\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 06, pp. 1--7.\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 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/06\hfil Quasilinear integrodifferential equation] {Existence of solutions for quasilinear delay integrodifferential equations with nonlocal conditions} \author[K. Balachandran, F. P. Samuel \hfil EJDE-2009/06\hfilneg] {Krishnan Balachandran, Francis Paul Samuel} % in alphabetical order \address{Krishnan Balachandran \newline Department of Mathematics, Bharathiar University, Coimbatore 641 046, India} \email{balachandran\_k@lycos.com} \address{Francis Paul Samuel \newline Department of Mathematics, Bharathiar University, Coimbatore 641 046, India} \email{paulsamuel\_f@yahoo.com} \thanks{Submitted September 26, 2008. Published January 6, 2009.} \subjclass[2000]{34G20, 47D03, 47H10, 47H20} \keywords{Semigroup; mild and classical solution; Banach fixed point theorem} \begin{abstract} We prove the existence and uniqueness of mild and classical solution to a quasilinear delay integrodifferential equation with nonlocal condition. The results are obtained by using $C_0$-semigroup and the Banach fixed point theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The existence of solution to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski \cite{B5}. In that paper, he established the existence and uniqueness of mild, strong and classical solutions of the nonlocal Cauchy problem \begin{gather} u'(t)+Au(t) = f(t,u(t)), t \in (0,a]\label{1e1}\\ u(0)+g(t_1,t_2,\dots ,t_p,u(t_1),u(t_2)\dots ,u(t_p)= u_0,\label{1e2} \end{gather} where $0 < t_1 < \dots < t_p \leq a$, $-A$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$, $u_0 \in X$ and $f: [0,a] \times X \to X$, $g:[0,a]^p \times X^p \to X$ are given functions. The symbol $g(t_1,\dots ,t_p, u(\cdot))$\ is used in the sense that in the place of $\cdot$'' we can substitute only elements of the set $(t_1,\dots ,t_p)$. For example $g(t_1,\dots ,t_p, u(\cdot))= C_1 u(t_1)+\dots +C_p u(t_p),$ where $C_i$ ($i =1,2\dots ,p$) are given constants. Subsequently many authors extended the work to various kind of nonlinear evolution equations \cite{B2,B3,B6,B7}. Several authors have studied the existence of solutions of abstract quasilinear evolution equations in Banach space \cite{A1,B4,D1,S1}. Bahuguna \cite{B1}, Oka \cite{O1} and Oka and Tanaka \cite{O2} discussed the existence of solutions of quasilinear integrodifferential equations in Banach spaces. Kato \cite{K1} studied the nonhomogeneous evolution equations and Chandrasekaran \cite{C1} proved the existence of mild solutions of the nonlocal Cauchy problem for a nonlinear integrodifferential equation. Dhakne and Pachpatte \cite{D2} established the existence of a unique strong solution of a quasilinear abstract functional integrodifferential equation in Banach spaces. An equation of this type occurs in a nonlinear conversation law with memory \begin{gather} u(t,x)+\Psi(u(t,x))_x = \int_0^t b(t-s)\Psi(u(t,x))_x\ ds + f(t,x),\quad t \in [0,a], \label{1e3}\\ u(0,x)= \phi(x),\quad x \in \mathbb{R}.