\beta t}~d(B(\theta_{-t} \omega)) =0 ,~ \forall~ \beta~ > 0,$$ where $$d(B)=\sup_{x\in B}\|x\|_{X}.$$

Definition 5. Let \(\mathcal{D}\) be a collection of random subsets of X and \(K=\{K(\tau,\omega):\tau\in \mathbb{R},\omega\in\Omega\}\in \mathcal{D}\), then \(K\) is called an absorbing set of \(\Phi\in \mathcal{D}\), if for all~\(\tau\in \mathbb{R}, \omega\in\Omega\) and \(B\in \mathcal{D}\) , there exists, \(T=T(\tau,\omega,B)> 0\) such that. $$\Phi(t,\tau,\theta_{-t}\omega,B(\tau,\theta_{-t}\omega))\subseteq K(\tau,\omega),~\forall~t~\geq T$$

Definition 6. Let \(\mathcal{D}\) be a collection of random subsets of X, then \(\Phi\) is said to be \(\mathcal{D}\) a-pullback asymptotically compact in X, if for p-a.e \(\omega\in\Omega\) , \(\{\Phi(t_{n},\theta_{-t_{n}}\omega~,x_{n})\}_{n=1}^{\infty}\) has a convergent subsequence in X when \(t_{n}\mapsto\infty\) and \(x_{n}\in{B(\theta_{-t_{n}}\omega)}\) with \(\{B(\omega)\}_{\omega\in\Omega}\in \mathcal{D}\).

Definition 7. Let \(\mathcal{D}\) be a collection of random subsets of X and \(\mathcal{A}=\{\mathcal{A}(\tau,\omega):\tau\in \mathbb{R},\omega\in\Omega\}\in \mathcal{D}\), then \(\mathcal{A}\) is called a \(\mathcal{D}\)-random attractor (or \(\mathcal{D}\)-pullback attractor ) for \(\Phi\), if the following conditions are satisfied, for all \(t\in \mathbb{R}^{+},\tau\in \mathbb{R}\) and \(\omega\in\Omega\)

1. \(\mathcal{A}(\tau,\omega)\) is compact, and \(\omega\mapsto d(x,\mathcal{A}(\omega))\) is measurable for every \(x\in X\)
2. \({\mathcal{A}(\tau,\omega)}\) is invariant, that is $$\Phi(t,\tau,\omega,\mathcal{A}(\tau,\omega))=~\mathcal{A}(\tau+t,\theta_{t}\omega),\forall~t~\geq \tau .$$
3. \(\mathcal{A}(\tau,\omega)\) attracts every set in \(\mathcal{D}\), that is for every \(B=\{B(\tau,\omega):\tau\in \mathbb{R},\omega\in\Omega\}\in \mathcal{D}\), $$\lim_{t~\mapsto\infty}~d_{X}(\Phi(t,\tau,\theta_{-t}\omega,B(\tau,\theta_{-t}\omega)),\mathcal{A}(\tau,\omega))=0.$$ Where \(d_{X}\) is the Hausdorff semi-distance given by $$d_{X}(Y,Z)=\sup_{y\in Y} \inf_{z\in Z}\|y-z\|_{X}$$ for any \(Y\in X\) and \(Z\in X \).

Lemma 8. Let \(\mathcal{D}\) be a neighborhood-closed collection of \((\tau,\omega)\)- parameterized families of nonempty subsets of X and \(\Phi\) be a continuous cocycle on X over \(\mathbb{R}\) and \((\Omega,\mathcal{F},P,(\theta_{t})_{t\in\mathbb{R}})\). Then \(\Phi\) has a pullback \(\mathcal{D}\)-attractor \(\mathcal{A}\) in \(\mathcal{D}\) if and only if \(\Phi\) is pullback \(\mathcal{D}\)-asymptotically compact in X and \(\Phi\) has a closed, \(\mathcal{F}\)-measurable pullback \(\mathcal{D}\)-absorbing set \(K \in\mathcal{D}\), the unique pullback \(\mathcal{D}\)-attractor \(\mathcal{A}={\mathcal{A}(\tau,\omega)}\) is given \(\mathcal{A}(\tau,\omega)=\mathbb{\bigcap}_{r\geq0}\overline{\mathbb{\bigcup}_{t\geq r}\Phi(t,\tau,\theta_{-t}\omega,K(\tau,\theta_{-t}\omega)})~ \tau\in\mathbb{R}~,\omega\in\Omega \).

3. Existence and uniqueness of solutions

In this section, we present the existence and uniqueness of solutions for the system (1)-(2). It is well known that the operator \(A=-\triangle\) with the domain \(D(A)= H^{2}(\mathbb{R}^{n})\).

We recall some important results, let \(H_{0}= L^{2}(\mathbb{R}^{n})\), \(H_{1}= H^{1}(\mathbb{R}^{n})\) and \(H_{2}=\mathfrak{R}_{\mu}=L_{\mu}^{2}(\mathbb{R}^{+},H^{1}(\mathbb{R}^{n}))\). And denote \(H^{*}_{1}= H^{-1}(\mathbb{R}^{n})\) the dual space of \(H_{1}\), as usual, we identify \(H^{*}_{0}\), the dual space of \(H_{0}\). Then we get

\(E=E(\mathbb{R}^{n})=H_{0}\times H_{1} \times H_{2}=H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\times\mathfrak{R}_{\mu}\), endowed with the usual norms on E, $$\|\mathbf{\varphi}\|^{2}_{ E}=\|\varphi\|^{2}_{H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\times\mathfrak{R}_{\mu }}.$$ Due to the Ornstein-Uhlenbeck process deducing by the Brownian motion, which holds the \(It\hat{o}\) differential equation and hence the solution is given by

Where the random variable \(|z(\omega)|\) is tempered and there is an invariant set \(\bar{\Omega}\subseteq\Omega\) of full P measure such that \(z(\theta_{t}\omega)=z(t,\omega)\) is continuous in t for every \(\omega\in\bar{\Omega}\). This equation has a random fixed point in the sense of random dynamical systems generating a stationary a solution is known as the stationary Ornstein-Uhlenbeck process (see [16, 17, 29,35] for more details).

Next we need to transform the stochastic system into deterministic with a random parameter, then show that it generates a random dynamical system. In fact, we define a cocycle for problem (12)-(14). Let

by (15) and (4), (5), we can obtain the following random evolution equation with the initial-boundary conditions Which, in contrast to the stochastic differential Equation (1)-(2), can by analysis pathwise with deterministic calculus, define $$\varphi= \left(% \begin{array}{cc} u \\ v \\ \eta\\ \end{array} \right), $$ $$L\varphi=\left(% \begin{array}{ccc} \varepsilon u -v \\ \varepsilon(\varepsilon-\alpha A)u+\beta Au -(\varepsilon -\alpha A)v +\int_{0}^{\infty}\mu(s)A\eta(s)ds \\ \varepsilon u-v +\eta_{s} \\ \end{array} \right) $$ and $$Q\left(\varphi,\omega,t\right)= \left(% \begin{array}{cc} cuz(\theta_{t}\omega) \\ cu(\varepsilon-\alpha A)z(\theta_{t}\omega)+c^{2}uz^{2}(\theta_{t}\omega)+cvz(\theta_{t}\omega)-f(u)+g(x,t)\\ cuz(\theta_{t}\omega) \\ \end{array} \right) $$ Then the following equation is equivalent to the system (15)-(17) In line with [36], we know that -L is the infinitesimal generator of \(C^{0}\) semigroup \(e^{-Lt}\) on E for \(t>0\), by the assumptions (6)-(11). It is easy to check \(Q(\varphi,t,\omega):E\rightarrow E\) is locally Lipschitz continuous with respect to \(\varphi\), by the classical semigroup theory concerning the (local) existence and uniqueness solutions of evolution differential equation, we have the following theorem.

Theorem 9. Under the condition (6)-(11) and for each \(\tau\in\mathbb{R},\omega\in\Omega\) and for any \(\varphi_{\tau}\in E\), there exists \(T>0\) such that (18) has a unique mild function \(\varphi(t,\tau,\omega,\varphi_{\tau})\in C([\tau,\tau+T);E)\) and \(\varphi(t)\) satisfies the integral equation

\(\varphi(t,\tau,\omega,\varphi_{\tau})\) is jointly continuous into t and measurable in \(\omega\).

From Theorem 9, we know that for P-a.s. each \(\omega\in\Omega\), the following results hold for all \(\mathrm{T}>0\)

1. if \(\varphi_{\tau}(\omega)\in E\) then \(\varphi(t,\omega,\varphi_{\tau}(\tau))\in C([\tau,\tau+\mathrm{T});E)\),
2. \(\varphi(t,\tau,\omega,\varphi_{\tau})\) is jointly continuous into t and measurable in \(\varphi_{\tau}(\omega)\),
3. the solution mapping of (18) satisfies the properties of continuous cocycle.

We notice that a unique solution \(\varphi(t,\tau,\omega,\varphi_{\tau})\) of (18) can define a continuous random dynamical system over \(\mathbb{R}\) and (\(\Omega,\mathcal{F},P,(\theta_{t})_{t\in\mathbb{R}}\)). Hence the solution mapping generates a random dynamical system. Moreover, We also define the following transformation: Similar to (18), we get that where $$\psi= \left(% \begin{array}{cc} u \\ v \\ \eta\\ \end{array} \right), $$ $$H\psi=\left(% \begin{array}{ccc} \varepsilon u -v \\ \varepsilon(\varepsilon-\alpha A)u+\beta Au -(\varepsilon -\alpha A)v +\eta \\ \varepsilon u-v + \eta_{s} \\ \end{array} \right) $$ and $$Q(\psi,\omega,t)= \left(% \begin{array}{cc} 0 \\ cvz(\theta_{t}\omega)-f(u)+q(x,t)\\ 0 \\ \end{array} \right) $$ We introduce the isomorphism \(T_{\epsilon}Y= (u,u_{t},\eta)^{\top}\) , \(Y= (u,v,\eta)^{\top}\in E\) which has inverse isomorphism \(T_{-\epsilon}Y= (u,v-\varepsilon u,\eta)^{\top}\), it follows that \((\theta,\psi)\) with mapping is a random dynamical system from a above discussion, we show that the two RDS are equivalent.

