\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 31, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/31\hfil Integrodifferential equations with nonlocal condition] {Existence and uniqueness of solutions of nonlinear mixed integrodifferential equations with nonlocal condition in Banach Spaces} \author[M. B. Dhakne, H. L. Tidke \hfil EJDE-2011/31\hfilneg] {Machindra B. Dhakne, Haribhau L. Tidke} % in alphabetical order \address{Machindra B. Dhakne \newline Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, \newline Aurangabad-431 004, India} \email{mbdhakne@yahoo.com} \address{Haribhau L. Tidke \newline Department of Mathematics, School of Mathematical Sciences, North Maharashtra University, Jalgaon-425 001, India} \email{tharibhau@gmail.com} \thanks{Submitted April 28, 2010. Published February 18, 2011.} \subjclass[2000]{45N05, 47B38, 47H10} \keywords{Existence and uniqueness; mild and strong solutions; \hfill\break\indent mixed Volterra-Fredholm; integrodifferential equation; Banach fixed point theorem; nonlocal condition} \begin{abstract} In this article, we study the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition in Banach spaces. Furthermore, we study continuous dependence of mild solutions. Our analysis is based on semigroup theory and Banach fixed point theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $X$ be a Banach space with norm $\|\cdot\|$. Let $B_{r}=\{x\in X: \|x\|\leq r\}\subset X$ be a closed ball in $X$ and $E=C([t_0,t_0+\beta];B_{r})$ denote the complete metric space with metric $$d(x,y)=\|x-y\|_{E}=\sup_{t\in [t_0,t_0+\beta]}\{\|x(t)-y(t)\|:x, y\in E\}.$$ Motivated by the work in \cite{b2,b6}, we consider the nonlinear mixed Volterra-Fredholm integrodifferential equation \begin{gather} x'(t)+Ax(t)=f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds, \int_{t_0}^{t_0+\beta}h(t,s,x(s))ds),\quad t\in [t_0,t_0+\beta] \label{e1.1}\\ x(t_0)+g(t_1,t_2,\dots ,t_p,x(\cdot)=x_0, \label{e1.2} \end{gather} where $0\leq t_00$ such that $\|g(t_1,t_2,\dots ,t_p,x_1(\cdot))-g(t_1,t_2,\dots ,t_p,x_2(\cdot))\|\leq G \|x_1-x_2\|_{E}$ for $x_1, x_2\in E$. \item[(H2)] $-A$ is the infinitesimal generator of a $C_0$ semigroup $T(t)$, $t\geq 0$ in $X$ such that $\|T(t)\|\leq M,$ for some $M\geq 1$. \item[(H3)] There are constants $L_1, K_1, H_1$ and $G_1$ such that \begin{gather*} L_1=\max_{t_0\leq t\leq t_0+\beta}\|f(t,0,0,0)\|,\\ K_1=\max_{t_0\leq s\leq t\leq t_0+\beta}\|k(t,s,0)\|,\\ H_1=\max_{t_0\leq s, t\leq t_0+\beta}\|h(t,s,0)\|,\\ G_1=\max_{x\in E}\|g(t_1,t_2,\dots ,t_p,x(\cdot))\|. \end{gather*} \item[(H4)] The constants $\|x_0\|, M, G_1,L, K, K_1, H, H_1, \beta$ and $r$ satisfy the following two inequalities: \begin{gather*} M[\|x_0\|+G_1+Lr \beta+LKr{\beta}^2+LK_1{\beta}^2+LHr{\beta}^2+LH_1{\beta}^2+L_1\beta] \leq r,\\ [MG+ML\beta+MLK{\beta}^2+MLH{\beta}^2]<1. \end{gather*} \end{itemize} With these preparations we are now in a position to state our main results to be proved in the present paper. \begin{theorem}\label{thm1} Assume that \begin{itemize} \item[(i)] hypotheses {\rm (H1)--(H4)} hold, \item[(ii)] $f:[t_0,t_0+\beta]\times X\times X\times X\to X$ is continuous in $t$ on $[t_0,t_0+\beta]$ and there exists a constant $L>0$ such that $\|f(t,x_1,y_1,z_1)-f(t,x_2,y_2,z_2)\|\leq L(\|x_1-x_2\|+\|y_1-y_2\|+\|z_1-z_2\|),$ for $x_i,y_i,z_i\in B_r$, $i=1,2$. \item[(iii)] $k, h:[t_0,t_0+\beta]\times [t_0,t_0+\beta]\times X\to X$ are continuous in $s, t$ on $[t_0,t_0+\beta]$ and there exist positive constants $K,H$ such that \begin{gather*} \|k(t,s,x_1)-k(t,s,x_2)\|\leq K(\|x_1-x_2\|), \\ \|h(t,s,x_1)-h(t,s,x_2)\|\leq H(\|x_1-x_2\|), \end{gather*} for $x_i,y_i\in B_r$, $i=1,2$. \end{itemize} Then problem \eqref{e1.1}--\eqref{e1.2} has a unique mild solution on $[t_0,t_0+\beta]$. \end{theorem} \begin{theorem}\label{thm2} Assume that \begin{itemize} \item[(i)] hypotheses {\rm (H1)--(H4)} hold, \item[(ii)] $X$ is a reflexive Banach space with norm $\|\cdot\|$ and $x_0\in D(A)$,the domain of $A$, \item[(iii)] $g(t_1,t_2,\dots ,t_p,x(\cdot))\in D(A)$, \item[(iv)] There exists a constant $L>0$ such that \begin{align*} \|f(t_1,x_1,y_1,z_1)-f(t_2,x_2,y_2,z_2)\|&\leq L(|t_1-t_2|+\|x_1-x_2\|+\|y_1-y_2\|\\ &\quad+\|z_1-z_2\|), \end{align*} \item[(v)] There exist constants $K, H>0$ such that \begin{gather*} \|k(t_1,s,x_1)-k(t_2,s,x_2)\|\leq K(|t_1-t_2|+\|x_1-x_2\|), \\ \|h(t_1,s,x_1)-h(t_2,s,x_2)\|\leq H(|t_1-t_2|+\|x_1-x_2\|), \end{gather*} \end{itemize} Then $x$ is a unique strong solution of \eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$. \end{theorem} \begin{theorem}\label{thm3} Suppose that the functions $f, g, k$ and $h$ satisfy hypotheses {\rm (H1)-(H4)} and assumptions (ii), (iii) of Theorem \ref{thm1}. Then, for each pair of elements $x^*_0, x^{**}_0\in X$, and for the corresponding mild solutions $x_1, x_2$ of problem \eqref{e1.1} with $x_1(t_0)+g(t_1,t_2,\dots ,t_p,x_1(\cdot))=x^*_0$ and $x_2(t_0)+g(t_1,t_2,\dots ,t_p,x_2(\cdot))=x^{**}_0$, the inequality $\|x_1-x_2\|_{E}\leq \frac{M}{(1-MG)}\|x^*_0-x^{**}_0\| \exp{(\frac{ML\beta}{(1-MG)}(1+K\beta+H\beta))}$ is true, whenever $G<1/M$. \end{theorem} \section{Proofs of theorems} \begin{proof}[Proof of Theorem \ref{thm1}] Define an operator $F: E\to E$ by \begin{align*} (Fz)(t)&=T(t-t_0)x_0-T(t-t_0)g(t_1,t_2,\dots ,t_p,z(\cdot))\\ &\quad+\int_{t_0}^t T(t-s)f(s,z(s),\int_{t_0}^s