\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 33, pp. 1--6.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/33\hfil Nonlocal Cauchy problem] {Nonlocal Cauchy problem for quasilinear integrodifferential equations in Banach spaces} \author[M. Chandrasekaran \hfil EJDE-2007/33\hfilneg] {Mariappan Chandrasekaran} \address{Mariappan Chandrasekaran \newline Department of Mathematics\\ Sri Shakthi Institute of Engineering and Technology\\ Coimbatore, Tamil Nadu, India} \email{m\_chands\_2000@yahoo.com} \thanks{Submitted February 6, 2007. Published February 27, 2007.} \subjclass[2000]{34K05, 34K30} \keywords{Nonlocal conditions; mild Solutions; nonlinear equation; \hfill\break\indent Schaefer theorem} \begin{abstract} The aim of this paper is to prove the existence of mild solutions of the nonlocal Cauchy problem for a nonlinear integrodifferential equation. The results are established by using the method of semigroup and the Schaefer theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The purpose of this paper is to study the existence of mild solution of the following nonlinear integrodifferential equation with nonlocal condition \begin{gather} \frac {dy(t)}{dt} = A(t,y)y +\int_0^tf(t,s,y(s))ds, \quad t\in J=[0,b], \label{e1.1}\\ y(0) +g(y) = y_0, \label{e1.2} \end{gather} where $ f:\Delta \times E\to E $ and $A : J\times E\to E$ are continuous functions, $g:C(J,E)\to E$, $y_0\in E$ and $E$ is a real Banach space with the norm $\|\cdot\|$. Here $\Delta=\{(t,s):0\leq s\leq t\leq b\}$. Such problems with the classical initial condition or nonlinear boundary conditions have been studied by Conti \cite{c1}, Conti and Iannaccci \cite{c2}, Kartsatos \cite{k1}, Anichini \cite{a1}, Anichini and Conti \cite{a2} and Marino and Pietramala \cite{m1}. The nonlocal condition, which is a generalization of the classical initial condition, was motivated by physical problems. The pioneering work on nonlocal conditions is due to Byszewski \cite{b4}. In the few past years several papers have been devoted to studying the existence of solutions of differential equations with nonlocal conditions. Among others, we refer to the papers of Balachandran and Chandrasekarn \cite{b2}, Balachandran and Illamaran \cite{b1}, Byszewski \cite{b4,b5} and Ntouyas and Tsamatos \cite{n1}. The results generalise \cite[Theorem 3.1]{b3}. In this paper we study the existence of solutions for the problem \eqref{e1.1}-\eqref{e1.2}) by using the classical fixed point theorem for compact maps due to Schaefer \cite{s1}. \section{Preliminaries and basic hypothesis} In the remainder of the paper $C(J,E)$ is the Banach space of continuous functions from $J$ into $E$ with the norm $$ \|y\|_{\infty} = \sup {\{\|y(t)\| : t\in J\}} $$ and $B(E)$ denotes the Banach space of bounded linear operators from $E$ into $E$ with the norm $$ \|N\|_{B(E)} := \sup {\{\|Ny\| : \|y\| = 1\}}. $$ The following lemmas are crucial in the proof of our main theorem. \begin{lemma}[{\cite[p. 36]{h1}}] \label{lem2.1} Suppose that $\phi_1, \phi_2 \in C(J,R)$, $\phi_3 \in L^1(J,R)$, $\phi_3(t)\geq 0$ a.e. on $J$ and $\phi_1(t)\leq\phi_2(t) + \int_0^t\phi_3(s)\phi_1(s)ds$. Then $$ \phi_1(t)\leq\phi_2(t) + \int_0^t \phi_3(s)\phi_2(s) \times \exp ( { \int_s^t}\phi_3(\tau)d\tau)ds. $$ \end{lemma} \begin{lemma}[{\cite{s1} and \cite[p. 29]{s2}}] \label{lem2.2} Let $X$ be a Banach space and let $N:X \to X$ be a continuous compact map. If the set $$ \Omega := \{ y\in X : \lambda y = N(y) \text{ for some }\lambda \geq 1\} $$ is bounded, then $N$ has a fixed point. \end{lemma} Let us list the following hypotheseses: \begin{itemize} \item[(H1)] $A:J\times E \to B(E)$ is a continuous function such that for all $r>0$, there exists $r_1 = r_1(r)>0$ such that $\|v\|\leq r$ implies $\|A(t,v)\|_{B(E)} \leq r_1$, for all $v\in E$. \item[(H2)] $f:J\times E \to E$, $(t,u) \to f(t,v)$ is a continuous function. \item[(H3)] There exists a constant $L>0 $ such that $\|g(y)\| \leq L $ for each $y\in E $. \item[(H4)] $ \|f(t,s,y)\| \leq p(t)\psi (\|y\|)$ for almost all $t\in J$ and all $y\in E $, where $p\in L^1 (J,R_+)$ and $\psi : R_+\to (0,\infty) $ is continuous and increasing with $$ M \int_0^b p(s)ds< \int _c^\infty \frac{du}{\psi(u)}, $$ where $c=M\|y_0\| +ML$ and $$ M= \sup \{ U_y(t,s)\|_{B(E)}: (t,s)\in J\times J \}. $$ \item[(H5)] For each bounded $B\subset C(J,E), y\in B $, and $t\in J$ the set $$ \big\{U_y(t,0)y_0 - U_y(t,0)g(y) + \int_0^tU_y(t,s)\int_0^sf(s,\tau ,y(\tau))d\tau ds \big\} $$ is relatively compact. \end{itemize} \begin{remark} \label{rmk2.1} \rm From (H1) we are able to claim the existence for any fixed $u\in C(J,E)$ of a unique function $U_n:J\times J\to B(E) $ defined and continuous on $J\times J $ such that \begin{equation} U_u(t,s) = I + \int_s^tA_u(w)U_u(w,s)dw \label{e2.1} \end{equation} where $I$ stands for the identity operator on $E$ and $A_u(t) := A(t,u(t))$. From \eqref{e2.1} one has \begin{gather*} U_u(t,t) = I, \quad U_u(t,s)U_u(s,r) = U_u(t,r),\quad (t,s,r)\in J\times J \times J\,;\\ \frac{\partial U_u(t,s)}{\partial t} = A_u(t)U_u(t,s) \quad\text{for almost all }t\in J, \quad \forall s\in J. \end{gather*} \end{remark} \begin{remark} \label{rmk2.2} \rm From (H1) it follows that $u\in C(J,E)$ implies $A_u\in C(J,B(E))$ and $\|u_n-u^\bullet\|_{\infty} \to 0$ implies $$ \|A_{u_n}- A_{u^\bullet}\|_{\infty} := \max \{\|\|A_{u_n}(t)- A_{u^\bullet}(t)\|_{B(E)} :t\in J \}\to 0, \quad\text{as } n\to \infty. $$ \end{remark} A continuous solution $y(t)$ of the integral equation $$ y(t) = U_y(t,0)y_0 - U_y(t,0)g(y) + \int_0^tU_y(t,s)\int_0^sf(s,\tau ,y(\tau))d\tau ds $$ is called a mild solution of \eqref{e1.1}--\eqref{e1.2}. \section{An existence theorem} \begin{theorem} \label{thm3.1} Let $g:C(J,E)\to E$ be a continuous function and assume that (H1)--(H5) are satisfied. Then problem \eqref{e1.1}-\eqref{e1.2} has at least one mild solution on $J$. \end{theorem} \begin{proof} We transform the problem \eqref{e1.1}--\eqref{e1.2} into a fixed point problem. Consider the map $N: C(J,E)\to C(J,E) $ defined by $$ (Ny)(t) := U_y(t,0)y_0 - U_y(t,0)g(y) + \int_0^tU_y(t,s) \int_0^sf(s,\tau ,y(\tau))d\tau ds, \quad t\in J $$ We remark that the fixed points of $N$ are mild solutions to \eqref{e1.1}-\eqref{e1.2}. We shall show that $N$ is a continuous compact map. The proof will be given in several steps. \noindent \textbf{Step 1.