4. Random absorbing set

In this section, we will show the existence of a random absorbing set for the RDS ~~\(\varphi\left(t,\tau,\omega,\varphi_{\tau}(\omega)\right),t\geq 0\) in the space E. Let \(\varphi=(u,v,\eta)^{\top}=\left(u,u_{t}+\varepsilon u-cuz(\theta_{t}\omega),\eta\right)^{\top}\), where \(\varepsilon\) is chosen as

Lemma 10. For any \(\varphi=(u,v,\eta)^{\top}\in E\) we have

Proof. This is easily obtained by simple computation.

Lemma 11. Assume that (6)-(11) hold, then for each \(\tau\in\mathbb{R},\omega\in\Omega\), there exists tempered random absorbing ball \(B_{0}(\tau,\omega)=\{\varphi\in E:\left\|\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}(\theta_{-\tau}\omega))\right\|_{E}\leq M(\tau,\omega)\}\), \(B_{E}(M(\tau,\omega))\in\mathcal{ D(}E)\), such that for any set \(B\in \mathcal{D}(E)\), there exists \(T_{B}=T_{B}(\tau,\omega,B)>0~,\tau\in\mathbb{R},\omega\in\Omega, B \in \mathcal{D}\), so as \(\forall t\geq T_{B}\) and \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B(\tau-t,\theta_{-t}\omega)\), the solution of a system (31) satisfies

that is

Proof. For any \(\tau\in\mathbb{R},\omega\in\Omega ,t\geq\tau\), let \(\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}(\theta_{-\tau}\omega))=(u_{\tau},v_{\tau},\eta_{\tau})\in E,\) be a mild solution of (18) with initial value \(\varphi_{\tau-t}\).
Taking the inner product \((\cdot,\cdot)_{E}\) of (18) with \(\varphi(\tau)\), we find that

Let us estimate the right hand side of (29) By the Cauchy-Schwartz inequality, we find that here we estimate nonlinear term (30), by (6)-(10) and the H\"{o}lder inequality, we get that Due to (4),(6), (1-8) and Poincaré inequality, there exists positive constant \(\mu_{1},\mu_{2}\) such that it follows from (10) for each given \(\mu_{3}>0\) where \(\tilde{F}(u)=\int_{\mathbb{R}^{n}}F(u)dx\). Collecting (31)-(41) and (30), we show that where \(\mu_{4}\) depends on \(\mu_{3},\frac{\varepsilon}{2\sqrt{\lambda_{0}}},\frac{\alpha\sqrt{\lambda_{0}}}{2}\). Then substituting all together into (29) yield Since \(\sigma =\min[\frac{\varepsilon}{4},\frac{\varepsilon k}{2}]\) and \(\|\varphi\|^{2}=(\|u\|_{1}^{2}+\|v\|^{2}+\|\eta\|_{\mu,1}^{2})\), then we have the following equivalent system Let \(\Gamma(\omega)=\sigma-\mu_{4} |c||z(\theta_{t}\omega)|\) and \(|z(\theta_{t}\omega)|\) is tempered, by (13) and (14), we can choose the following inequality $$\varrho(t,\omega)=\beta(1+|c|^{2}|z(\theta_{t}\omega)|^{2}+\frac{2|c|^{4} |z(\theta_{t}\omega)|^{4}}{\varepsilon\lambda_{0}}+\|g(x,t)\|^{2}),$$ where \(\beta>0\) depend only on \(\mu_{2} ,c_{\mu_{3}},C,\varepsilon,\alpha,\lambda_{0}\). By applying Gronwall's inequality to (44) over \([\tau-t,\tau]\) and replacing \(\omega\) to \(\theta_{-\tau}\omega\), we have Suppose that Using (8) and Young inequality, the embedding theorem, we have for any bounded set \(B\) of \(E\), where \(\sup_{\varphi\in B}\|\varphi\|_{E}\leq M(\tau,\omega)\), if \(\varphi(\tau)\in B\), then For any set \(\{B(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}\in \mathcal{D},\) \(\varphi_{\tau}= (u_{\tau}(x),u_{1,\tau}(x)+\varepsilon u_{\tau}(x)-cu_{\tau}z(\theta_{t}\omega))^{\top}\in\{B(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}\in \mathcal{D}(E)\).\\ We have \begin{align*} \displaystyle\lim_{t\rightarrow\infty}\sup\left(\left\|\varphi_{\tau}(\theta_{-\tau}\omega)\right\|_{E}^{2}+2C_{8}\left(\|u_{\tau}\|^{2}+\|u_{\tau}\|^{P+2}_{H^{1}}\right)\right)e^{-2\sigma t}=0,\end{align*} When \(g(x,t)\) is only satisfied (11) which is a tempered random variable, then by (47)-(48), there exists \(B_{0}(\omega)=\{\varphi\in E:\|\varphi_{\tau}(\theta_{-\tau}\omega)\|_{E}\leq M^{2}(\tau,\omega)\}\) is closed measurable absorbing ball in D(E) and \(T=T(\tau ,B,\omega)>0\) such that \(\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})=\varphi_{\tau}\in B_{0}(\omega)\) satisfy the following result p-a.s \(\omega\in\Omega\) $$\left\|\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\right\|^{2}_{E}\leq M^{2}(\tau,\omega).$$

Next, we conduct uniform estimates on the tail parts of the solutions for large space variables when the time is sufficiently large in order to prove the pullback asymptotic compactness of the cocycle associated with equation (15) on the unbounded domain \(\mathbb{R}^{n}\). We first, choose a smooth function \(\rho\) defined on \(\mathbb{R^{+}}\) such that \(0\leq\rho(s)\leq1\) for all \(s\in\mathbb{R}\) and Then there exist constants \(\mu_{\mathrm{7}}\) and \(\mu_{\mathrm{8}}\) such that \(|\rho^{\prime}(s)|\leq\mu_{\mathrm{1}} ,|\rho^{\prime\prime}(s)|\leq\mu_{\mathrm{2}}\) for any \(s\in\mathbb{R}\), given \(r\geq1\), denote by \(\mathbb{H}_{r}=\{x\in\mathbb{R}^{n}:|x|< r\}\) and \(\{\mathbb{R}^{n}\backslash\mathbb{H}_{r}\}\) the complement of \(\mathbb{H}_{r}\). To prove asymptotic compactness of the random dynamical system we prove the following Lemma.

Lemma 12. Under conditions (6)-(11) and B=\(\{B(\tau,\omega)\}_{\tau\in\mathbb{R},\omega\in\Omega}\in \mathcal{D} \) and \(\varphi_{\tau}(\omega)\in B\). Then there exist \(\tilde{T}=\tilde{T}(\tau,B,\omega)>0\) and \(R=R(\tau,B,\omega)>1\) so that the solution \(\varphi(t,\tau,\theta_{-t}\omega,\varphi_{\tau}(\theta_{-t}\omega))\) of (15) satisfies for P-a.e \(\omega\in\Omega ,~\forall~t\geq \tilde{T} , r\geq R\)

Proof. Multiplying the second term of (15) with \(\rho\left[\frac{|x|^{2}}{r^{2}}\right]v\) in \(L^{2}(\mathbb{R}^{n})\) and integrating over \(\mathbb{R}^{n}\), we obtain

In order to estimate the left hand side, we must substituting \(v\) in the first term of (15), then we obtain the following results By second Equation (4), we have ~Integrating by parts, assumption (4),(6) and (7) and Young inequality, we can show that then and by (55), (56) and (57), it follows that and For the nonlinear term, according to (9), (10), (41) and applying Young inequality, after detailed computations, we obtain By the Cauchy-Schwartz inequality, the Young inequality and \(\|\nabla v\|^{2}\geq\lambda_{1}\|v\|^{2}\), we deduce that Combining with (51)-(63) and (51), we see that \begin{eqnarray*} &&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\left(|v|^{2}+\varepsilon^{2}|u|^{2}+(\beta-\alpha\varepsilon)|\nabla u|^{2}+|\eta(s)|_{\mu,1}^{2}+2\tilde{F}(u)\right)dx \\ &&\leq \varepsilon\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\left[|v|^{2}-\varepsilon^{2} |u|^{2} -(\beta-\alpha\varepsilon-\frac{\varepsilon\left(2m_{0} -\mu_{1}\delta\right)}{\delta})|\nabla u|^{2}\right]dx\\ &&-\varepsilon k\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\tilde{F}(u)dx-\delta\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]|\eta(s)|_{\mu,1}^{2}dx\\ &&+|c||z(\theta_{t}\omega)|\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\left[-\frac{(\alpha+1)}{2}|v|^{2}+\varepsilon^{2} |u|^{2}\right]\\ &&+|c||z(\theta_{t}\omega)|\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\left[(\beta-\alpha\varepsilon-\frac{\alpha\lambda_{1}\varepsilon}{2}-\frac{\varepsilon\left(2m_{0}-\mu_{3}\delta\right)}{\delta})|\nabla u|^{2}+\delta|\eta(s)|_{\mu,1}^{2}\right]dx\\ &&+\frac{\sqrt{2}}{r}\mu_{7}\left[\alpha(\|\nabla v\|^{2}+\|v \|^{2})+(\beta-\alpha\varepsilon)(\|\nabla u\|^{2}+\|v \|^{2})\right]\\ &&+\frac{\sqrt{2}}{r}\mu_{7}\left[\alpha|c||z(\theta_{t}\omega)(\|\nabla u\|^{2}+\|v \|^{2}+\|\eta\|_{\mu,1}^{2}+\|v \|^{2}\right]\\ &&+\frac{1}{2}\left(3\varepsilon|c||z(\theta_{t}\omega)|+c^{2}|z(\theta_{t}\omega)|^{2}\right)\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\left[|u|^{2}+|v|^{2}\right]dx\\ &&+\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]\frac{|g(x,t)|^{2}}{4\alpha\lambda_{1}}dx+\left( \mu_{2}\varepsilon -c_{\mu_{3}}|c||z(\theta_{t}\omega)|\right)\int_{\mathbb{R}^{n}}\rho\left[\frac{|x|^{2}}{r^{2}}\right]dx . \end{eqnarray*} Letting (provided \(\varepsilon\) is small enough) By all the above inequality, we can write that Setting then it follows that Integrating (67) over \([\tau,t]\), we find that, for all \(t\geq \tau\) where \(\sigma_{1}=\sigma-\frac{1}{2}\left(3\varepsilon|c||z(\theta_{t}\omega)|+ c^{2}|z(\theta_{t}\omega)|^{2}\right).\) By replacing \( \omega ~by~ \theta_{-t}\omega\), we have Since \(\mathbb{X}_{\tau}=(u_{\tau},v_{\tau},\eta_{\tau})^{\top}\in B(\tau,\theta_{-t}\omega),B\in \mathcal{D}\) is tempered, by (47)-(48), we find that the first term on the right-hand side of (69) goes to zero as \(t\rightarrow -\infty\). Hence, there exist \(T_{1}(\tau ,B,\omega)>0\) and \(\mathbb{R}_{1} = \mathbb{R}_{1}(\tau,\omega,B)\) such that for all such that \(t\geq T_{1}\) by condition (11), (13)-(14) and Lemma 10, there are \(T_{2}=T_{2}(\tau ,B,\omega)>0\) and \(\mathbb{R}_{1}=\mathbb{R}_{1}(\tau ,\omega)\geq 1\) such that for all \(t\geq T_{2}\) and \(R\geq\mathbb{R}_{1}\) By Lemma 10, there are \(T_{3}=T_{3}(\tau ,B,\omega)> 0\) and \(\mathbb{R}_{2}=\mathbb{R}_{2}(\tau ,\omega)\geq 1\) such that for all \(t\geq T_{3}\) and \(r\geq\mathbb{R}_{2}\) By letting then, from (70)-(72), it follows that

