\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 150, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/150\hfil Implicit impulsive differential equations] {Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations} \author[S. Abbas, M. Benchohra \hfil EJDE-2011/150\hfilneg] {Sa\"id Abbas, Mouffak Benchohra} \address{Sa\"id Abbas \newline Laboratoire de Math\'ematiques, Universit\'e de Sa\"{\i}da, B. P. 138, 20000, Sa\"{\i}da, Alg\'erie} \email{abbasmsaid@yahoo.fr} \address{Mouffak Benchohra \newline Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\es \\ B.P. 89, 22000, Sidi Bel-Abb\es, Alg\'erie} \email{benchohra@univ-sba.dz} \thanks{Submitted August 19, 2011. Published November 8, 2011.} \subjclass[2000]{26A33, 34A08} \keywords{Hyperbolic differential equation; fractional order; \hfill\break\indent Riemann-Liouville integral; mixed regularized derivative; impulse; fixed point} \begin{abstract} In this article we investigate the existence and uniqueness of solutions for the initial value problems, for a class of hyperbolic impulsive fractional order differential equations by using some fixed point theorems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as the differential calculus since, starting from some speculations of Leibniz (1697) and Euler (1730), it has been developed up to nowadays. The idea of fractional calculus and fractional order differential equations and inclusions has been a subject of interest not only among mathematicians, but also among physicists and engineers. Indeed, we can find numerous applications in rheology, control, porous media, viscoelasticity, electrochemistry, electromagnetism, etc. \cite{GlNo,Hi,Ma,MeScKiNo,OlSp}. There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Kilbas \emph{et al.} \cite{KST}, Miller and Ross \cite{MiRo}, Podlubny \cite{Pod}, Samko \emph{et al.} \cite{SaKiMa}, the papers of Abbas and Benchohra \cite{AbBe1,AbBe2,AbBe3}, Abbas \emph{et al.} \cite{AbAgBe,AbBeGo,AbBeNi}, Belarbi \emph{et al.} \cite{BBO}, Benchohra \emph{et al.} \cite{BeHaGr, BeHaNt, BeHeNtOu}, Diethelm \cite{DiFo}, Kilbas and Marzan \cite{KiMa}, Mainardi \cite{Ma}, Podlubny \emph{et al.} \cite{PoPeViOlDo}, Vityuk and Golushkov \cite{ViGo}, Yu and Gao \cite{YuGa}, Zhang \cite{Zh} and the references therein. The theory of impulsive differential equations have become important in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There has been a significant development in impulse theory in recent years, especially in the area of impulsive differential equations and inclusions with fixed moments; see the monographs of Benchohra \emph{et al.} \cite{BHN}, Lakshmikantham \emph{et al.} \cite{LaBaSi}, the papers of Abbas and Benchohra \cite{AbBe2,AbBe3}, Abbas \emph{et al.} \cite{AbAgBe, AbBeGo} and the references therein. The Darboux problem for partial hyperbolic differential equations was studied in the papers of Abbas and Benchohra \cite{AbBe1,AbBe2}, Abbas \emph{et al.