k(s,\tau,z(\tau))d\tau, \int_{t_0}^{t_0+\beta} h(s,\tau,z(\tau))d\tau)ds, \end{align*} for $t\in [t_0,t_0+\beta]$. Now, we show that $F$ maps $E$ into itself. For $z\in E, t\in [t_0,t_0+\beta]$ and using hypotheses (H2)-(H4) and assumptions (ii), (iii), we have \begin{align*} &\|(Fz)(t)\|\\ &\leq \|T(t-t_0)x_0\|+\|T(t-t_0)g(t_1,t_2,\dots ,t_p,z(\cdot))\|\\ &\quad+\|\int_{t_0}^t T(t-s)f(s,z(s),\int_{t_0}^s k(s,\tau,z(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,z(\tau))d\tau)ds\|\\ &\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ \|f(s,z(s),\int_{t_0}^s k(s,\tau,z(\tau))d\tau,\\ &\quad\int_{t_0}^{t_0+\beta} h(s,\tau,z(\tau))d\tau)-f(s,0,0,0)\|+\|f(s,0,0,0)\|]ds\\ &\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ L(\|z(s)-0\|+\|\int_{t_0}^s k(s,\tau,z(\tau))d\tau-0\|\\ &\quad+\|\int_{t_0}^{t_0+\beta} h(s,\tau,z(\tau))d\tau-0\|)+\|f(s,0,0,0)\|]ds\\ &\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ Lr+L\int_{t_0}^s \|k(s,\tau,z(\tau))-k(s,\tau,0)+k(s,\tau,0)\|d\tau\\ &\quad+L\int_{t_0}^{t_0+\beta} \|h(s,\tau,z(\tau))-h(s,\tau,0)+h(s,\tau,0)\|d\tau+L_1]ds\\ &\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ Lr+L\beta (Kr+K_1)+L\beta(Hr+H_1)+L_1]ds\\ &\leq M[\|x_0\|+G_1+Lr\beta+LKr{\beta}^2+LK_1{\beta}^2 +LHr{\beta}^2+LH_1{\beta}^2+L_1\beta] \leq r. \end{align*} Thus, $F$ maps $E$ into itself. Now, for every $z_1, z_2\in E$, $t\in [t_0,t_0+\beta]$ and using hypotheses (H1), (H2), (H4) and assumptions (ii), (iii), we obtain \begin{align*} &\|(Fz_1)(t)-(Fz_2)(t)\|\\ &\leq \|T(t-t_0)\|\|g(t_1,t_2,\dots ,t_p,z_1(\cdot))-g(t_1,t_2,\dots ,t_p,z_2(\cdot))\|\\ &\quad+\int_{t_0}^t \|T(t-s)\|\|[f(s,z_1(s),\int_{t_0}^s k(s,\tau,z_1(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,z_1(\tau))d\tau)\\ &\quad -f(s,z_2(s),\int_{t_0}^s k(s,\tau,z_2(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,z_2(\tau))d\tau)]\|ds\\ &\leq MG\|z_1-z_2\|_{E}+\int_{t_0}^t ML[\|z_1(s)-z_2(s)\|\\ &\quad+\int_{t_0}^s \|k(s,\tau,z_1(\tau))-k(s,\tau,z_2(\tau))\|d\tau\\ &\quad +\int_{t_0}^{t_0+\beta} \|h(s,\tau,z_1(\tau))-h(s,\tau,z_2(\tau))\|d\tau]ds\\ &\leq MG\|z_1-z_2\|_{E}+ML\|z_1-z_2\|_{E} \int_{t_0}^t [1+K\int_{t_0}^s d\tau+H\int_{t_0}^{t_0+\beta} d\tau]ds\\ &\leq MG\|z_1-z_2\|_{E}+ML\|z_1-z_2\|_{E}\beta [1+K\beta+H\beta]\\ &\leq q \|z_1-z_2\|_{E}, \end{align*} where $q=MG+ML\beta+MLK{\beta}^2+MLH{\beta}^2$ and hence, we obtain \begin{align*} \|Fz_1-Fz_2\|_{E}\leq q\|z_1-z_2\|_{E}, \end{align*} with $00$ and \begin{align*} P&=M[\epsilon_1+\|Ax_0\|+\epsilon_{2}+\|Ag(t_1,t_2,\dots ,t_p,x(\cdot))\|+LK_2\beta+LH_2\beta\\ &\quad+L_2+L\beta+LK{\beta}^2+LH{\beta}^2+LK_2\beta, \end{align*} which is independent of $\theta$ and $t\in [t_0,t_0+\beta]$. Thanks to Gronwall's inequality, we obtain $\|x(t+\theta)-x(t)\|\leq P\theta e^{ML\beta}, \quad\text{for } t\in [t_0,t_0+\beta].