5. Decomposition of equations

In order to obtain regularity estimates later, we decompose the Equation (4) by decomposing the nonlinear term. At first, we will give the following decomposition on nonlinearity \(f=f_{0}+f_{1}\), where \(f_{0},f_{1}\in\mathbb{C}^{1}\) satisfy the following conditions for some proper constant: there is a constant \(C>0\) such that and where
\(F_{i}(s)=\int_{0}^{s}f_{i}(r)dr,i=0,1.\)
We decompose the solution \(\varphi=(u,v,\eta^{t})\) into the two parts $$\varphi=\varphi_{1}+\varphi_{2}$$ where \(\varphi_{1}=(\tilde{u},~\tilde{v},\xi)\),\(\varphi_{2}=(\bar{u},\bar{v},\zeta)\) solves the following equation, respectively, and

To prove the existence of a compact random attractor for the Random Dynamical System \(\Phi\), we get the solutions of systems (77) and (78) similar to solution of a system (26), which one decays exponentially and another are bounded in higher regular space. In order to get the regularity estimate, we will prove some a priori estimates for the solutions of systems (77) on \(\mathbb{R}^{n}\times[\tau,\infty]\)

Let \(\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})=\Phi_{\varepsilon}(t,\tau,\omega)\varphi(\tau,\omega)\) be the solution of (15)-(\ref{equ17}) or (18) with \(\varphi(\tau,\omega)\in B_{0}\), set \(\varphi=\varphi_{1}+\varphi_{2}\) are the basis of absorbing set \(\Phi\), suppose that \(T_{1}=T_{1}(\tau ,B_{0},\omega)\) for any \(\tau\in\mathbb{R},\omega\in\Omega\), where \(T_{1}=T(\tau ,B_{0},\omega)>0\) is the pullback absorbing time in Lemma 11, then it holds \(B_{1}(\tau,\omega)\subseteq B_{0}(\omega)\) such that

Lemma 13. Assume that \((75)\) hold. For any \(\tau\in \mathbb{R},\omega\in\Omega,t\geq0\), there exists \(M_{0}(\omega)>0\) such that \(\varphi_{1}(r)=\varphi_{1}(r,\tau-t,\omega,\varphi_{\tau-t})\) a solution of the system (77) satisfies.

Proof. Let \(\varphi_{1}=(\tilde{u},\tilde{v},\xi)^{\top}=(\tilde{u},\tilde{u}_{t}+\varepsilon \tilde{u},\xi)^{\top}\) be a solution of system (77), then it follows that

Taking the inner product of (\ref{equ81}) in \(L^{2}(\mathbb{R}^{n})\) with \(\varphi_{1}\) in E, we show that by Lemma 10 we have $$\left(\tilde{H}\varphi_{1},\varphi_{1}\right)\geq \frac{\varepsilon}{2}\left(\|\tilde{u}\|_{1}^{2}+\|\tilde{v}\|^{2}\right) +\frac{\alpha}{2}\|\tilde{v}\|^{2}+\frac{\varepsilon}{4}\|\xi\|_{\mu,1}^{2},$$ where \(\varepsilon\) satisfy (30). Now we estimate the third term of (83) such that According to \((75)_{2}\) and \((75)_{3}\), we get Thus, combining with (82), (83) and (81), it follows that where \(\rho=\varepsilon c_{\vartheta}\) and \(\tilde{\sigma}=\min(\frac{\varepsilon}{2},\frac{\alpha}{2},\frac{\varepsilon}{4},k_{0}\varepsilon )\) hence Together \((75)_{1}, (86)\) and (88), using Gronwall's inequality over \([r,\tau-t]\), such that, by definition of \(B_{0}(\omega)\) and Lemma 11,

Lemma 14. For any \(\tau\in\mathbb{R},\omega\in\Omega\) and \(t>0\), there exist positive constant \(\sigma_{1}\geq0\), such that \(\varphi_{1}(r)\in B ,r\in\mathbb{R},\omega\in\Omega ,B \in \mathcal{D}\), the solution of system (77) satisfies

Proof. Let \(\varphi_{1}=(\tilde{u},\tilde{v},\xi)^{\top}=(\tilde{u},\tilde{u}_{t}+\varepsilon \tilde{u},\xi)^{\top}\) be a solution of (77) similar to Lemma 13. Then from \((75)\), there exists \(\tilde{\sigma}(\omega)\geq0\) and \(\bar{\varrho}(\omega)\) such that $$F_{0}(\tilde{u})\geq 0 ,f_{0}(\tilde{u})\tilde{u}\geq 0 ,\forall~\tilde{u}\in\mathbb{R},$$ next due to \((75)_{1}\) and for every \(u^{1}\in H^{1}(\mathbb{R}^{n})\), by embedding theorem \(H^{1}(\mathbb{R}^{n})\subset L^{6}(\mathbb{R}^{n})\subset L^{4}(\mathbb{R}^{n})\subset L^{2}(\mathbb{R}^{n})\) and (82), we conclude

\label{equ90b} due to (82) and (91), we can obtain the following result, take \(\sigma_{1}(\omega)=\min~[\tilde{\sigma},\frac{\tilde{\sigma}}{2\bar{\varrho}(\omega)}].\) By Gronwall's inequality to (92) over \([r,\tau-t]\) and replacing \(\omega\) to \(\theta_{-r}\omega\), we find by \((75)_{1}\), we get the following estimate Thus, collecting all (88) and (93)-(94), we arrive at (90), where $$\bar{M}^{2}(\omega)=\left(\tilde{M}(\omega)+C_{p}\tilde{M}^{6}(\omega)\right) e^{2\sigma_{1}(\omega)(r+t-\tau)}+\rho\int_{\tau-t}^{r}e^{-2\sigma_{1}(s,\omega)}ds.$$

Lemma 15. Under the conditions of (6)-(11), (75)-(76). For any \((\tau,r)\in\mathbb{R},\omega\in\Omega\), there exists random variable radius \(\varrho_{2}(\tau,\omega)>0\) such that solution of the system (78) satisfies the following estimates, for all \(t\geq r,r\geq\tau-t\),

where

Proof. Let \(\varphi_{2}=(\bar{u},\bar{v},\zeta)^{\top}=(\bar{u},\bar{u}_{t}+\varepsilon\bar{u}-c\bar{u}z(\theta_{t}\omega),\zeta)^{\top}\), then the system (78) is equivalent to the following system with initial data

thus, we can rewritten (97), by the following equation where $$\bar{H}\varphi_{2}=\left(% \begin{array}{ccc} \varepsilon \bar{u} -\bar{v} \\ \varepsilon(\varepsilon-\alpha A)\bar{u}+\beta A\bar{u} -(\varepsilon -\alpha A)\bar{v} + \zeta \\ \varepsilon \bar{u}-\bar{v} + \zeta_{s} \\ \end{array} \right), $$ and Taking scalar product of (98) with \(A^{\nu}\varphi_{2}(r)\), then positively According to (96) and Lemma 10, we find next, we will estimate the right-hand side of (100), yield From (31)-(97) and (98) one by one, we get, for nonlinear term we have \begin{eqnarray*} (f(u)-f_{0}(\tilde{u}),A^{\nu}\bar{v})&=&(f(u)-f_{0}(\bar{u}),A^{\nu}(\bar{u}+\varepsilon \bar{u}-c\bar{u} z(\theta_{t}\omega)))\\ &\leq&\frac{d}{dt}\int_{\mathbb{R}^{n}}(f(u)-f_{0}(\tilde{u}))A^{\nu}\bar{u} dx+\int_{\mathbb{R}^{n}}(f(u)-f_{0}(\tilde{u}))A^{\nu}\bar{u} dx\\ &&-\int_{\mathbb{R}^{n}}(f'(u)u_{t}-f'_{0}(\tilde{u})\tilde{u}_{t})A^{\nu}\bar{u} dx-\int_{\mathbb{R}^{n}}(f(u)-f_{0}(\tilde{u}))A^{\nu}\bar{u}z(\theta_{t}\omega)dx. \end{eqnarray*} Next, due to (8), (75)-(76), the Cauchy-Schwartz inequality and the Young inequality, we arrive at hence, the following inequalities holds and note that \(\nu\leq\frac{5-p}{2},\) such as and Thus, by collecting all (101)-(114) and (100), we show that By Gronwall's inequality to (115) on \([\tau-t,r]\) and replacing \(\omega\) to \(\theta_{-\tau}\omega\), we have \begin{eqnarray*} \left\|A^{\frac{\nu}{2}}\varphi_{2}(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}(\theta_{-\tau}\omega))\right\|_{E}^{2}&\leq&\left(\|A^{\frac{\nu}{2}}\varphi_{2}(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|_{E}^{2} +2(f(u(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}))\right.\nonumber\\ &&\left.-f_{0}(\tilde{u}\left(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}\right)\right)\end{eqnarray*} We put Similar to above equation, by (116) and (118), we can get $$~~~~~~~~~~\|A^{\nu}\varphi_{2}(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|_{E}^{2}\leq\varrho^{2}_{2}(\tau,\omega).~~~~~~~~~~~~~~~~~~~~~~~$$