} \cite{AbBeVi}, Vityuk \cite{Vi}, Vityuk and Golushkov \cite{ViGo}, Vityuk and Mykhailenko \cite{ViMy0,ViMy} and by other authors. In the present article we are concerned with the existence and uniqueness of solutions to fractional order initial-value problem (IVP) for the system \begin{gather}\label{e1} \overline{D}_{\theta}^{r}u(x,y)=f(x,y,u(x,y), \overline{D}_{\theta}^{r}u(x,y));\quad \text{for }(x,y)\in J,\ x\neq x_k,\ k=1,\dots,m, \\ \label{e2} u(x_k^+,y)=u(x_k^-,y)+I_k(u(x_k^-,y)); \quad \text{for }y\in [0,b],\ k=1,\dots,m, \\ \label{e3} \left\{\begin{gathered} u(x,0)=\varphi (x); \quad x\in [0,a],\\ u(0,y)=\psi (y); \quad y\in[0,b],\\ \varphi(0)=\psi(0), \end{gathered}\right. \end{gather} where $J:=[0,a]\times [0,b]$, $a,b>0$, $\theta=(0,0)$, $\overline{D}_{\theta}^{r}$ is the mixed regularized derivative of order $r=(r_1,r_2)\in(0,1]\times (0,1]$, $0=x_00$. \end{definition} Analogously, we define the integral $$I_{0,y}^{\alpha}u(x,y)=\frac{1}{\Gamma (\alpha)}\int_0^{y} (y-s)^{\alpha-1}u(x,s)ds,$$ for almost all $x\in [0,a]$ and almost all $y\in [0,b]$. \begin{definition}[\cite{KST,SaKiMa}] \label{def2.2} \rm Let $\alpha\in (0,1]$ and $u\in L^1(J)$. The Riemann-Liouville fractional derivative of order $\alpha$ of $u(x,y)$ with respect to $x$ is defined by $$(D_{0,x}^{\alpha}u)(x,y)=\frac{\partial}{\partial x}I_{0,x}^{1-\alpha}u(x,y),$$ for almost all $x\in [0,a]$ and all $y\in [0,b]$. \end{definition} Analogously, we define the derivative $$(D_{0,y}^{\alpha}u)(x,y)=\frac{\partial}{\partial y}I_{0,y}^{1-\alpha}u(x,y),$$ for almost all $x\in [0,a]$ and almost all $y\in [0,b]$. \begin{definition}[\cite{KST,SaKiMa}] \label{def2.3} \rm Let $\alpha \in (0,1]$ and $u\in L^1(J)$. The Caputo fractional derivative of order $\alpha$ of $u(x,y)$ with respect to $x$ is defined by the expression $$^{c}D_{0,x}^{\alpha}u(x,y)=I_{0,x}^{1-\alpha}\frac{\partial}{\partial x}u(x,y),$$ for almost all $x\in [0,a]$ and all $y\in [0,b]$. \end{definition} Analogously, we define the derivative $$^{c}D_{0,y}^{\alpha}u(x,y)=I_{0,y}^{1-\alpha}\frac{\partial}{\partial y}u(x,y),$$ for almost all $x\in [0,a]$ and almost all $y\in [0,b]$. \begin{definition}[\cite{ViGo}] \label{def2.4} Let $r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$, $\theta=(0,0)$ and $u\in L^1(J)$. The left-sided mixed Riemann-Liouville integral of order $r$ of $u$ is defined by $$(I_{\theta}^{r}u)(x,y) =\frac{1}{\Gamma (r_1)\Gamma (r_2)}\int_0^{x} \int_0^y (x-s)^{r_1-1}(y-t)^{r_2-1}u(s,t)\,dt\,ds.$$ \end{definition} In particular, $$( I_{\theta}^{\theta}u)(x,y)=u(x,y), \ ( I_{\theta}^{\sigma}u)(x,y) =\int_0^{x}\int_0^{y}u(s,t)\,dt\,ds;$$ for almost all $(x,y)\in J$, where $\sigma=(1,1)$. For instance, $I_{\theta}^{r}u$ exists for all $r_1,r_2\in(0,\infty)$, when $u\in L^1(J)$. Note also that when $u\in C(J)$, then $(I_{\theta}^{r}u)\in C(J)$, moreover $$(I_{\theta}^{r}u)(x,0)=(I_{\theta}^{r}u)(0,y)=0; \quad x\in [0,a], \; y\in [0,b].