\label{1e4} \end{gather} It is clear that if nonlocal condition (\ref{1e2}) is introduced to (\ref{1e3}), then it will also have better effect than the classical condition $u(0,x)= \phi(x)$. Therefore, we would like to extend the results for (\ref{1e1})-(\ref{1e2}) to a class of integrodifferential equations in Banach spaces. The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of quasilinear delay integrodifferential equation with nonlocal conditions of the form \begin{gather} u'(t)+A(t,u)u(t)= f(t,u(t),u(\alpha(t)))+\int_0^t k(t,s,u(s),u(\beta(s)))ds, \label{1e5}\\ u(0)+g(u)= u_0,\label{1e6} \end{gather} where $t \in [0,a]$, $A(t,u)$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$, $u_0 \in X$, $f : I \times X \times X \to X$, $k : \Delta \times X \times X \to X$, $g : C(I : X)\to X$, $\alpha, \beta:I \to I$ are given functions. Here $I = [0,a]$ and $\Delta = \{(t,s) : 0 \leq s \leq t \leq a\}$. The results obtained in this paper are generalizations of the results given by Pazy \cite{P1}, Kato \cite{K2,K3} and Balachandran and Uchiyama \cite{B4}. \section{Preliminaries} Let $X$ and $Y$ be two Banach spaces such that $Y$ is densely and continuously embedded in $X$. For any Banach spaces $Z$ the norm of $Z$ is denoted by $\|\cdot\|$ or $\|\cdot \|_{Z}$. The space of all bounded linear operators from $X$ to $Y$ is denoted by $B(X,Y)$ and $B(X,X)$ is written as $B(X)$. We recall some definitions and known facts from Pazy \cite{P1}. \begin{definition} \label{def2.1} \rm Let $S$ be a linear operator in $X$ and let $Y$ be a subspace of $X$. The operator $\tilde{S}$ defined by $D(\tilde{S})= \{x\in D(S)\cap Y : Sx \in Y\}$ and $\tilde{S}x = Sx$ for $x \in D(\tilde{S})$ is called the part of $S$ in $Y$. \end{definition} \begin{definition} \label{def2.2} \rm Let $B$ be a subset of $X$ and for every $0 \leq t \leq a$ and $b\in B$, let $A(t, b)$ be the infinitesimal generator of a $C_0$ semigroup $S_t,_b(s), s\geq 0$, on $X$. The family of operators $\{A(t, b)\}, (t, b)\in I \times B$, is stable if there are constants $M\geq 1$ and $\omega$ such that \begin{gather*} \rho(A(t, b)) \supset (\omega, \infty) \quad\text{for } (t, b) \in I \times B, \\ \|\prod_{j =1}^k R(\lambda : A(t_j, b_j))\| \leq M(\lambda - \omega)^{-k} \end{gather*} for $\lambda > \omega$ every finite sequences $0 \leq t_1 \leq t_2 \leq \dots \leq t_k\leq a$, $b_j \in B,\ 1 \leq j\leq k$. The stability of $\{A(t, b)\}, (t, b) \in I \times B$ implies (see \cite{P1}) that $\|\prod_{j =1}^k S_{t_j,b_j}(s_j)\| \leq M \exp \big\{\omega\sum_{j=1}^k s_j\big\},\quad s_j \geq 0$ and any finite sequences $0 \leq t_1 \leq t_2 \leq \dots \leq t_k\leq a$, $b_j \in B$, $1 \leq j\leq k$. $k = 1,2,\dots$ \end{definition} \begin{definition} \label{def2.3} \rm Let $S_{t,b}(s), s\geq 0$ be the $C_0$-semigroup generatated by $A(t, b)$, $(t ,b) \in I \times B$. A subspace $Y$ of $X$ is called $A(t, b)$-admissible if $Y$ is invariant subspace of $S_{t,b}(s)$ and the restriction of $S_{t,b}(s)$ to $Y$ is a $C_0$-semigroup in $Y$. \end{definition} Let $B\subset X$ be a subset of $X$ such that for every $(t, b) \in I \times B$, $A(t, b)$ is the infinitesimal generator of a $C_0$-semigroup $S_{t,b}(s), s \geq0$ on $X$. We make the following assumptions: \begin{itemize} \item[(E1)] The family $\{A(t, b)\},(t, b) \in I \times B$ is stable. \item[(E2)] $Y$ is $A(t, b)$-admissible for $(t, b)\in I \times B$ and the family $\{\tilde{A}(t, b)\}, (t, b) \in I \times B$ of parts $\tilde{A}(t, b)$ of $A(t, b)$ in $Y$, is stable in $Y$. \item[(E3)] For $(t, b)\in I \times B$, $D(A(t, b)) \supset Y$, $A(t, b)$ is a bounded linear operator from $Y$ to $X$ and $t \to A(t, b)$ is continuous in the $B(Y, X)$ norm $\|.\|$ for every $b\in B$. \item[(E4)] There is a constant $L > 0$ such that $\|A(t, b_1) - A(t, b_2)\|_Y{_\to}_ X\ \leq L \|b_1 - b_2\|_X$ holds for every $b_1, b_2 \in B$ and $0\leq t \leq a$.\\ \end{itemize} Let $B$ be a subset of $X$ and $\{A(t, b)\}, (t, b)\in I \times B$ be a family of operators satisfying the conditions (E1)-(E4). If $u \in C(I : X)$ has values in $B$ then there is a unique evolution system $U(t,s;u), 0 \leq s \leq t \leq a$, in $X$ satisfying, (see \cite[Theorem 5.3.1 and Lemma 6.4.2, pp. 135, 201-202]{P1} \begin{itemize} \item[(i)] $\|U(t,s;u)\| \leq M e^{\omega(t-s)}$ for $0 \leq s \leq t \leq a$. where $M$ and $\omega$ are stability constants. \item[(ii)] $\frac{\partial^+}{\partial t} U(t,s;u)w =A(s, u(s))U(t,s;u)w$ for $w\in Y$, for $0 \leq s \leq t \leq a$. \item[(iii)] $\frac{\partial}{\partial s} U(t,s;u)w = -U(t,s;u)A(s, u(s))w$ for $w\in Y$, for $0 \leq s\leq t \leq a$. \end{itemize} Further we assume that \begin{itemize} \item[(E5)] For every $u \in C(I : X)$ satisfying $u(t)\in B$ for $0 \leq t \leq a$, we have $U(t,s;u)Y\subset Y,\quad 0 \leq s\leq t \leq a$ and $U(t,s;u)$ is strongly continuous in $Y$ for $0 \leq s\leq t \leq a$. \item[(E6)] $Y$ is reflexive. \item[(E7)] For every $(t,b_1, b_2) \in I \times B \times B$, $f(t, b_1, b_2) \in Y$. \item[(E8)] $g: C(I : B) \to Y$ is Lipschitz continuous in $X$ and bounded in $Y$, that is, there exist constants $G > 0$ and $G_1 > 0$ such that \begin{gather*} \|g(u)\|_Y \leq G, \\ \|g(u) - g(v)\|_Y \leq G_1 \max_{t\in I} \|u(t) - v(t)\|_X. \end{gather*} \end{itemize} For the