6. Random attractors

In this section, we establish the existence of a \(\mathcal{D}\)-random attractor for the random dynamical system \(\Phi\) associated with system (18) on \(\mathbb{R}^{n}\), that is, by Lemma 10, \(\Phi\) has a closed random absorbing set in \(\mathcal{D}\), which along with the \(\mathcal{D}\)-pullback asymptotic compactness, they imply the existence of a unique \(\mathcal{D}\)-random attractor. Next due to decomposition of solutions we shall prove the \(\mathcal{D}\)-pullback asymptotic compactness of \(\Phi\) (see[10, 37]).
For \(\tau\in\mathbb{R},\omega\in\Omega,t\geq0\), we get

Lemma 16. Let \(E_{\nu}=H_{2\nu+1}\times H_{2\nu}\times L^{2}_{\mu}(\mathbb{R}^{+},H_{2\nu+1})\rightarrow L^{2}_{\mu}(\mathbb{R}^{+},H_{2\nu+1})\) is projection operator setting, \(Y=\psi(r,B_{\nu}(\tau,\omega))\) is a random bounded absorbing set, by Lemma 15, \(\psi(r)\) is the solution of the system (77), and by Lemma 15, there is a positive random radius \(\varrho_{\nu}(\tau,\omega)\) depending on \(r\), such that

Denote by \(B_{\nu}\) the closed ball of \(H_{1+2\nu}\times H_{2\nu}\) of random variable radius \(\varrho_{\nu}(\tau,\omega)\), let we apply on a finite domain. \(B_{\nu}\) is compact subset of \(H_{1+2\nu}\times H_{2\nu}\). Thus, we chose that a set \(\tilde{B}_{\nu}(\tau,\omega)\) hence, \(\nu\) is as in (98). From (3) and (121), we find The next Lemma we investigate the main result about the existence of a random attractor for random dynamical system \(\Phi\).

Lemma 17. we assume that \(\psi(t,\tau,\omega)\) is a solution of system (78) and the conditions of Lemma 14 hold, for each \(t\geq0\), there exists a random set \(\tilde{B}_{\nu}(\omega)\in \mathcal{D}(E_{\nu})\) with $$\|\tilde{B}_{\nu}(\tau,\omega)\|_{E_{\nu}}=\sup_{\tilde{\psi}\in \tilde{B}_{\nu}(\tau,\omega)}\|\tilde{\psi}\|_{E}\leq \tilde{M}(\tau,\omega)$$ is relatively compact in E. Then we show the following attraction property of \(\bar{\mathcal{A}}(\tau,\omega)\), for every \(B(\tau,\theta_{-t}\omega)\in \mathcal{D}(E)\), if there exist and positive number \(\sigma\) and \(\tilde{M}(\tau,\omega)\geq0\) so as for each \(\tau\in \mathbb{R},\omega\in\Omega\) it satisfy

Proof. Let \(\varphi_{\tau-t}(\theta_{-\tau }\omega)\in B_{1}(\tau-t,\theta_{-t}\omega)\) and by (119)-(121) and Lemma 17, it concludes that \(\tilde{B}_{\nu}(\tau~,\omega)\) is relatively compact in \(L_{\mu}^{2}(\mathbb{R}^{+},H_{1})\), let \(B_{\nu}(\omega)\subset E_{\nu}\subset E\) be the ball of \(E_{\nu}\) of radius \(M(\tau,\omega)\) defined by (28), where \(\nu\) is as in (98). Lastly, we get compact set \(A_{0}(\varsigma,w)=\tilde{B}_{\nu}\times B_{\nu}\subset E.\)
Since Lemma 10, Lemma 14 and \(\varphi_{\tau-t}(\theta_{-\tau }\omega)\in B_{0}(\tau-t,\theta_{-t}\omega)\), there exists a random set \(M(\tau,\omega)\in B_{0}\subseteq B(\tau,\omega)\in \mathcal{D}(E)\), such that

next, follows from Lemma 13, for \(\varphi_{\tau-t}(\theta_{-\tau }\omega)\in B_{1}(\tau-t,\theta_{-t}\omega)\), there exists positive a random variable \(M_{0}(\tau,\omega)\in B_{1}(\tau,\omega)\in \mathcal{D}(E)\) and \(M_{1}(\tau,\omega)\in B_{1}(\omega)\in \mathcal{D}(E)\) such that, by Lemma 15, let \(\varphi_{\tau-t}(\theta_{-\tau }\omega)\in B_{1}(\tau-t,\theta_{-t}\omega)\), there exists positive a random variable \(\varrho^{2}_{2}(\tau,\omega)\in B_{1}(\omega)\in \mathcal{D}(E)\), such that let \(\nu\geq0\) is fixed, by above recursion of finite steps at most \(\frac{1}{\nu}+2\), there exists random set \(\tilde{\varrho}_{\nu}\in \tilde{B}_{\nu}(\omega)\in \mathcal{D}(E_{\nu})\) as for as due to Lemma 16 and (119)-(121), \(\varrho_{\nu}(\tau,\omega)\in B_{1}(\omega)\in \mathcal{D}(E)\) we have and Thus, by Lemma 11, there exists \(T=T(\tau,\omega,B)\geq0\) such that \(\varphi(t,\tau-t,\theta_{-\tau}\omega,B(\tau-t,\theta_{-\tau}\omega)\subseteq B_{0}(\omega))~\forall t\geq T\) Let \(t\geq T\) and \(T= t-r\geq T(\tau,\omega,B_{0})\geq 0\), using cocycle property (3) of \(\Phi\), we show that for each \(\varphi(\tau,\tau-t,(\theta_{-\tau}\omega),\varphi_{\tau-t}(\theta_{-\tau}\omega))\in \varphi(t,\tau-t,\theta_{-\tau}\omega,B(\tau-t,\theta_{-t}\omega))\), for \(t\geq r+T(\tau,\omega,B_{0})\), where \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in B(\tau-t,\theta_{-t}\omega)\). By (98) and Lemma 15, we get Therefore, thanks to Lemma 14, we conclude that \begin{eqnarray*} \inf_{\tilde{\psi}\in\tilde{\mathcal{A}}(\tau,\omega)}\|\varphi(\tau,\tau-t,(\theta_{-\tau}\omega),\varphi_{\tau-t}(\theta_{-\tau}\omega))-\tilde{\psi}\|^{2}_{E}&\leq&\|\psi(\tau,\tau-t,(\theta_{-\tau}\omega),\psi_{\tau-t}(\theta_{-\tau}\omega))\|^{2}_{E}\nonumber\\ &\leq& \tilde{M}^{2}(\tau,\omega)e^{-\sigma t}~,~\forall t>\tilde{T}+T(\tau,\omega,B_{0}), \end{eqnarray*} so \begin{eqnarray*} d_{H}(\Phi(t,\tau-t,\theta_{-t}\omega,B(\tau-t,\theta_{-\tau}\omega)),\tilde{\mathcal{A}}(\tau,\omega))\leq\tilde{M}(\tau,\omega)e^{-\sigma_{1} t}~\rightarrow 0~ at~t\rightarrow +\infty. \end{eqnarray*}

Theorem 18. Suppose that (6)-(11) hold. Then the continuous cocycle \(\Phi\) associated with the problem (15)-(17) or (18) has a unique \(\mathcal{D}\)-pullback attractor \(\mathcal{A}\subseteq\tilde{\mathcal{A}}(\tau,\omega)\bigcap B_{0}(\omega)\),~ \(\mathcal{A}=\{\mathcal{A}(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}\in \mathcal{D} \) in \(\mathbb{R}^{n}.\)

Proof. Hence that the continuous cocycle \(\Phi\) has a closed random absorbing set \(\{A(\omega)\}_{\omega\in\Omega}\) in \(\mathcal{D}\), by Lemma 10, Lemma 11 and Lemma 16, the continuous cocycle \(\Phi\) is \(\mathcal{D}\)-pullback asymptotically compact in \(\mathbb{R}^{n}\). Since that the existence of a unique \(\mathcal{D}\)- random attractor for \(\Phi\) follows from Lemma 8 immediately.

Author Contributions

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

Competing Interests

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

1. Introduction

This study analyzes fishery management in contest of an endangered predator population competing with human being for commercially important prey. In earlier studies, natural predators were implicitly incorporated in the fishery model. We, however, explicitly model the predator-prey relationship thinking that endangered predators can also be found in many fisheries where the expansion of the predator population and the rate of harvesting are necessary. Traditionally it is impossible to control the predator population when they are endangered. We focus on harvesting control effort over the habitat of preys for maintaining the predator-prey relationship and protected the economic importance of the fishery.

Brauer et al. in [1] and Myerscough et al. in [2] studied a general model of prey-predator interaction under constant harvesting and developed the dynamics model of harvesting. Dai et al. in [3] gave complete mathematical analysis of a prey-predator model with Holling Type I predator response Holling, [4], where both the interacting species are independently harvested. Azar et al. [5] made a comparative study between constant catch and constant harvesting effort in a prey-predator model and examined a few significant phenomena such as a constant catch on the predator may destabilize a system that is stable when a constant harvesting effort is applied. Recently, Kar et al. [6] presented a mathematical model of non selective harvesting model in a prey-predator fishery. In their work [7] they have described taxation as a control tool in their model.

Extensive and unregulated harvest of marine fishes can lead to the depletion of several fish species. Several fish species can be depleted by irrational and un regulated harvesting of marine fishes. A possible solution to these problems is to create of marine reserves restricting fishing and other related activities. This study is the modified model of Dubey et al. [8] and to analyze the optimal harvesting policy.

2. Mathematical model formulation

In a two-patch environment, we consider the following predator-prey system: Here \(x(t),y(t)\) represents the prey population in the i-th patch, at time \(t \le 0\). We consider, the patches with a barrier only as far as the prey population is concerned and the predator population has no barriers. Thus, the total predator population for both patches is \(z(t)\). The constitutes of Patch 2, a reserve area for the prey and fishing is not allowed in this zone, though Patch 1 is an open-access fishery zone. We assumed that the prey populations are migrated randomly between two patches. In the absence of predator population, the growth rate of prey population logistically. Description of state variables and parameters is given in Table 1.