$$ \begin{example}\label{examp2.5} \rm Let $\lambda,\omega\in(-1,\infty)$ and $r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$, then $$I_{\theta}^{r}x^{\lambda}y^{\omega} =\frac{\Gamma(1+\lambda)\Gamma(1+\omega)} {\Gamma(1+\lambda+r_1)\Gamma(1+\omega+r_2)}x^{\lambda+r_1} y^{\omega+r_2},$$ for almost all $(x,y)\in J$. \end{example} By $1-r$ we mean $(1-r_1, 1-r_2)\in (0,1]\times (0,1]$. Denote by $D^2_{xy}:=\frac{\partial ^2}{\partial x\partial y}$, the mixed second order partial derivative. \begin{definition}[\cite{ViGo}] \label{def2.6} \rm Let $r\in (0,1]\times (0,1]$ and $u\in L^1(J)$. The mixed fractional Riemann-Liouville derivative of order $r$ of $u$ is defined by the expression $D_{\theta}^{r}u(x,y)=(D^2_{xy}I_{\theta}^{1-r}u)(x,y)$ and the Caputo fractional-order derivative of order $r$ of $u$ is defined by the expression $^{c}D_{\theta}^{r}u(x,y)=(I_{\theta}^{1-r}D^2_{xy}u)(x,y)$. \end{definition} The case $\sigma=(1,1)$ is included and we have $$(D_{\theta}^{\sigma}u)(x,y)=(^{c}D_{\theta}^{\sigma}u)(x,y) =(D ^2_{xy}u)(x,y),$$ for almost all $(x,y)\in J$. \begin{example}\label{examp2.7} \rm Let $\lambda,\omega\in(-1,\infty)$ and $r=(r_1,r_2)\in (0,1]\times (0,1]$, then $$D_{\theta}^{r}x^{\lambda}y^{\omega} =\frac{\Gamma(1+\lambda)\Gamma(1+\omega)} {\Gamma(1+\lambda-r_1)\Gamma(1+\omega-r_2)}x^{\lambda-r_1} y^{\omega-r_2},$$ for almost all $(x,y)\in J$. \end{example} \begin{definition}[\cite{ViMy}] \label{def2.8}\rm For a function $u:J\to \mathbb{R}^n$, we set $$q(x,y)=u(x,y)-u(x,0)-u(0,y)+u(0,0).$$ By the mixed regularized derivative of order $r=(r_1,r_2)\in (0,1]\times (0,1]$ of a function $u(x,y)$, we name the function $$\overline{D}_{\theta}^{r}u(x,y)=D_{\theta}^{r}q(x,y).$$ \end{definition} The function $$\overline{D}_{0,x}^{r_1}u(x,y)=D_{0,x}^{r_1}[u(x,y)-u(0,y)],$$ is called the partial $r_1-$order regularized derivative of the function $u(x,y):J\to \mathbb{R}^n$ with respect to the variable $x$. Analogously, we define the derivative $$\overline{D}_{0,y}^{r_2}u(x,y)=D_{0,y}^{r_2}[u(x,y)-u(x,0)].$$ Let $a_1\in[0,a]$, $z^+=(a_1,0)\in J$, $J_z=[a_1,a]\times[0,b]$, $r_1, r_2>0$ and $r=(r_1,r_2)$. For $u\in L^1(J_z,\mathbb{R}^n)$, the expression $$(I_{z^+}^{r}u)(x,y)=\frac{1}{\Gamma (r_1)\Gamma (r_2)} \int_{a_1^+}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}u(s,t)\,dt\,ds,$$ is called the left-sided mixed Riemann-Liouville integral of order $r$ of $u$. \begin{definition}[\cite{ViGo}] \label{def2.9} \rm For $u\in L^1(J_z,\mathbb{R}^n)$ where $D^2_{xy}u$ is Lebesque integrable on $[x_k,x_{k+1}]\times[0,b], \ k=0,\dots,m$, the Caputo fractional-order derivative of order $r$ of $u$ is defined by the expression $(^{c}D_{z^+}^{r}f)(x,y)=( I_{z^+}^{1-r}D ^2_{xy}f)(x,y)$. The Riemann-Liouville fractional-order derivative of order $r$ of $u$ is defined by $(D_{z^+}^{r}f)(x,y)=(D ^2_{xy}I_{z^+}^{1-r}f)(x,y)$. \end{definition} Analogously, we define the derivatives \begin{gather*} \overline{D}_{z^+}^{r}u(x,y)=D_{z^+}^{r}q(x,y), \\ \overline{D}_{a_1,x}^{r_1}u(x,y)=D_{a_1,x}^{r_1}[u(x,y)-u(0,y)],\\ \overline{D}_{a_1,y}^{r_2}u(x,y)=D_{a_1,y}^{r_2}[u(x,y)-u(x,0)]. \end{gather*} \section{Existence of solutions} In what follows set $$J_k:=(x_k,x_{k+1}]\times [0,b].