} $U_u(t,s)$ is continuous with respect to $u$; i.e., $\|u_n-u^\bullet\|_{\infty} \to 0$ implies $$ \|U_{u_n}- U_{u^\bullet}\|_{\infty} :={\sup_{(t,s)\in J\times J}} \{\|U_{u_n}(t,s)- U_{u^\bullet}(t,s)\|_{B(E)} \}\to 0 \quad \text{as } n\to {\infty}. $$ Indeed, let $\|u_n - u^{\bullet} \|_{\infty}\to 0$. Then there exists $r>0$ such that $\|u_n\|_{\infty},\| u^{\bullet} \|_{\infty}\leq r$. Moreover, if $s\leq t$ we have \begin{align*} \|U_{u_n} - U_{u^*}\|_{\infty} &\leq \int_s^t\|U_{u_n}(w,s)\|_{B(E)}\|A_{u_n}(w) -A_{u^*}(w)\|_{B(E)}dw\\ &\quad + \int_s^t\|A_{u^*}\|_{B(E)}\|U_{u_n}(w,s) -U_{u^*}(w,s)\|_{B(E)}dw\\ & \leq M \int_s^t\|A_{u_n}(w) -A_{u^*}(w)\|_{B(E)}dw\\ &\quad + \int_s^t\|A_{u^*}\|_{B(E)}\|U_{u_n}(w,s) -U_{u^*}(w,s)\|_{B(E)}dw\,. \end{align*} Using Lemma \ref{lem2.1} we obtain \begin{align*} &\|U_{u_n} - U_{u^*}\|_{\infty} \\ &\leq M\int_s^t\|A_{u_n}(w) -A_{u^*}(w)\|_{B(E)}dw + M\int_s^t\|A_{u^*}(w)\|_{B(E)}\\ &\quad \times \big[\int_s^t \|A_{u_n}(\tau ) -A_{u^*}(\tau)\|_{B(E)}d\tau \big] \exp (\int_w^t\|A_{u^*}(z)\|_{B(E)}dz )dw \\ & \leq bM\|A_{u_n}-A_{u^*}\|_{\infty}+b^2M\|A_{u^*}\|_{\infty} \|A_{u_n}-A_{u^*}\|_{\infty} \exp(b\|A_{u^*}\|_\infty)\\ & \leq \|A_{u_n}-A_{u^*}\|_{\infty } Mb(1+br_1\exp(br_1)). \end{align*} The conclusion follows from Remark \ref{rmk2.2}. \noindent\textbf{Step 2.} $N$ maps bounded sets into relatively compact sets; i.e. $N$ is a compact map. Let $B_r = \{ y\in C(J,E) : \|y\|_{\infty} \leq r \}$. Then for each $t\in J$ we have $$ (Ny)(t) = U_y(t,0)y_0 - U_y(t,0)g(y) + \int_0^tU_y(t,s)\int_0^sf(s,\tau ,y(\tau))d\tau ds, \quad t\in J. $$ By (H3) and (H4), for each $t\in J$, we have \begin{align*} \|Ny\| &\leq \|U_y(t,0)\|_{B(E)}\|y_0\|+ \|U_y(t,0)\|\|g(y)\| \\ &\quad + \int_0^t\|U_y(t,s)\int_0^sf(s,\tau ,y(\tau))d\tau \|ds \\ &\leq M\|y_0\| +ML + Mb {\sup_{y\in[0,r]}} \psi (y) ( \int_0^t p(s)ds ) \end{align*} or $$ \|Ny\|_{\infty} \leq M\|y_0\| +ML + M b\sup_{t\in J} ( \int_0^t p(s)ds) \max_{y\in B} \sup_{y\in [0,r]} \psi (y):= l. $$ Now let $t_1 ,t_2,\in J, t_1 0$; hence $N(B_r)$ is an equicontinuous family of functions. Therefore by the Ascoli Arzela theorem $N(B_r)$ is relatively compact. \noindent\textbf{Step 3.} The set $ \Omega = \{ y\in C(J,E) : \lambda y = N(y) , \lambda >1\}$ is bounded. Let $y\in \Omega $. Then $\lambda y =N(y) $ for some $ \lambda >1$. Then \[ y(t) = \lambda ^{-1} U_y(t,0)y_0 - \lambda ^{-1} U_y(t,0)g(y) +\lambda ^{-1} \int_0^tU_y(t,s)\int_0^sf(s,\tau ,y(\tau))d\tau ds \quad t\in J. \] By (H3) and (H4) this implies that for each $t\in J$ we have $$ \|y(t)\|\leq M\|y_0\| +ML +M\int_0^tp(s)\psi (\|y(t)\|)ds. $$ Let us take the right-hand side of the above inequality as $v(t) $; then we have $v(0) = M\|y_0\| +ML$ and $\|y(t)\|\leq v(t)$, $t\in J$. Using