Table 1. Description of state variables and parameters.

Parameter	Description
\(r_1\) and \(r_2 \)	Intrinsic growth rates prey in the unreserved and reserved area
\(k_1\) and \(k_2\)	Environmental carrying capacity unreserved and reserved area respectively
\(\epsilon\)	dispersal rate
\(E\) and \(q \)	Harvesting effort and catchability coefficient
\(\gamma\) and \(\delta_3\)	Predator death rate and intra specific competition coefficients

If \(\epsilon = 0\), then no member of the prey population can leave its patch. From the j-th patch to the i-th patch the net exchange is proportional to the difference \(x\sim y\) of fish population densities in each patch.

3. Preliminary results

3.1. Boundedness

Now easily we can show that all solutions of system (1) -(3) are uniformly bounded.

Theorem 1. All the solutions \((x(t),y(t),z(t))\) of the system (1) -(3) in \(\mathbb{R}_+^3\) are bounded.

Proof. To prove the theorem, we consider the following function $$w(t)=\frac{c_1}{\alpha_1}x(t)+\frac{c_2}{\alpha_2}y(t)+z.$$ Therefore, time derivative is found to be

For each \(\mu>0\), the following inequality holds if computing the square separately in \(x\) and \(y\). Therefore, It is clear that the right-hand side of inequality (5) is bounded for all \((x,y,z)\in\Re_+^3\), provide \(E\) is bounded.
Therefore, we take \(v>0\) such that $$\frac{dw}{dt}+\mu w\le v.$$ Using the theory of differential inequalities developed by Birkhoff et al. in [9] we obtain, Letting \(t\to\infty,\) yields \(0< w< \frac{v}{\mu}\). Hence, all solutions of the model system (1) -(3) for all \(\mathbb{R}_+^3\) are attained to the region \(D\), where $$D=\left\{(x,y,z)\in\mathbb{R}_+^3: w=\frac{v}{mu}+\epsilon. \:\:\ \text{for any} \epsilon>0\right\}$$

3.2. Dissipativeness

Theorem 2. If \( r_2\ge\epsilon\) then the system (1) -(3) is dissipative.

Proof. By usual straight forward arguments, we can show that the solution of the system (1) -(3) always exists and is positive, In fact from the Equations (1) -(3) of the model system that \(lim_{t\to\infty} x(t)\le 1\) from Equations (2) we notice that \(\dot y=r_2y(1-\frac{y}{k_2})+\epsilon(x-y)-\alpha_2yz\le y(r_2-\epsilon).\) By similar arguments, we have, \(lim_{t\to\infty} y(t)\le(r_2-\epsilon)=\bar{y}\) where \(\bar{y}\) denotes an upper bound of \(y(t)\) which will be positive if \(r_2>\epsilon.\)

4. Equilibria analysis

Theorem 3. The possible steady states of the system of Equations (1) -(3) are:

1. Trivial equilibrium point \(E_0(0,0,0)\),
2. Axial equilibrium point \(E_1(x_1,y_1,0),\)
3. Interior equilibrium point \(E^*(x^*,y^*,z^*).\)

Proof.

1. Trivial equilibrium point always exists.
2. We get from (1) -(3)
   \begin{eqnarray} &&r_1x\left(1-\frac{x_1}{k_1}\right)+\epsilon(y_1-x_1)-qEx_1=0, \label{model-2-eq7} \end{eqnarray}

   (7)
   \begin{eqnarray} &&r_2y_1\left(1-\frac{y_1}{k_2}\right)+\epsilon(x_1-y_1)=0, \label{model-2-eq8} \end{eqnarray}

   (8)
   \begin{eqnarray} &&c_1x+c_2y=0. \label{model-2-eq9} \end{eqnarray}

   (9)
   Solving (7)-(8), we have, \(E_1\left(\frac{-k_1}{c_2}\left(\frac{\epsilon c_1}{c_2}+\epsilon-qE-r_1\right), \frac{c_1k_1}{c_2}\left(\frac{\epsilon c_1}{c_2}+\epsilon-qE-r_1\right),0\right)\)
3. We get from (1) -(3)
   \begin{eqnarray} &&r_1x^*\left(1-\frac{x^*}{k_1}\right)+\epsilon(y^*-x^*)-\alpha_1x^*z^*-qEx^*=0, \label{model-2-eq10} \end{eqnarray}

   (10)
   \begin{eqnarray} &&r_2y^*\left(1-\frac{y^*}{k_2}\right)+\epsilon(x^*-y^*)-\alpha_1y^*z^*=0, \label{model-2-eq11} \end{eqnarray}

   (11)
   \begin{eqnarray} &&z^*(-\gamma-\delta z^*)+c_1x^*+c_2y^*=0, \label{model-2-eq12} \end{eqnarray}

   (12)
   Solving (10) -(12) we will find interior equilibrium point.

5. Stability analysis

5.1. local stability

Now, we investigate the local asymptotically stability of the model (1) -(3) around the feasible equilibrium points.

5.1.1. Stability for \(E_0\)

$$\lambda[\lambda^2-(r_1+r_2-2\epsilon-qE)\lambda+(r_1-qE)(r_2-\epsilon)-r_2\epsilon]=0.$$ The equilibrium point \(E_0\) is a saddle point with locally stable manifold in \(xy\)- plane and with locally unstable manifold in \(z\)-direction if $$\frac{1}{q}\left(r_1+r_2-2\epsilon\right)< E< \frac{r_1r_2-(r_1+r_2)\epsilon}{q(r_2-\epsilon)}.$$

5.1.2. Stability for \(E_1\)

The characteristic equation for \(E_1\) is given by $$\lambda\left[\lambda^2+\left\{\frac{r_1x}{k_1}+\frac{r_2y}{k_2}+\epsilon\big(\frac{x}{y}+\frac{y}{x}\big)\right\}\lambda+\frac{r_1x}{k_1}\left(\frac{r_2y}{k_2}+\epsilon\frac{x}{y}\right)+\frac{r_2y^2}{xK_2}\right]=0.$$ Therefore, \(E_1\) is a saddle point with locally stable manifold in \(xy\)- plane and with locally unstable manifold in the \(z\)-direction.

5.1.3. Stability for \(E_2\)

The characteristic equation for \(E_2\) is given by where, \begin{align*} a&=\frac{r_1x^*}{k_1}+\frac{r_2y^*}{k_2}+\epsilon\left(\frac{y^*}{x^*}+\frac{x^*}{y^*}\right)+z^*\delta>0\\ b&=y^*\delta(\alpha-z^*\delta)+\frac{r_2y^*}{k_2}\left(\frac{r_1x^*}{k_1}+\frac{\epsilon y^*}{x^*}\right)+(\alpha_1c_1x^*+\alpha_2c_2y^*)z^*+\frac{\epsilon r_1x^{*2}}{k_1y^*}>0\\ c&=z^*\left[y^*\left(\frac{r_1x^*}{k_1}+\frac{\epsilon y^*}{x^*}\right)+\left(\frac{r_2\delta}{k_2}+\alpha_2c_2\right)+c_1\alpha_1x^*\left(\frac{r_2x^*}{k_2}+\frac{\epsilon y^*}{x^*}\right)+\frac{\epsilon r_1\delta x^{*2}}{k_1y^*}+\epsilon(\alpha_2c_1y^*+\alpha_1c_2x^*)\right]>0. \end{align*} We see that all eigenvalues of Equation (13) have negative real parts if and only if \(a>0, \quad c>0\) and \(ab-c>0\) which satisfies the Routh-Hurwitz criterion. Here, \(a>0, \quad c>0\) and it is easy to examined that \(ab-c>0\). Hence, \(E^*(x^*,y^*,z^*)\) is locally asymptotically stable.

5.2. Global stability analysis

From the point of view of ecological managers it may be found an equilibrium point where the model system is globally asymptotically stable in order to plan harvesting strategy and keep sustainable ecological development. Therefore, in the interior equilibrium point \(E^*(x^*,y^*,z^*)\), we have discussed the global stability.

Theorem 4. The model system (1) -(3) is globally asymptotically stable in the positive equilibrium point \(E^*(x^*,y^*,z^*)\) if \(\epsilon\big(\sqrt{\frac{\sigma_1}{x^*}}-\sqrt{\frac{\sigma_2}{y^*}}\big)^2< 2\sqrt{\sigma_1\sigma_2\frac{r_1r_2}{k_1k_2}}.\)

Proof. Using the standard Lyapunov function we have,

Along a solution, the derivative of (1) -(3) takes the form: \begin{align*} \dot V&=\sigma_1(x-x^*)\left(r_1-\frac{r_1}{k_1}x-\alpha_1z+E\frac{y-x}{x}\right)+\sigma_2(y-y^*)\left(r_2-\frac{r_2}{k_2}y-\alpha_2z+\epsilon\frac{x-y}{y}\right)\\ &\,\,\,\,+\sigma_3(z-z^*)(-\gamma-\delta z+c_1x+c_2y)\\ &=-\sigma_1\frac{r_1}{k_1}\left(x-x^*\right)^2+\sigma_2(y-y^*)^2-(z-z^*)\{\sigma_1\alpha_1(x-x^*)+\sigma_2\alpha_2(y-y^*)\}\\ &\,\,\,\,-\sigma_3\delta(y-y^*)\{c_1(x-x^*)+c_2(y-y^*)\}-\epsilon\Gamma(x,y), \end{align*} where $$\Gamma(x,y)=\sigma_1y\frac{(x-x^*)}{xx^*}+\sigma_2x\frac{(y-y^*)}{yy^*}-\big(\frac{\sigma_1}{x}+\frac{\sigma_2}{y}\big)(x-x^*)(y-y^*)$$ If we choose \(\sigma_1=\frac{c_1}{\alpha_1}\quad\sigma_2=\frac{c_2}{\alpha_2},\quad\sigma_3=1\), then we have $$\dot V=-\sigma_1\frac{r_1}{k_1}(x-x^*)^2+\sigma_2\frac{r_2}{k_2}(y-y^*)^2-\delta(z-z^*)^2-\epsilon\Gamma(x,y).$$ Now it is easy to show that $$\Gamma(x,y)\ge \left(2\sqrt{\frac{\sigma_1\sigma_2}{x^*y^*}}-\frac{\sigma_1}{x^*}-\frac{\sigma_2}{y^*}\right)(x-x^*)(y-y^*) =-\left(\sqrt{\frac{\sigma_1}{x^*}}-\sqrt{\frac{\sigma_2}{y^*}}\right)^2(x-x^*)(y-y^*).$$ Thus we have, $$\dot V=-\sigma_1\frac{r_1}{k_1}(x-x^*)^2+\sigma_2(y-y^*)^2-\delta(z-z^*)^2-\epsilon\left(\sqrt{\frac{\sigma_1}{x^*}}-\sqrt{\frac{\sigma_2}{y^*}}\right)^2(x-x^*)(y-y^*).$$ Therefore \(\epsilon\left(\sqrt{\frac{\sigma_1}{x^*}}-\sqrt{\frac{\sigma_2}{y^*}}\right)^2< 2\sqrt{\sigma_1\sigma_2\frac{r_1r_2}{k_1k_2}}.\) then for \(\delta>0\) and \((x,y,z)\ne (x^*,y^*,z^*),\quad \dot V< 0\). Therefore, we say that, \(E^*(x^*,y^*,z^*)\) is globally asymptotically stable.