$$ To define the solutions of \eqref{e1}-\eqref{e3}, we shall consider the space \begin{align*} PC(J)=&\big\{u: J\to\mathbb{R}^n: u \in C(J_k, \mathbb{R}^n) ; \ k=0,1, \dots,m, \text{ and} \\ &\text{ there exist }u(x_k^-,y) \text{ and } u(x_k^+,y); \ k=1,\dots,m, \\ &\text{ with } u(x_k^-,y)=u(x_k,y) \text{ for each } y\in [0,b]\big\}. \end{align*} This set is a Banach space with the norm $$\|u\|_{PC}=\sup_{(x,y)\in J}\|u(x,y)\|.$$ Set $$J':=J\backslash\{(x_1,y),\dots,(x_{m},y),\ y\in [0,b]\}.$$ \begin{definition} \label{def3.1} \rm A function $u\in PC(J)$ such that $u, \overline{D}_{x_k,x}^{r_1}u, \overline{D}_{x_k,y}^{r_2}u, \overline{D}_{z_k^+}^{r}u;\ k=0,\dots,m$, are continuous on $J'$ and $I_{z^+}^{1-r}u\in AC(J')$ is said to be a solution of \eqref{e1}-\eqref{e3} if $u$ satisfies \eqref{e1} on $J'$ and conditions \eqref{e2}, \eqref{e3} are satisfied. \end{definition} For the existence of solutions for \eqref{e1}-\eqref{e3} we need the following lemmas. \begin{lemma}[\cite{ViMy}] \label{lem1} Let the function $f:J\times\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}^n$ be continuous on its variables. Then the problem \begin{gather}\label{e1'} \overline{D}_{\theta}^{r}u(x,y)=f(x,y,u(x,y), \overline{D}_{\theta}^{r}u(x,y));\quad \text{if }(x,y)\in J:=[0,a]\times [0,b], \\ \label{e2'} \begin{gathered} u(x,0)=\varphi (x); \quad x\in [0,a],\\ u(0,y)=\psi (y); \quad y\in[0,b],\\ \varphi(0)=\psi(0), \end{gathered} \end{gather} is equivalent to the problem $$g(x,y)=f(x,y,\mu(x,y)+I_{\theta}^{r}g(x,y),g(x,y)),$$ and if $g\in C(J)$ is the solution of this equation, then $u(x,y)=\mu(x,y)+I_{\theta}^{r}g(x,y)$, where $$\mu(x,y)=\varphi(x)+\psi(y)-\varphi(0).$$ \end{lemma} \begin{lemma}[\cite{AbBeGo}] \label{lem2} Let $0< r_1,r_2\leq 1$ and let $h: J \to\mathbb{R}^n$ be continuous. A function $u$ is a solution of the fractional integral equation $$\label{e4'} u(x,y)=\begin{cases} \mu(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_0^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds;\\ \text{ if } (x,y)\in [0,x_1]\times[0,b], \$3pt] \mu(x,y)+\sum_{i=1}^{k}(I_{i}(u(x_{i}^{-},y))-I_{i}(u(x_{i}^{-},0)))\\ +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}\int_{x_{i-1}}^{x_{i}}\int_0^{y} (x_{i}-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds\\ + \frac{1}{\Gamma(r_1)\Gamma(r_2)} \int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}h(s,t)\,dt\,ds;\\ \text{ if } (x,y)\in (x_k,x_{k+1}]\times[0,b],\ k=1,\dots,m, \end{cases}$$ if and only if u is a solution of the fractional initial-value problem \begin{gather}\label{e5'} ^{c}D_{z_k^{+}}^{r}u(x,y)= h(x,y), \quad (x,y)\in J', \; k=1,\dots,m,\\ \label{e6'} u(x_k^{+},y)= u(x_k^{-},y)+I_k(u(x_k^{-},y)), \quad y\in [0,b], \; k=1,\dots,m. \end{gather} \end{lemma} By Lemmas \ref{lem1} and \ref{lem2}, we conclude the following statement. \begin{lemma} \label{lem3} Let the function