the nondecresing character of $\psi $ we get $$ v' \leq Mp(t) \psi (v(t)), \quad t\in J. $$ This implies for each $t\in J$ that $$ \int_{v(0)}^{v(t)}{\frac{du}{\psi(u)}} \leq M \int_0^b p(s)ds < {\int_{v(0)}^{\infty}} {\frac{du}{\psi(u)}}\,. $$ This inequality implies that there exists a constant $d$ such that $v(t)\leq d, t\in J$, and hence $\|y\|_{\infty} \leq d$ where $d$ depends only on the functions $p$ and $\psi$. This shows that $\Omega $ is bounded. Set $ X:= C(J,E)$. As a consequence of Lemma \ref{lem2.2} we deduce that $N$ has a fixed point which is mild solution of \eqref{e1.1}-\eqref{e1.2}. \end{proof} Acknowledgement: The author wish to thank Dr.S.Thangavelu, Chairman, Sri Shakthi Institute of Engineering and Technology, Coimbatore for giving support for this research work. \begin{thebibliography}{00} \bibitem{a1} G. Anichini, \emph{Nonlinear problems for systems fo differential equations}, Nonlinear Anal.1 (1997), 691-699. \bibitem{a2} G. Anichini and G. Conti, \emph{Boundary value problems with nonlinear boundary conditions}, Nonlinearity 1 (1998), 1-10. \bibitem{b1} 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{b2} K. Balachandran and M. Chandrasekaran, \emph{Existence of solutions of a delay differential equation with nonlocal condition}, Indian J.Pure Appl. Math.27 No.5 (1996), 443-449. \bibitem{b3} M. Benchohra and S. K. Ntouyas, \emph{Existence of solutions of nonlinear differential equations with nonlocal conditions}, Journal of Mathematical Analysis and Applications, 252 (2000), 477-483. \bibitem{b4} L. Byszewski, \emph{Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem}, Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz. 18 (1993), 109 -112. \bibitem{b5} L. Byszewski, \emph{Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem}, Journal of Mathematical Analysis and Applications, 162 (1991), 494-505. \bibitem{c1} R. Conti, \emph{Recent trends in the theory of boundary value probles for ordinary differential equations}, Boll. Un. Mat. Ital.22 (1967), 135 -178. \bibitem{c2} G. Conti and R. Iannacci, \emph{Nonzore solutions of nonlinear systems of differential equations via fixed point theorems for multivalued maps}, Nolinear Anal. 8 (1982), 415-421. \bibitem{h1} J.K.Halle, \emph{Ordinary Differential Equations}, Interscience, New York, 1969. \bibitem{k1} A. Kartsatos, \emph{Non zero solutions to boundary value problems for nonlinear systems}, Pacific J.Math.53 (1974), 425-433. \bibitem{m1} G. Marino and P. Pietramala, \emph{Boundary value problems with nonlinear boundary conditions in Banach spaces}, Comment. Math. Univ. Carolina 31 (1990), 711-721. \bibitem{n1} S. K. Ntouyas and P. Ch. Tsamatos, \emph{Global existence for semilinear evolution equations with nonlocal conditions}, J. Math. Anal. Appl. 210 (1997), 679-687. \bibitem{s1} H. Schaefer, \emph{Uber die Methode der a priori Schranken}, Math. Ann. 129 (1955), 415-416. \bibitem{s2} D. R. Smart, \emph{Fixed point Theorems}, Cambridge Uni. Press, Cambridge.UK, 1974. \end{thebibliography} \end{document}