6. Bionomic equilibrium and and optimal harvesting policy

The bionomic equilibrium is said to be achieved when the total revenue is earned by the difference of pricing and harvesting cost. Let us consider the constant fishing cost per unit effort is \(c\) and the constant price per unit landed fish in the open access area is \(p\). Therefore, the economic rent is given as follows Now, if \(c>pqx\), i.e., if the fishing cost exceeds the revenue, then the economic rent obtained from the fishery becomes negative and the fishery will be closed. Therefore, the bionomic equilibrium existence, it is assumed that \(c>pqx\). The positive bionomic equilibrium solutions of \(\dot x=\dot y=\dot z=\Pi=0\) is \((x_\infty,y_\infty,z_\infty,E_\infty).\) Solving these equations we get, $$x_\infty=\frac{c}{pq}$$ $$y_\infty=\frac{\left[\alpha_2\left(-\gamma+\frac{cc_1}{pq}\right)+\delta(\epsilon-r_2)\right]+\sqrt{\left[\alpha_2(-\gamma+\frac{cc_1}{pq}+\delta(\epsilon-r_2))^2+\frac{4r_2\epsilon c}{pq}(\alpha_2c_2+\frac{\delta r_2}{k_2})\right]}}{2(\alpha_2c_2+\frac{\delta r_2}{k_2})}$$ $$z_\infty=\frac{1}{\delta}\left(-\gamma+\frac{cc_1}{pq}+c_2y_\infty\right)$$ and $$E_\infty=\frac{1}{\delta}\left[r_1\left(1-\frac{c}{k_1pq}\right)-\epsilon-\frac{\alpha_1}{\delta}\left(-\gamma+\frac{cc_1}{pq}\right)+\left(\frac{\epsilon pq}{c}-\frac{\alpha_1c_2}{\delta}\right)y_{\infty}\right].$$ If \(E>E_\infty\) , then the total harvesting cost the fish population will exceed the total amount of revenues collected from the fishery. Therefore, we assumed that some fishermen will face loss and withdraw themselves from fishing. Hence \(E>E_\infty\) is possible to be maintained indefinitely. The fishery is more profitable when \(\frac{d\lambda_1}{dt}=-\frac{\partial H}{\partial x}\) and hence in an open access fishery it would be attracted more and more fishermen which increasing the harvesting. Hence \(E< E_\infty\) is not possible to maintain indefinitely. Now our objective is to maximizes the present value of a continuous time-stream of revenues. We select a harvesting strategy. Consider \(\sigma\) be the instantaneous annual discount rate, The problem (16), subject to the Equations (1)-(3), by applying Pontryagin's maximum Principle with control constant \(0< E< E_{max}\) can be solved. The feasible upper limit on the harvesting effort \(E_{max}\). Then the Hamiltonian problem is given by where \(\lambda_1,\quad\lambda_2\) and \(\lambda_3\) are adjoint variables and \(\mu(t)=(e_{\sigma t}(pqx-c)E)-\lambda_1qx\) is called the switching function. The optimal control \(E(t)\) the maximizes the linear control variable of Hamiltonian \(H\) must satisfying conditions:

1. \(E=E_{max}\) when \(\mu(t)>0\) i.e., when \(\lambda_1(t)e^{\sigma t}< p-\frac{c}{qx};\)
2. \(E=0\) when \(\mu(t)< 0\) i.e., when \(\lambda_1(t)e^{\sigma t}>p-\frac{c}{qx};\)

\(\lambda_1(t)e^{\sigma t}\) is the traditional shadow price and p-is the net economic revenue on a unit harvest which shows that \(E=E_{max}\) according as the shadow price is less than or greater than the net economic revenue on a unit harvest. Economically, the first condition is that after passing all the expenses if the profit is positive, then it is beneficial to harvest up to the limit of available effort and second condition is that when the shadow price exceeds the fishermen's net economic revenue on a unit harvest, then the fishermen will not exert any effort. When \(\mu(t)=0\) , i.e., when the shadow price on a unit harvest equals the net economic revenue, then the Hamiltonian \(H\) of the control variable \(E(t)\) i.e., \(\frac{\partial H}{\partial E}=0\) becomes independent. It is the necessary condition to be optimal over the control set \(0< E^*< E_{max}\) for the singular control \(E^*(t)\). Thus the optimal harvesting policy is Again \(\mu(t)=0,\) implies that This implies at the steady state effort level, the total user harvesting cost per unit effort must be equal to the discounted value of the future profit. Now, we have the adjoint equations as follows: \begin{eqnarray} &&\frac{d\lambda_1}{dt}=-\frac{\partial H}{\partial x}=-\left[pqEe^{\sigma t}+\lambda_1\{r_1\left(1-\frac{2x}{k_1}\right)-\epsilon-\alpha_1z-qE\}+\lambda_2\epsilon+\lambda_3zc_1\right]\nonumber\\ &&\frac{d\lambda_2}{dt}=-\frac{\partial H}{\partial y}=-\left[\lambda_1\epsilon+\lambda_2\{r_2\left(1-\frac{2y}{k_2}\right)-\epsilon-\alpha_2y\}+\lambda_3zc_2\right]\nonumber\\ &&\frac{d\lambda_3}{dt}=-\frac{\partial H}{\partial z}=-[-\lambda_1\alpha_1y-\lambda_2\alpha_2y-\lambda_3\delta z].\nonumber \end{eqnarray} Here \(x,y,z\) and \(E\) can be treated as constants to find optimal equilibrium solution of the system. Therefore, adjoint equations are given as follows: Now from Equations (21),(22) and (23) eliminating \(\lambda_2\) and \(\lambda_3\) we get
\(D^3-[\frac{r_1x}{k_1}+\frac{r_2y}{k_2}+\epsilon(\frac{x}{y}+\frac{y}{x})+\delta z]D^2+[\big(\frac{r_2y}{k_2}+\epsilon\frac{x}{y}+\delta z\big)\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\delta(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})\}+\alpha_1c_1x+\alpha_2c_2y-\epsilon\delta\frac{c_1}{c_2}\}z-\epsilon\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})-\epsilon^2]D-[\{\delta z(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})+\alpha_2c_2yz\}\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})+\epsilon\}(\alpha_1c_2x-\epsilon\delta)z]=Me^{\sigma t},\)
where \(M=-pqE[\sigma^2+\sigma(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})(1+y)+(\delta^2+\alpha_2c_2y)z]\). The auxiliary equation is
\(m^3-[\frac{r_1x}{k_1}+\frac{r_2y}{k_2}+\epsilon(\frac{x}{y}+\frac{y}{x})+\delta z]m^2+[\big(\frac{r_2y}{k_2}+\epsilon\frac{x}{y}+\delta z\big)\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\delta(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})\}+\alpha_1c_1x+\alpha_2c_2y-\epsilon\delta\frac{c_1}{c_2}\}z-\epsilon\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})-\epsilon^2]m-[\{\delta z(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})+\alpha_2c_2yz\}\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\}\frac{x}{y})+\epsilon(\alpha_1c_2x-\epsilon\delta)z]=0.\nonumber \)
Consider the root of the above equation are \(m_1, m_1\) and \(m_3\), then the general solution becomes \(\lambda_1(t)=A_1e^{m_1t}+A_2e^{m_2t}+A_3e^{m_3t}+\frac{M}{N}e^{-\sigma t},\) where
\(N=-\sigma^3-[\frac{r_1x}{k_1}+\frac{r_2y}{k_2}+\epsilon(\frac{x}{y}+\frac{y}{x})+\delta z]\sigma^2+[\big(\frac{r_2y}{k_2}+\epsilon(\frac{x}{y}+\delta z\big)\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\delta(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})\}+\alpha_1c_1x+\alpha_2c_2y-\epsilon\delta\frac{c_1}{c_2}\}z-\epsilon\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})-\epsilon^2]\sigma-[\{\delta z(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})+\alpha_2c_2yz\}\{\frac{r_1x}{k_1}+\epsilon(\frac{y}{x}+\frac{c_1}{c_2})\}+\{\frac{c_1}{c_2}(\frac{r_2y}{k_2}+\epsilon\frac{x}{y})+\epsilon\}(\alpha_1c_2x-\epsilon\delta)z]\ne 0.\)
The shadow price \(\lambda_1(t)e^{\sigma t}\) remains bounded as \(t\to \infty\) if and only if \(A_1=A_2=A_3=0\) and then \(\lambda_1(t)e^{\sigma t}=\frac{M}{N}=\text{constant.}\) Now substituting \(\lambda_1(t)\) in (19) we get, Together with equation \(\dot x=\dot y=\dot z=0\) and Equation (23), gives the optimal equilibrium populations \(x=x_\infty, y=y_\infty\) and \(z=z_\infty,\) when \(\sigma\to\infty,\) Equation (23) leads to obvious result \(pqx_\infty=c\) that implies \(\Pi(x_{\infty},y_{\infty},z_{\infty}, E_{\infty})=0.\) This shows that infinite discount rate leads to a economic revenue which is completely dissipation. Using (23), we have \(\Pi=(pqx-c)E=\frac{MqxE}{N}\). Since \(M\) is of \(O{(\sigma)}\) where \(N\) is \(O(\sigma^2)\) we see that \(\Pi\) is \(O(\sigma^{-1}\). Thus, the decreasing function of \(\sigma(\le 0)\) is \(\Pi\). Therefore, we conclude that \(\sigma=0\) leads to maximization of \(\Pi.\)

7. Numerical simulation

Analytical studies can never be completed without numerical verification of the derived results. In this section, we present computer simulations of some solutions of the system (1) -(3). Beside verification of our analytical findings, these numerical simulations are very important from practical point of view. We use four different set of numerical values for support of analytical results mentioned in Table 2.