$ Therefore, $x$ is Lipschitz continuous on $[t_0,t_0+\beta]$. The Lipschitz continuity of $x$ on $[t_0,t_0+\beta]$ combined with $(iv)$ and $(v)$ of Theorem \ref{thm2} implies \begin{align*} t\to f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds,\int_{t_0}^{t_0+\beta} h(t,s,x(s))ds) \end{align*} is Lipschitz continuous on $[t_0,t_0+\beta]$. By \cite[Corollary 4.2.11]{p1}, we observe that the equation \begin{gather*} y'(t)+Ay(t)=f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds, \int_{t_0}^{t_0+\beta} h(t,s,x(s))ds),\quad t\in [t_0,t_0+\beta]\\ y(t_0)=x_0-g(t_1,t_2,\dots ,t_p,x(\cdot)) \end{gather*} has a unique strong solution $y(t)$ on $[t_0,t_0+\beta]$ satisfying the equation \begin{align*} y(t)&=T(t-t_0)x_0-T(t-t_0)g(t_1,t_2,\dots ,t_p,x(\cdot))\\ &\quad+\int_{t_0}^t T(t-s)f(s,x(s),\int_{t_0}^s k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)ds\\ &=x(t),\quad t\in [t_0,t_0+\beta]. \end{align*} Consequently, $x(t)$ is the strong solution of initial value problem \eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$. This completes the proof of Theorem \ref{thm2}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3}] Suppose that $x_1(t)$ and $x_2(t)$ satisfy \eqref{e1.1} on $[t_0,t_0+\beta]$ with $x_1(t_0)+g(t_1,t_2,\dots ,t_p,x_1(\cdot))=x^*_0$ and $x_2(t_0)+g(t_1,t_2,\dots ,t_p,x_2(\cdot))=x^{**}_0$, respectively and $x_1, x_2\in E$. Using the equation \eqref{e2.1}, hypotheses {\rm (H1)--(H4)} and assumptions (ii), (iii), we obtain \begin{align*} &\|x_1(t)-x_2(t)\|\\ &\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E} +\int_{t_0}^t ML\Big[\|x_1(s)-x_2(s)\|\\ &\quad +\int_{t_0}^s K\|x_1(\tau)-x_2(\tau)\|d\tau +\int_{t_0}^{t_0+\beta} H\|x_1(\tau)-x_2(\tau)\|d\tau \Big]ds\\ &\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E}\\ &\quad+\int_{t_0}^t ML\Big[\|x_1(s)-x_2(s)\|+\int_{t_0}^s K\sup_{\tau\in [t_0,s]}\|x_1(\tau)-x_2(\tau)\|d\tau\\ &\quad+\int_{t_0}^{t_0+\beta} H\sup_{\tau\in [t_0,t_0+\beta]} \|x_1(\tau)-x_2(\tau)\|d\tau \Big]ds\\ &\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E}+\int_{t_0}^t ML\big[1+\beta K+\beta H\big]\|x_1-x_2\|_{E}ds. \end{align*} Therefore, we obtain $\|x_1-x_2\|_{E}\leq \frac{M}{(1-MG)}\|x^*_0-x^{**}_0\|+\int_{t_0}^t \frac{ML\beta}{(1-MG)}(1+K\beta+H\beta)\|x_1-x_2\|_{E}ds.