f:J\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n be continuous. Then problem \eqref{e1}-\eqref{e3} is equivalent to the problem $$\label{eq1} g(x,y)=f(x,y,\xi(x,y),g(x,y)),$$ where  \xi(x,y)=\begin{cases} \mu(x,y)+\frac{1}{\Gamma(r_1)\Gamma(r_2)}\int_0^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds;\\ \text{ if } (x,y)\in [0,x_1]\times[0,b], \\[3pt] \mu(x,y)+\sum_{i=1}^{k}(I_{i}(u(x_{i}^{-},y))-I_{i}(u(x_{i}^{-},0)))\\ +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{i=1}^{k}\int_{x_{i-1}}^{x_{i}}\int_0^{y} (x_{i}-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds\\ + \frac{1}{\Gamma(r_1)\Gamma(r_2)} \int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}g(s,t)\,dt\,ds;\\ \text{ if } (x,y)\in (x_k,x_{k+1}]\times[0,b],\ k=1,\dots,m, \end{cases}   \mu(x,y)=\varphi(x)+\psi(y)-\varphi(0).  And if g\in C(J) is the solution of \eqref{eq1}, then u(x,y)=\xi(x,y). \end{lemma} Further, we present conditions for the existence and uniqueness of a solution of problem \eqref{e1}-\eqref{e3}. We will us the following hypotheses. \begin{itemize} \item[(H1)] The function f:J\times \mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n is continuous; \item[(H2)] For any u,v,w,z\in \mathbb{R}^n and (x,y) \in J, there exist constants l>0 and 00 such that  \|I_k(u)-I_k(\overline u)\|\leq l^*\|u-\overline u\|, \quad \text{for } u, \overline u \in \mathbb{R}^n, \; k=1,\dots,m.  \end{itemize} \begin{theorem}\label{thm3} Assume {\rm (H1)--(H3)} are satisfied. If $$\label{e6} 2ml^*+\frac{2la^{r_1}b^{r_2}}{(1-l_{*})\Gamma(r_1+1)\Gamma(r_2+1)}<1,$$ then there exists a unique solution for IVP \eqref{e1}-\eqref{e3} on J. \end{theorem} \begin{proof} Transform the problem \eqref{e1}-\eqref{e3} into a fixed point problem. Consider the operator N:PC(J)\to PC(J) defined by \label{e7} \begin{split} N(u)(x,y) &= \mu(x,y)+\sum_{00 such that  \|\mu\|_{\infty}+2m\psi^{*}(\overline M) +\frac{2a^{r_1}b^{r_2}\big(p^{*}+q^{*}\|\mu\|_{\infty}+ 2mq^{*}\psi^{*}(\overline M)\big)}{\big(1-d^* -\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\big) \Gamma(1+r_1)\Gamma(1+r_2)} < \overline{M},  where p^{*}=\sup_{(x,y)\in J}p(x,y),  q^{*}=\sup_{(x,y)\in J}q(x,y) and\\ d^{*}=\sup_{(x,y)\in J}d(x,y). \end{itemize} \begin{theorem} \label{thm3.7} Assume {\rm (H1), (H4), (H5), (H6)} hold. If $$\label{e''} d^*+\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}<1,$$ then \eqref{e1}-\eqref{e3} has at least one solution on J. \end{theorem} \begin{proof} Transform problem \eqref{e1}-\eqref{e3} into a fixed point problem. Consider the operator N defined in \eqref{e7}. We shall show that the operator N is continuous and compact. \textbf{Step 1:} N is continuous. Let \{u_{n}\}_{n\in \mathbb{N}} be a sequence such that u_{n}\to u in PC(J). Let \eta>0 be such that \|u_{n}\|_{PC} \leq \eta. Then for each (x,y)\in J we have $$\label{e007} \begin{split} &\|N(u_n)(x,y)-N(u)(x,y)\|\\ &\leq \sum_{k=1}^{m}(\|I_k(u_n(x_k^{-},y))-I_k(u(x_k^{-},y))\| +\|I_k(u_n(x_k^{-},0))-I_k(u(x_k^{-},0))\|)\\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m} \int_{x_{k-1}}^{x_k}\int_0^{y} (x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g_n(s,t)-g(s,t)\|\,dt\,ds \\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)} \int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1} \|g_n(s,t)-g(s,t)\|\,dt\,ds, \end{split}$$ where g_n,g\in C(J) such that \begin{gather*} g_n(x,y)=f(x,y,u_n(x,y),g_n(x,y)), \\ g(x,y)=f(x,y,u(x,y),g(x,y)). \end{gather*} Since u_n\to u as n\to \infty and f is a continuous function, we obtain  g_n(x,y)\to g(x,y) \quad\text{as n\to \infty, for each (x,y)\in J}.  Hence, \eqref{e007} gives \[ \|N(u_{n})-N(u)\|_{PC} \leq 2ml^*\|u_n-u\|_{PC} +\frac{2a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)} \|g_n-g\|_{\infty}\to 0$ as $n\to \infty$. \textbf{Step 2:} $N$ maps bounded sets into bounded sets in $PC(J)$. Indeed, it is sufficinet show that for any $\eta^{*}>0$, there exists a positive constant $M^{*}$ such that, for each $u\in B_{\eta^{*}}=\{u\in PC(J):\|u\|_{PC}\leq \eta^{*}\}$, we have $\|N(u)\|_{PC}\leq M^{*}$. For $(x,y)\in J$, we have $$\label {e0007} \begin{split} &\|N(u)(x,y)\|\\ &\leq \|\mu(x,y)\|+\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y))\| +\|I_k(u(x_k^{-},0))\|)\\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m} \int_{x_{k-1}}^{x_k}\int_0^{y} (x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds\\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)} \int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds, \end{split}$$ where $g\in C(J)$ such that $g(x,y)=f(x,y,u(x,y),g(x,y))$. By (H4), for each $(x,y)\in J$, we have $$\|g(x,y)\|\leq p(x,y)+q(x,y)\|\xi(x,y)\|+d(x,y)\|g(x,y)\|.$$ On the other hand, for each $(x,y)\in J$, \begin{align*} \|\xi(x,y)\| &\leq \|\mu(x,y)\|+\sum_{k=1}^{m}(\|I_k(u(x_k^{-},y))\| +\|I_k(u(x_k^{-},0))\|)\\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)}\sum_{k=1}^{m} \int_{x_{k-1}}^{x_k}\int_0^{y} (x_k-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds\\ &\quad +\frac{1}{\Gamma(r_1)\Gamma(r_2)} \int_{x_k}^{x}\int_0^{y}(x-s)^{r_1-1}(y-t)^{r_2-1}\|g(s,t)\|\,dt\,ds\\ &\leq \|\mu\|_{\infty}+ 2m\psi^{*}(\eta^{*}) +\frac{2a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\|g\|_{\infty}. \end{align*} Hence, for each $(x,y)\in J$, we have \begin{align*} \|g\|_{\infty}&\leq p^{*}+q^{*}\Big(\|\mu\|_{\infty}+ 2m\psi^{*}(\eta^{*})+\frac{2a^{r_1}b^{r_2}} {\Gamma(1+r_1)\Gamma(1+r_2)}\|g\|_{\infty}\Big)+d^{*}\|g\|_{\infty}. \end{align*} Then, by \eqref{e''}, we have \begin{align*} \|g\|_{\infty}\leq\frac{p^{*}+q^{*}\big(\|\mu\|_{\infty}+ 2m\psi^{*}(\eta^{*})\big)}{1-d^* -\frac{2q^*a^{r_1}b^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}}:=M. \end{align*} Thus, \eqref{e0007} implies \begin{align*} \|N(u)\|_{PC}\leq\|\mu\|_{\infty}+2m\psi^{*}(\eta^{*}) +\frac{2Ma^{r_1}b^{r_2}} {\Gamma(1+r_1)\Gamma(1+r_2)}:= M^{*}. \end{align*} \textbf{Step 3:} $N$ maps bounded sets into equicontinuous sets in $PC(J)$. Let \\ $(\tau_1,y_1), (\tau_2,y_2)\in J$, $\tau_1<\tau_2$ and \$y_1