Table 2. Set of parameter values for numerical simulations; \(S\equiv \)Parameter sets.

\(r_1\)	\(r_2\)	\(k_1\)	\(k_2\)	\(\epsilon\)	\(\alpha_1\)	\(\alpha_2\)	\(\gamma_1\)	\(\delta\)	\(c_1\)	\(c_2\)	\(E\)	\(q\)
\(3\)	\(1.5\)	\(50\)	\(40\)	\(0.5\)	\(0.2\)	\(0.2\)	\(0.6\)	\(0.05\)	\(0.03\)	\(0.04\)	\(2\)	\(0.01\)

From the theory established earlier the interior equilibrium point \(E_2(19.50, 10.20, 7.86)\) is globally asymptotically stable. From the Figure 1, we may conclude that the steady state \(E_2(19.50, 10.20, 7.86)\) is globally asymptotically stable. Hence the theory established earlier is verified.

8. conclusion

This research deals with the harvesting problem in a prey-predator fishery model the reserved zone for prey species in the Sundarban. The positive steady state of both local and global stability has been established. To get global stability, it is necessary that the dispersal rate to be bounded above by related constant. In the exploited model system, we have examined the possibilities of the existence of bionomic equilibria. By using Pontryagin's maximum principle, we have optimized the harvesting policy. We have found that the shadow prices satisfy the transversality condition when they are constant. The total user cost of harvest per unit effort is equal to the steady state effort. We have shown that zero discounting maximizes the economic revenue and that an infinite discount rate is completely dissipate.

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.

References

1. Brauer, F., & Soudack, A. C. (1982). Coexistence properties of some predator-prey systems under constant rate harvesting and stocking. Journal of Mathematical Biology, 12(1), 101-114.[Google Scholor]
2. Myerscough, M. R., Gray, B. F., Hogarth, W. L., & Norbury, J. (1992). An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking. Journal of Mathematical Biology, 30(4), 389-411. [Google Scholor]
3. Dai, G., & Tang, M. (1998). Coexistence region and global dynamics of a harvested predator-prey system. SIAM Journal on Applied Mathematics, 58(1), 193-210. [Google Scholor]
4. Holling, C. S. (1965). The functional response of predators to prey density and its role in mimicry and population regulation. The Memoirs of the Entomological Society of Canada, 97(S45), 5-60. [Google Scholor]
5. Azar, C., Holmberg, J., & Lindgren, K. (1995). Stability analysis of harvesting in a predator-prey model. Journal of Theoretical Biology, 174(1), 13-19. [Google Scholor]
6. Kar, T. K., & Chaudhuri, K. S. (2002). On non-selective harvesting of a multispecies fishery. International Journal of Mathematical Education in Science and Technology, 33(4), 543-556. [Google Scholor]
7. Kar, T. K., & Chaudhuri, K. S. (2003). On non-selective harvesting of two competing fish species in the presence of toxicity. Ecological Modelling, 161(1-2), 125-137.[Google Scholor]
8. Dubey, B., Chandra, P., & Sinha, P. (2003). A model for fishery resource with reserve area. Nonlinear Analysis: Real World Applications, 4(4), 625-637. [Google Scholor]
9. Birkhoff, G., & Rota, G. C. (1982). Ordinary Differential Equations. 1989. Ginn, Boston.[Google Scholor]

1. Introduction

Let \(\mathcal{A}\) denote the class of all functions of the form which are analytic in the open unit disk \(U=\lbrace z:|z|< 1\rbrace\). For a function \(f\in \mathcal{A}\), Raducanu and Orhan [1] introduced the following operator: \[ D_{\alpha\nu}^{0}f(z)=f(z)\] \[ D_{\alpha\nu}^{1}f(z)=\alpha \nu z^{2}f''(z)+(\alpha -\nu)zf'(z)+{(1-\alpha+\nu)}f(z)\] If \(f\) is given by (1), then from the definition of the operator \(D_{\alpha \nu}^{n}f\), the Equation (2) can be rewritten as: where \((n\in N_{0}=N\cup \lbrace 0\rbrace)\).

Remark 1.

1. When \(\alpha =1, \nu=0\), we get the Sălăgean differential operator introduced by Sălăgean in [2].
2. When \(\nu=0\), we obtain differential operator defined by Al-Oboudi in [3].

Let \(\mathcal{A}_{p}\) denote the class of functions of the form which are analytic and p-valent in the open unit disk \(U=\lbrace z:|z|< 1\rbrace\). We can write the following equalities for the functions \(f\in \mathcal{A}_{p}:\) \[ D_{\alpha \nu}^{0,p}f(z)=f(z)\] If \(f\) is given by Equation 4, then from Equation 5 and Equation 6, we see that

Remark 2.

1. If \(\nu=0\), \( D_{\alpha \nu}^{n,p}f=D_{\alpha,p}^{n}f\) defined by Bulut in [4]
2. If \(p=1\), \( D_{\alpha \nu}^{n,p}f=D_{\alpha \nu}^{n}f\) introduced by Raducanu and Orhan in [1]
3. If \(p=1,\alpha=1,\nu=0\), \(D_{\alpha \nu}^{n,p}f=D^{n}f\) defined by Sălăgean in [2]
4. If \(p=1,\nu=0\), \(D_{\alpha \nu}^{n,p}f=D_{\alpha}^{n}f\) defined by Al-Oboudi in [3].

Let \(\mathcal{T}_{p}\) denote the subclass of \(\mathcal{A}_{p}\) consisting of functions of the form If \(f\) is given by Equation 8, then from Equation 5 and Equation 6, we get

Definition 1. A function \(f\in \mathcal{T}_{p}\) is in the class, \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if

for \( 0\le\nu \le\alpha \le 1,0\le \vartheta< 1,\)\(0\le\gamma< 1,\)\(0< \beta\le1,\)\(0< \varphi< 1,p\in N\),\(D_{\alpha \nu}^{n,p}f(z)\) as in 9.

In this paper, basic properties of the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) are studied such as: coefficient inequalities, growth and distortion theorem, closure property, \(\delta\)-neighborhoods, extreme points, radii of close-to-convexity, starlikeness and convexity for these subclasses.

Remark 3. If \(\nu=0\), \(\vartheta=\alpha\), \(\varphi=\mu\), the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) reduces to the class \(R_{p}^n(\alpha,\beta,\gamma,\mu)\) investigated by Bulut [4]

Definition 2. A function \(f\in \mathcal{T}_{p}\) is in the class \(S_{p}^{n,(\delta_0)}(\vartheta,\beta,\gamma,\varphi)\), if there exists a function \(g(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) such that $$\left|\frac{f(z)}{g(z)}-1\right|< 1-\delta_0 ... (z\in U ,0\le \delta_0 < 1)$$ for \(0\le \vartheta< 1,\) \(0\le\gamma< 1,\) \(0< \beta\le1,\) \(0< \varphi< 1.\)

Definition 3. For a function \(f\in \mathcal{T}_{p}\), \(\delta\ge 0\), \(\delta\)-neighborhood of \(f\) is defined as:

in particular, for a function \(h\in \mathcal{T}_{p}\), given by \(h(z)=z^{p}\) \((p\in N)\), we immediately have The concept of neighborhoods was first introduced by Goodman [5] and generalized by Ruschewey [6] and Altintas [7] (see also [8, 9].

2. Coefficient inequalities

Theorem 4. A function \(f\in \mathcal{T}_{p}\) is in the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if

for \( 0\le\nu \le\alpha \le 1, 0\le \vartheta< 1,\) \(0\le\gamma< 1,\) \(0< \beta\le1,\) \(0< \varphi< 1\), \(n\in N_{0},\) \(p\in N\). Furthermore, the result is sharp for the function given as \begin{align*} f(z)=z^p - \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}a_k, (k\ge p+1). \end{align*}

Proof. Suppose that \(f\in S_{p}^n(\vartheta, \beta, \gamma, \varphi),\) then from inequality 10, we have \begin{eqnarray*} \left| \frac {(D_{\alpha \nu}^{n,p}f(z))' -pz^{p-1}}{\vartheta(D_{\alpha \nu}^{n,p}f(z))'+(\beta-\gamma)} \right|&=&\left| \frac {pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1} -pz^{p-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)} \right|\\ &= &\left| \frac {\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)} \right|\\&<&\varphi, (z\in U,n\in N_{0}) \end{eqnarray*} it is well known that \(\Re z\le \left|z\right|\), therefore, we obtain \begin{align*} \Re\left\{\frac {\sum_{K=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)}\right\} < \varphi. \end{align*} If we choose \(z\) real and let \(z \rightarrow 1^-,\) then we get \begin{eqnarray*} \sum_{K=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}&\le& \varphi\{\vartheta(p-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k})+(\beta-\gamma)\} \end{eqnarray*} which is precisely the assertion 13. On contrary, suppose that the inequality 13 hold true and let \(z\in \delta U=\lbrace z\in C:\left|z\right|=1\rbrace.\) Then, from 10, we have \begin{eqnarray*} &&\left|(D_{\alpha \nu}^{n,p}f(z))' -pz^{p-1} \right|- \varphi\left|\vartheta(D_{\alpha \nu}^{n,p}f(z))'+(\beta-\gamma) \right|\le \sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}\\&&- \varphi(\vartheta p+\beta-\gamma)+ \varphi \vartheta \sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}\\ &&=\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}(1+\varphi\vartheta)a_{k}- \varphi(\vartheta p+\beta-\gamma)\le 0. \end{eqnarray*} By maximum modulus theorem, we have \(f \in S_{p}^n(\vartheta,\beta,\gamma,\varphi).\)

Corollary 5. If \(f \in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), then \( a_{p+1}\le \frac{\varphi(\vartheta p+\beta-\gamma)}{(p+1)\left[1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi\vartheta)}.\)

3. Growth and distortion theorem

Theorem 6. For each \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have \( \left|z\right|^p-\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)(p+1)}\left|z\right|^{p+1} \le\left|f(z)\right| \le\left|z\right|^p+\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)(p+1)}\left|z\right|^{p+1}.\)

Proof. Let \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi),z\in U\), the bound on \(f(z)\) is given by

from Theorem 4, we have by using (15) in (14), we obtain again using (15), we have Consequently, combining (16) and (17) we obtain the desired result.