$ Using Gronwall's inequality, we obtain $\|x_1-x_2\|_{E}\leq \frac{M}{(1-MG)}\|x^*_0-x^{**}_0\| \exp{(\frac{ML\beta}{(1-MG)}(1+K\beta+H\beta))},$ provided that $G<\frac{1}{M}$. From this inequality, it follows that the continuous dependence of solutions depends upon the initial data. This completes the proof of the Theorem \ref{thm3}. \end{proof} \section{Application} To illustrate the applications of some of our main results, we consider the nonlinear mixed Volterra- Fredholm partial integrodifferential equation $$\begin{gathered} w_{t}(u,t)-w_{uu}(u,t)=P(t,w(u,t),\int_0^t k_1(t,s,w(u,s))ds, \int_0^{\beta} h_1(t,s,w(u,s))ds),\\ 0< u< 1, \quad 0\leq t\leq \beta \end{gathered}\label{e4.1}$$ with initial and boundary conditions \begin{gather} w(0,t)=w(1,t)=0,\quad 0\leq t \leq \beta,\label{e4.2}\\ w(u,0)+\sum_{i=1}^p w(u,t_i)=w_0(u),\quad 00$such that $|\sum_{i=1}^p w(u,t_i)-\sum_{i=1}^p w(v,t_i)|\leq G^* \sup_{t\in [0,\beta]}|u(t)-v(t)|$ for$u, v\in E_1=C([0,\beta]; B^*_{r^*})$, where$B^*_{r^*}=\{x\in R:|x|\leq r^*\}$. \item There are constants$L^*_1, K^*_1, H^*_1$and$G^*_1$such that \begin{gather*} L^*_1=\max_{0\leq t\leq \beta}|P(t,0,0,0)|,\\ K^*_1=\max_{t_0\leq s\leq t\leq t_0+\beta}|k_1(t,s,0)|,\\ H^*_1=\max_{t_0\leq s, t\leq t_0+\beta}|h_1(t,s,0)|,\\ G^*_1=\max_{x\in E_1}|\sum_{i=1}^p w(u,t_i)|,\quad 00$ such that $|P(t,x_1,y_1,z_1)-P(t,x_2,y_2,z_2)|\leq L^*(|x_1-x_2|+|y_1-y_2|+|z_1-z_2|),$ for $x_i,y_i,z_i\in B^*_{r^*}$, $i=1,2$. \item $k, h:[0,\beta]\times [0,\beta]\times \mathbb{R}\to \mathbb{R}$ are continuous in $s, t$ on $[0,\beta]$ and there exist respectively constants $K^*, H^*>0$ such that \begin{gather*} |k_1(t,s,x_1)-k_1(t,s,x_2)|\leq K^*(|x_1-x_2|),\\ |h_1(t,s,x_1)-h_1(t,s,x_2)|\leq H^*(|x_1-x_2|), \end{gather*} for $x_i,y_i\in B^*_{r^*}$, $i=1,2$. \item $-A$ is the infinitesimal generator of a $C_0$ semigroup $T(t)$, $t\geq 0$ in $X$ such that $\|T(t)\|\leq M^*,$ for some $M^*\geq 1$. \item The constants $|w_0(u)|, M^*, G^*_1,L^*, K^*, K^*_1, H^*, H^*_1, \beta$ and $r$ satisfy the following two inequalities: \begin{align*} &M^*[|w_0(u)|+G^*_1+L^*r \beta+L^*K^*r{\beta}^2+L^*K^*_1{\beta}^2\\ &+L^*H^*r{\beta}^2+L^*H^*_1{\beta}^2+L^*_1\beta]\leq r^*, \end{align*} and $[M^*G^*+M^*L^*\beta+M^*L^*K^*{\beta}^2+M^*L^*H^*{\beta}^2]<1.$ \end{enumerate} First, we reduce the equations \eqref{e4.1}--\eqref{e4.3} into \eqref{e1.1}--\eqref{e1.2} by making suitable choices of $A, f, g, k$ and $h$. Let $X=L^2[0,1]$. Define the operator $A: X \to X$ by $Az = -z''$ with domain $D(A)=\{z \in X : z, z'$ are absolutely continuous, $z''\in X$ and $z(0)=z(1)=0\}$. Define the functions $f:[0,\beta]\times X\times X\times X\to X$, $k:[0,\beta]\times [0,\beta]\times X\to X$, $h:[0,\beta]\times [0,\beta]\times X \to X$ and $g:[0,\beta]^p \times X\to X$ as follows \begin{gather*} f(t,x,y,z)(u)=P(t,x(u),y(u),z(u)),\\ k(t,s,x)(u)=k_1(t,s,x(u)),\\ h(t,s,x)(u)=h_1(t,s,x(u)),\\ g(t_1,t_2,\dots ,t_p,x(\cdot)u=\sum_{i=1}^p w(u,t_i) \end{gather*} for $t\in [0,\beta], x,y,z\in X$ and \$0