conditions (E9) and (E10) let $Z$ be taken as both $X$ and $Y$. \begin{itemize} \item[(E9)] $k:\Delta \times Z \to Z$ is continuous and there exist constants $K_1 > 0$ and $K_2 > 0$ such that \begin{gather*} \int_0^t\|k(t,s,u_1,v_1)- k(t,s,u_2,v_2)\|_Z ds \leq K_1 (\|u_1(t) - u_2(t)+v_1(t)-v_(t)\|_Z), \\ K_2 = \max\{\int_0^t\|k(t,s,0,0)\|_Z \ ds :(t,s)\in \Delta\}. \end{gather*} \item[(E10)] $f : I \times Z \times Z \to Z$ is continuous and there exist constants $K_3 > 0$ and $K_4 > 0$ such that \begin{gather*} \|f(t,u_1,v_1) - f(t, u_2,v_2)\|_Z \leq K_3 ( \|u_1 - u_2\|_Z + \|v_1 - v_2\|_Z) \\ K_4 = \max_{t\in I} \|f(t,0,0)\|_Z. \end{gather*} Let us take $M_0 = \max \{\|U(t,s;u)\|_{B(Z)}, 0 \leq s\leq t \leq a, u\in B\}$. \item[(E11)] $\alpha, \beta : I \to I$ is absolutely continuous and there exist constants $b > 0$ and $c > 0$ such that $\alpha'(t)\geq b$ \ and $\beta'(t) \geq c$ respectively for $t \in I$. \item[(E12)] \begin{gather*} M_0\Big[\|u_0\|_Y + G+r[K_3a(1+1/b)+K_1a(1+1/c)]+a(K_4+K_2)\Big] \leq r \\ \begin{aligned} q &=\Big[Ka \|u_0\|_Y +GKa+M_0G_1+M_0[K_3a(1+1/b)+K_1a(1+1/c)]\\ &\quad +Ka[r(K_3a(1+1/b)+K_1a(1+1/c))]+a(K_4+K_2)\Big]<1. \end{aligned} \end{gather*} \end{itemize} Next we prove the existence of local classical solutions of the quasilinear problem \eqref{1e5}--\eqref{1e6}. For a mild solution of \eqref{1e5}--\eqref{1e6} we mean a function $u\in C(I : X)$ with values in B and $u_0 \in X$ satisfying the integral equation \begin{aligned} u(t)&=U(t,0;u)u_0-U(t,0;u)g(u)+\int_0^tU(t,s;u)[f(s,u(s),u(\alpha(s)))\\ &\quad +\int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau]ds. \end{aligned} \label{2e1} A function $u \in C(I : X)$ such that $u(t)\in D(A(t, u(t)))$ for $t \in(0, a]$, $u\in C^1((0, a]: X)$ and satisfies \eqref{1e5})--\eqref{1e6} in $X$ is called a classical solution of (\ref{1e5}) -(\ref{1e6}) on $I$. Further there exists a constant $K > 0$ such that for every $u, v \in C(I : X)$ with values in $B$ and every $w \in Y$ we have $$\label{2e2} \|U(t,s;u)w - U(t,s;v)w\| \leq K \|w\|_Y \int_s^t \|u(\tau) - v(\tau)\|d\tau.$$ \section{Existence Result} \begin{theorem} \label{thm3.1} Let $u_0 \in Y$ and let $B = \{u\in X: \|u\|_Y \leq r\}$, $r > 0$. If the assumptions {\rm (E1)--(E12)} are satisfied, then \eqref{1e5}--\eqref{1e6} has a unique classical solution $u \in C([0,a]:Y)\cap C^1((0,a]:X)$ \end{theorem} \begin{proof} Let $S$ be a nonempty closed subset of $C([0,a]:X)$ defined by $S = \{u:u\in C([0, a]: X),\|u(t)\|_Y \leq r \ for\ 0\leq t\leq a\}$. Consider a mapping $P$ on $S$ defined by \begin{align*} (Pu)(t) &= U(t,0;u)u_0 - U(t,0;u)g(u)+\int_0^t U(t,s;u)\Big[f(s,u(s), u(\alpha(s)))\\ &\quad +\int_0^sk(s,\tau, u(\tau), u(\beta(\tau)))d\tau\Big]ds. \end{align*} We claim that $P$ maps $S$ into $S$. For $u\in S$, we