Theorem 7. For each \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have \( p\left|z\right|^{p-1}-\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)}\left|z\right|^{p} \le\left|f'(z)\right| \le p\left|z\right|^{p-1}+\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)}\left|z\right|^{p}.\)

Proof. Let \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi),z\in U\), the bound on the derivative of \(f(z)\) is given by \begin{align*} \left|f'(z)\right|\le p\left|z\right|^{p-1}+ (p+1)\left|z \right|^p\sum_{k=p+1}^\infty a_k, z\in U, \end{align*} and, in the same way as above, we get our desired result.

4. Closure properties

Theorem 8. Let the functions \begin{align*} f(z)=z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k} , && (a_k\ge 0)\\ g(z)=z^{p}-\sum_{k=p+1}^{\infty}b_{k}z^{k} , && (b_k\ge 0), \end{align*} be in the class \( S_{p}^n(\vartheta,\beta,\gamma,\varphi)\). Then for \(0\le \lambda\le1,\) the function \(h\) is defined as $$h(z)=(1-\lambda)f(z)+\lambda g(z)=z^{p}-\sum_{k=p+1}^{\infty}c_{k}z^{k},$$ where \( c_k:=(1-\lambda)a_k+\lambda b_k\ge0,\) is also in \( S_{p}^n(\vartheta,\beta,\gamma,\varphi).\)

Proof. Suppose that each of the functions \(f\) and \(g\) is in the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\). Then making use of inequality (13), we have \begin{eqnarray*} &&\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)c_{k}\\&&=(1-\lambda)\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k} \\&&+\lambda\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)b_{k} \\&& \le (1-\lambda)\varphi(\vartheta p+\beta-\gamma)+\lambda \varphi(\vartheta p+\beta-\gamma)\\&&= \varphi(\vartheta p+\beta-\gamma),\end{eqnarray*} which completes the proof.

5. \(\delta\)-Neighborhoods

Theorem 9. If

then \( S_{p}^n(\vartheta,\beta,\gamma,\varphi)\subset N_{\delta}^{p}(h,g).\)

Proof. For a function \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) of the form (8), Theorem 4 immediately yields $$ (p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)\sum_{k=p+1}^{\infty}a_{k}\le \varphi(\vartheta p+\beta-\gamma),$$ therefore,

On the other hand, we also find from (13) that that is which completes the proof.

Theorem 10. If \(g(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) and

then \(N_{\delta}^{p}(f,g)\subset S_{p}^{n,(\delta_0)}(\vartheta,\beta,\gamma,\varphi).\)

Proof. Suppose that \(f\in N_{\delta}^p(f,g)\), then by Definition 3, we have $$\sum_{k=p+1}^{\infty}k|a_k-b_k|\le \delta,$$ which readily implies the coefficient inequality given by $$\sum_{k=p+1}^{\infty}|a_k-b_k|\le \frac{\delta}{p+1} (p\in N).$$ Next, since \(g\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have from inequality (13) that \begin{align*} \sum_{k=p+1}^{\infty}b_{k}\le \frac{\varphi(\vartheta p+\beta-\gamma)}{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)}, \end{align*} so from the definition of the class, we have \begin{eqnarray*} \left|\frac{f(z)}{g(z)}-1\right|&<&\frac{\sum_{k=p+1}^{\infty}|a_k-b_k|}{1-\sum_{k=p+1}^\infty b_k}\\ &\le& \frac{\delta}{p+1}\frac{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)}{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)-\varphi(\vartheta p+\beta-\gamma)}\\ &= &1-\delta_0,\end{eqnarray*} provided that \(\delta_0\) is given precisely by (22). Thus, by the definition, \(f\in S_{p}^{n,\delta_0}(\vartheta,\beta,\gamma,\varphi)\) for \(\delta_0\) given by (22), this completes our proof.

6. Extreme points

Theorem 11. If \( f_{p}(z)=z^{p} , f_{k}(z)=z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k} (k\ge p+1) \) then, \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if it can be expressed in the form \(f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z),\) where \(\lambda_k\ge 0\) and \(\lambda_{p}\)=\(1 - \sum_{k=p+1}^{\infty}\lambda_{k}\).

Proof. Assume that \(f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z)\), then \begin{eqnarray*} f(z)&=&(1 - \sum_{k=p+1}^{\infty}\lambda_{k})z^{p}+\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k}\right\}\\&=&z^{p}-\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k}\right\}.\end{eqnarray*} Thus, \begin{eqnarray*}&&\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta) \lambda_{k} \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}\\&&= \varphi(\vartheta p+\beta-\gamma)\sum_{k=p+1}^{\infty}\lambda_{k}= \varphi(\vartheta p+\beta-\gamma)(1-\lambda_{p}) \le \varphi(\vartheta p+\beta-\gamma),\end{eqnarray*} which shows that \(f\) satisfies condition (13) and therefore,\(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi).\) Conversely, suppose that \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), since \[ a_{k} \le \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}, (k\ge p+1),\] we may set \[ \lambda_{k}=\frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}{\varphi(\vartheta p+\beta-\gamma)}a_{k}, \text{ and } \lambda_{p}=1- \sum_{k=p+1}^{\infty}\lambda_{k},\] then we obtain from \begin{eqnarray*}f(z)&=& z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}\\&=&(\lambda_{p}+\sum_{k=p+1}^{\infty}\lambda_{k})z^{p} - \sum_{k=p+1}^{\infty}\lambda_{k} \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^k\\ &=&\lambda_{p}z^{p}+ \sum_{k=p+1}^{\infty}\lambda_{k}(z^{p}- \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^k)\\&=& \lambda_{p}z^{p} + \sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z),\end{eqnarray*} which completes the proof.

Corollary 12. The extreme points of \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) are given by $$f_{p}(z)=z^{p} , f_{k}(z)=z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k} (k\ge p+1)$$

7. Radii of close-to-convexity,starlikeness and convexity

A function \(f\in \mathcal{T}_p\) is said to be \(p\)-valently close-to-convex of order \(\rho\) if it satisfies $$\Re \left\{ f'(z)\right\} >\rho$$ for some \(\rho (0\le\rho< p)\) and for all \(z\in U.\) Also, a function \(f\in \mathcal{T}_p\) is said to be \(p\)-valently starlike of order \(\rho\) if it satisfies $$\Re \left\{ \frac{zf'(z)}{f(z)}\right\} >\rho,$$ for some \(\rho (0\le\rho< p)\) and for all \(z\in U.\) Further, a function \(f\in \mathcal{T}_p\) is said to be \(p\)-valently convex of order \(\rho\) if it satisfies $$\Re \left\{ 1+\frac{zf''(z)}{f'(z)}\right\} >\rho,$$ for some \(\rho (0\le\rho< p)\) and for all \(z\in U.\)

Theorem 13. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) then \(f\) is \(p\)-valently close-to-convex of order \(\rho\) in \(\left|z\right|< r_1(\vartheta,\beta,\gamma,\varphi,\rho)\), where \begin{align*} r_1(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k}(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}

Proof. It is sufficient to show that \(\left|\frac{f'(z)}{z^{p-1}}-p\right|< p-\rho.\) Since \(\left|\frac{pz^{p-1}-\sum_{k=p+1}^\infty ka_Kz^{k-1}}{z^{p-1}}-p\right|< p-\rho,\) which implies that $$\left|\frac{f'(z)}{z^{p-1}}-p\right|\le \sum_{k=p+1}^{\infty}ka_k\left|z\right|^{k-p}< p-\rho,$$ implies

and by applying the result of Theorem 4, we get $$ \sum_{k=p+1}^{\infty}a_k \le\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k}}.$$ Hence,(23) is true if solving (24) for \(z\) we obtain $$\left|z\right|\le \left\{\frac{\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)}\right\}^{\frac{1}{k-p}}$$ which completes the proof.

Theorem 14. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) then \(f\) is \(p\)-valently starlike of order \(\rho\) in \(\left|z\right|< r_2(\vartheta,\beta,\gamma,\varphi,\rho)\), where \begin{align*} r_2(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)(k-\rho)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}

Proof. In order to prove, it suffices to show that \(\left|\frac{zf'(z)}{f(z)}-p\right| < p-\rho. \)

and by using inequality (13), we get $$ \sum_{k=p+1}^{\infty}a_k \le\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k}},$$ so, (25) holds true if $$\frac{(k-\rho)\left|z\right|^{k-\rho}}{p-\rho} \le \frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}{\varphi(\vartheta p+\beta-\gamma)},$$ and then \(f\) is starlike of order \(\rho.\)

Theorem 15. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), then \(f\) is \(p\)-valently convex of order \(\rho\) in \(\left|z\right|< r_3(\vartheta, \beta, \gamma, \varphi, \rho)\), where \begin{align*} r_3(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)p(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)(k-\rho)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}

Proof. To prove this, it suffices to show that \(\left|1+\frac{zf''(z)}{f'(z)}-p\right| < p-\rho .\) Since

it implies that \begin{align*} \left|1+\frac{zf''(z)}{f'(z)}-p\right|=\left|\frac{-\sum_{k=p+1}^{\infty}k(k-p)a_kz^{k-p}}{p-\sum_{k=p+1}^{\infty}ka_kz^{k-p}}\right|\le\frac{\sum_{k=p+1}^{\infty}k(k-p)a_k\left| z\right| ^{k-p}}{p-\sum_{k=p+1}^{\infty}ka_k\left| z\right| ^{k-p}} < p-\rho \end{align*} and by applying the result in Theorem 4, we get $$ \sum_{k=p+1}^{\infty}a_k \le\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k}}$$ so, (26) holds true if $$\frac{k(k-\rho)\left|z\right|^{k-p} }{p(p-\rho)} \le \frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}{\varphi(\vartheta p+\beta-\gamma)}$$ and then \(f\) is convex of order \(\rho.\)

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.