have \begin{align*} & \|Pu(t)\|_Y \\ &= \|U(t,0;u)u_0 - U(t,0;u)g(u)+ \int_0^t U(t,s;u)\Big[f(s,u(s),u(\alpha(s)))\\ &\quad + \int_0^s k(s,\tau, u(\tau)u(\beta(\tau)))d\tau\Big]ds\|\\ &\leq \|U(t,0;u)u_0\|+\|U(t,0;u)g(u)\|\\ &\quad + \int_0^t\ \|U(t,s;u)\|\Big[\|f(s,u(s),u(\alpha(s)))- f(s,0,0)\| + \|f(s,0,0)\|\\ &\quad + \|\int_0^s [k(s,\tau, u(\tau),u(\beta(\tau)))- k(s,\tau,0,0)] d\tau\|+ \|\int_0^s k(s,\tau,0,0)d\tau\|\Big]ds. \end{align*} Using assumptions (E8)-(E11), we get \begin{align*} \|Pu(t)\|_Y &\leq M_0 \|u_0\|_Y+M_0G+\int_0^tM_0\Big[ K_3(\|u(s)\|+\|u(\alpha(s))\|) +K_4 \\ &\quad + \int_0^s K_1(\|u(s)\|+u(\beta(\tau))\|)d\tau + \int_0^s K_2d\tau\Big]ds \\ &\leq M_0 \|u_0\|_Y + M_0G+M_0\Big[ K_3ar+ K_3 \int_0^t \|u(\alpha(s))\|(\alpha'(s)/b)ds \\ &\quad + K_4a + K_1ar+K_1\int_0^t(\|u(\beta(s))\|(\beta'(s)/c) ds+K_2a\Big] \\ &\leq M_0 \|u_0\|_Y+M_0G+M_0\Big[K_3ar+(K_3/b)\int_{\alpha (0)}^{\alpha (t)} \|u(s)\|ds+K_4a \\ &\quad + K_1ar+(K_1/c)\int_{\beta (0)}^{\beta (t)}(\|u(s)\| ds+K_2a\Big]\\ &\leq M_0\Big[\|u_0\|_Y + G+r[K_3a(1+1/b)+K_1a(1+1/c)]+a(K_4+K_2)\Big]\\ \end{align*} From assumption (E12), one gets $\|Pu(t)\|_Y \leq r$. Therefore $P$ maps $S$ into itself. Moreover, if $u,v \in S$, then \begin{align*} &\|Pu(t)- Pv(t)\|\\ &\leq \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)g(u)-U(t,0;v)g(v)\|\\ &\quad + \int_0^t\|U(t,s;u)\Big[f(s,u(s),u(\alpha(s))) + \int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau\Big]\\ &\quad - U(t,s;v)\Big[f(s,v(s),v(\alpha(s))) + \int_0^s k(s,\tau,v(\tau),v((\beta(\tau)))d\tau\Big]\|ds \\ &\leq \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)g(u)-U(t,0;v)g(u)\|\\ &\quad - \|U(t,0;v)g(u)-U(t,0;v)g(v)\|\\ &\quad + \int_0^t\Big\{\Big\|U(t,s;u)\Big[f(s,u(s),u(\alpha(s))) +\int_0^sk(s,\tau,u(\tau),u(\beta(\tau)))d\tau\Big]\\ &\quad - U(t,s;v)\Big[f(s,u(s),u(\alpha(s))) +\int_0^s k(s,\tau,u(\tau),u((\beta(\tau)))d\tau\Big]\|\\ &\quad + \|U(t,s;v)\Big[f(s,u(s),u(\alpha(s))) +\int_0^sk(s,\tau,u(\tau),u((\beta(\tau)))d\tau\Big]\\ &\quad - U(t,s;v)\Big[f(s,v(s),v(\alpha(s))) +\int_0^sk(s,\tau,v(\tau),v((\beta(\tau)))d\tau\Big]\|\Big\}ds \end{align*} Using assumptions (E8)-(E12), one can get \begin{align*} &\|Pu(t)- Pv(t)\|\\ &\leq Ka\|u_0\|_Y\max_{\tau\in I}\|u(\tau)-v(\tau)\| +GKa \max_{\tau\in I}\|u(\tau)-v(\tau)\|\\ &\quad + M_0G_1\max_{\tau\in I}\|u(\tau)-v(\tau)\|\\ &\quad + Ka \max_{\tau\in I}\|u(\tau)-v(\tau)\|\Big[K_3\int_0^t\|u(s)\|ds +K_3\int_0^t\|u(\alpha(s))\|(\alpha'(s)/b)ds\\ &\quad + K_4a +K_1ar+K_1\int_0^t\|u(\beta(s))\|(\beta'(s)/c) ds+K_2a\Big]\\ &\quad + M_0\Big[K_3\int_0^t\|u(s)-v(s)\|ds+K_3\int_0^t\|u(\alpha(s)) -v(\alpha(s))\|(\alpha'(s)/b)ds\\ &\quad + K_1a \max_{\tau\in I}\|u(\tau)-v(\tau)\|+K_1\int_0^t\|u(\beta(s)) -v(\beta(s))\|(\beta'(s)/c) ds \\ &\leq \Big[Ka\|u_0\|_Y+GKa +M_0G_1+M_0[K_3a(1+1/b)+K_1a(1+1/c)]\\ &\quad + Ka[r(K_3a(1+1/b)+K_1a(1+1/c))]+a(K_4+K_2)\Big] \max_{\tau\in I}\|u(\tau)-v(\tau)\|\\ &= q \max_{\tau\in I} \|u(\tau)-v(\tau)\| \end{align*} where $0 < q < 1$. From this inequality it follows that for any $t\in I$, $\|Pu(t)-Pv(t)\|\leq q \max_{\tau\in I} \|u(\tau)-v(\tau)\|,$ so that $P$ is a contraction on $S$. From the contraction mapping theorem it follows that $P$ has a unique fixed point $u\in S$ which is the mild solution of \eqref{1e5})--\eqref{1e6} on $[0,a]$. Note that $u(t)$ is in $C(I : Y)$ by (E6) see \cite[pp. 135, 201-202 Lemma 7.4]{P1}. In fact , $u(t)$ is weakly continuous as a $Y$-valued function. This implies that $u(t)$ is separably valued in $Y$, hence it is strongly measurable. Then $\|u(t)\|_ Y$ is bounded and measurable function in $t$. Therefore, $u(t)$ is Bochner integrable (see e.g. \cite[Chap.V]{Y1}). Using relation $u(t)= Pu(t)$, we conclude that $u(t)$ is in $C(I :Y)$. Now consider the evolution equation \begin{gather} v'(t)+B(t)v(t)= h(t), \quad t\in [0, a] \label{3e1}\\ v(0) = u_0-g(u)\label{3e2} \end{gather} where $B(t)= A(t, u(t))$ and $h(t) = f(t, u(t),u(\alpha(t)))+\int_0^t k(t,s,u(s),u(\beta(s))ds$, $t \in [0,a]$ and $u$ is the unique fixed point of $P$ in $S$. We note that $B(t)$ satisfies (H1)-(H3) in \cite[Sec. 5.5.3]{P1} and $h\in C(I : Y)$. Theorem 5.5.2 of \cite{P1} implies that there exists a unique function $v\in C(I : Y)$ such that $v \in C^1((0,a], X)$ satisfying \eqref{3e1} and \eqref{3e2} in $X$ and $v$ is given by \begin{align*} v(t)&=U(t,0;u)u_0-U(t,0;u)g(u)+\int_0^t U(t,s;u)[f(s,u(s), u(\alpha(s)))\\ &\quad + \int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau]ds, \end{align*} where $U(t,s;u)$ is the evolution system generated by the family $\{A(t, u(t))\}, t\in I$ of the linear operators in $X$. The uniqueness of $v$ implies that $v = u$ on $I$ and hence $u$ is a unique classical solution of \eqref{1e5})--\eqref{1e6} and $u \in C([0,a] : Y)\cap C^1((0,a] :X)$. \end{proof} \begin{thebibliography}{00} \bibitem {A1} H. Amann, \emph{Quasilinear evolution equations and parabolic systems,} Trans. Amer. Math. Soc. 29 (1986), 191-227. \bibitem {B1} D. Bahuguna, \emph{Quasilinear integrodifferential equations in Banach spaces,} Nonlinear Anal. 24 (1995), 175-183. \bibitem {B2} K. Balachandran and M. Chandrasekaran, \emph{Existence of solution of a delay differential equation with nonlocal condition,} Indian J. Pure Appl. Math. 27 (1996), 443-449. \bibitem {B3} K. Balachandran and S. Ilamaran, \emph{Existence and uniqueness of mild and strong solutions of a semilinear evolution equation with nonlocal conditions,} Indian J. Pure Appl. Math. 25 (1994), 411-418. \bibitem {B4} K. Balachandran and K. 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