\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Functional differential inclusions \hfil EJDE--2001/41} {EJDE--2001/41\hfil M. Benchohra \& S. K. Ntouyas \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 41, pp. 1--8. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence results for functional differential inclusions % \thanks{ {\em Mathematics Subject Classifications:} 34A60, 34K10. \hfil\break\indent {\em Key words:} Functional differential inclusions, measurable selection, \hfill\break\indent contraction multi-valued map, existence, fixed point, Banach space. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted March 30, 2001. Published June 4, 2001.} } \date{} % \author{M. Benchohra \& S. K. Ntouyas} \maketitle \begin{abstract} In this note we investigate the existence of solutions to functional differential inclusions on compact intervals. We use the fixed point theorem introduced by Covitz and Nadler for contraction multi-valued maps. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} This note is concerned with the existence of solutions defined on a compact real interval for the following initial and boundary-value problems of the functional differential inclusions \begin{eqnarray}\label{e1} &y'\in F(t,y_{t}), \quad\mbox{a. e. } t\in [0,T]& \\ \label{e2} &y(t)=\phi(t), \quad t\in [-r,0]& \end{eqnarray} and \begin{eqnarray}\label{e3} &y''\in F(t,y_{t}), \quad\mbox{a. e. } t\in [0,T]&\\ \label{e4} &y(t)=\phi(t), \quad t\in [-r,0],\quad y'(0)=\eta,& \end{eqnarray} where $ F:J\times C([-r,0],E)\to \mathcal{P}(E)$ is a multi-valued map, $\phi\in C([-r,0],E)$, $\eta \in E$, $\mathcal{P}(E)$ is the family of all subsets of a real separable Banach space $E$ with norm $\|\cdot\|$. For any continuous function $y$ defined on the interval $[-r,T]$ and any $t\in [0,T]$, we denote by $y_{t}$ the element of $C([-r,0],E)$ defined by $$ y_{t}(\theta)=y(t+\theta), \quad \theta\in [-r,0].$$ Here $y_{t}(\cdot)$ represents the history of the state from time $t-r$ to the time $t$. The approach used here is to reduce the existence of solutions to problems (\ref{e1})-(\ref{e2}) and (\ref{e3})-(\ref{e4}) to the search for fixed points of a suitable multi-valued map on the Banach space $C([-r,T],E)$. To prove the existence of fixed points, we use a fixed point theorem for Contraction multi-valued maps, introduced by Covitz and Nadler \cite{CoNa} (see also Deimling \cite{Dei}). For a review of recent results on boundary-value problems for functional differential equations we refer the reader to the books by Erbe, Qingai and Zhang \cite{ErQiZh} and by Henderson \cite{Hen}, to the papers by Ntouyas \cite{Nto}, by Nieto, Jiang and Jurang \cite{NiJiJu}, by Liz and Nieto \cite{LiNi}, and the references cited therein. The methods used in these problems are usually the topological transversality by Granas \cite{DuGr} and the monotone iterative method combined with upper and lower solutions \cite{LaLaVa}. \section{Preliminaries} In this section, we introduce notations, definitions, and preliminary facts from multi-valued analysis which are used throughout this note. Let $C([-r,0],E)$ be the Banach space consisting of all continuous functions from $[-r,0]$ to $E$ with the norm $$ \|\phi\|=\sup\{|\phi(\theta)|: -r\le \theta\le 0\}.$$ Similarly $C([0,T],E)$ denotes the Banach space of continuous functions on $[0,T]$ with norm $\|\cdot\|_{[0,T]}$. Let $L^1([0,T],E)$ denote the Banach space of measurable functions $y:[0,T]\to E$ which are Lebesgue integrable with norm $$ \|y\|_{L^1}=\int_{0}^{T}\, \|y(t)\|dt .$$ For a metric space $(X,d)$, we define $$\displaylines{ P(X)=\{Y\in \mathcal{P}(X): Y\neq \emptyset\}, \cr P_{cl}(X)=\{Y\in P(X): Y\mbox{ is closed }\},\cr P_{b}(X)=\{Y\in P(X): Y \mbox{ is bounded }\}. }$$ Let $H_{d}:P(X)\times P(X)\to\mathbb{R}_{+}\cup\{\infty\}$ be the operator $$ H_{d}(A,B)=\max\left\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(A,b)\right\},$$ where $d(A,b)=\inf_{a\in A}d(a,b)$ and $d(a,B)=\inf_{b\in B}d(a,b)$. Then $(P_{b,cl}(X),H_{d})$ is a metric space and $(P_{cl}(X),H_{d})$ is a generalized metric space. \paragraph{Definition} Let $N:X\to P_{cl}(X)$ be a multi-valued operator. Then \begin{itemize} \item $N$ is $\gamma$-Lipschitz if there exists $\gamma>0$ such that for each $x$ and $y$ in $X$, $H(N(x),N(y))\leq \gamma d(x,y)$. \item $N$ is a contraction if $N$ is $\gamma$-Lipschitz with $\gamma<1$. \item $N$ is completely continuous if $N(B)$ is relatively compact for every $B\in P_{b}(X)$. \item $N$ has a fixed point if there is $x\in X$ such that $x\in N(x)$. The fixed point set of the multi-valued operator $N$ will be denoted by $\mathop{\rm Fix} N$. \end{itemize} For more details on multi-valued maps and the proof of the results cited in this section, we refer the reader to the books by Deimling \cite{Dei}, by Gorniewicz \cite{Gor}, and by Hu and Papageorgiou \cite{HuPa}. Our results are based on the following fixed point theorem for contraction multi-valued operators intorduced by Covitz and Nadler in 1970 \cite{CoNa} (see also Deimling, \cite[Theorem 11.1]{Dei}). \begin{lemma}\label{l1} Let $(X,d)$ be a complete metric space. If $N:X\to P_{cl}(X)$ is a contraction, then $\mathop{\rm Fix}N\neq \emptyset$. \end{lemma} \section{Main Results} Now, we are able to state and prove our main theorems. The first result of this note concerns the initial value problem (\ref{e1})--(\ref{e2}). Before stating and proving this result, we give the definition of solution. \paragraph{Definition} A function $y: [-r,0]\to E$ is called solution of (\ref{e1})-(\ref{e2}) if $y\in C([-r,T],E)\cap AC([0,T],E)$ and satisfies the differential inclusion (\ref{e1}) a.e. on $[0,T]$ and the past conditions (\ref{e2}). \begin{theorem} \label{t1} Assume that: \begin{itemize} \item[(H1)] $F:[0,T]\times C([-r,0],E)\to P_{cl}(E)$ has the property that $F(\cdot,u): [0,T]\to P_{cl}(E)$ is measurable for each $u\in C([-r,0],E)$; \vskip 0.3cm \item[(H2)] $H(F(t,u),F(t,\overline u))\leq l(t)\|u-\overline u\|$, \ for each $t\in [0,T]$ and $u,\overline u\in C([-r,0],E)$, where $l\in L^1([0,T],\mathbb{R})$. \end{itemize} Then (\ref{e1})-(\ref{e2}) has at least one solution on $[-r,T]$. \end{theorem} \paragraph{Proof} Transform the problem into a fixed point problem. Consider the multi-valued operator, $N:C([-r,T],E)\to \mathcal{P}(C([-r,T],E))$ defined by: $$ N(y):=\Big\{h\in C([-r,T],E): h(t)= \Big\{\begin{array}{ll} \phi(t) & \mbox{ if $t\in [-r,0]$}\\ \phi(0)+\int_{0}^{t}g(s)\,ds & \mbox{if $t\in [0,T]$, } \end{array} \Big\} $$ where $$ g\in S_{F,y}=\Bigl\{g\in L^1([0,T],E) : g(t)\in F(t,y_{t}) \quad \hbox{for a.e. } t\in [0,T] \Bigr\}. $$ \paragraph{Remarks:} \begin{itemize} \item[(i)] It is clear that the fixed points of $N$ are solutions to (\ref{e1})-(\ref{e2}). \item[(ii)] For each $y\in C([-r,T],E)$ the set $S_{F,y}$ is nonempty since by (H1) $F$ has a measurable selection \cite[Theorem III.6]{CaVa}. \end{itemize} We shall show that $N$ satisfies the assumptions of Lemma \ref{l1}. The proof will be given in two steps. \paragraph{Step 1:} $N(y)\in P_{cl}(C(-r,T],E)$ for each $y\in C([-r,T],E)$. Indeed, let $(y_{n})_{n\geq 0}\in N(y)$ such that $y_{n}\to \tilde y$ in $C[-r,T],E)$. Then $\tilde y\in C[-r,T],E)$ and $$ y_{n}(t)\in \phi(0)+\int_{0}^{t}F(s,y_{s})\,ds \quad\hbox{for each } t\in [0,T].$$ Because $\int_{0}^{t}F(s,y_{s})\,ds$ is closed for each $t\in [0,T]$, then $$ y_{n}(t)\to \tilde y(t)\in \phi(0)+\int_{0}^{t}F(s,y_{s})\,ds,\quad \hbox{for } t\in [0,T].$$ So $\tilde y\in N(y)$. \paragraph{Step 2:} $H(N(y_{1}),N(y_{2}))\leq \gamma\|y_{1}-y_{2}\|$ for each $y_{1}, y_{2}\in C[-r,T],E)$ with $\gamma<1$. Let $y_{1},y_{2} \in C[-r,T],E)$ and $h_{1}\in N(y_{1})$. Then there exists $g_{1}(t)\in F(t,y_{t})$ such that $$ h_{1}(t)=\phi(0)+\int_{0}^{t}g_{1}(s)\,ds, \quad t\in [0,T].$$ From (H2) it follows that $$ H(F(t,y_{1t}), F(t,y_{2t}))\leq l(t)\|y_{1t}-y_{2t}\|.$$ Hence there is $w\in F(t,y_{2t})$ such that $$ \|g_{1}(t)-w\|\leq l(t)\|y_{1t}-y_{2t}\|, \quad t\in [0,T].$$ Consider $U:[0,T]\to \mathcal{P}(E)$, given by $$ U(t)=\{w\in E: \|g_{1}(t)-w\|\leq l(t)\|y_{1t}-y_{2t}\|\}.$$ Since the multi-valued operator $V(t)=U(t)\cap F(t,y_{2t})$ is measurable \cite[Prop. III.4]{CaVa} there exists $g_{2}(t)$ a measurable selection for $V$. So, $g_{2}(t)\in F(t,y_{2t})$ and $$ \|g_{1}(t)-g_{2}(t)\|\leq l(t)\|y_{1}-y_{2}\|, \quad \hbox{for each } t\in J.$$ For $t\in J$, let $h_{2}(t)=\phi(0)+\int_{0}^{t}g_{2}(s)\,ds$. Then \begin{eqnarray*} \|h_{1}(t)-h_{2}(t)\|&\leq&\int_{0}^{t}\|g_{1}(s)-g_{2}(s)\|\,ds\\ &\leq&\int_{0}^{t}l(s)\|y_{1}(s)-y_{2}(s)\|ds\\ &=& \int_{0}^{t}l(s)e^{-\tau L(s)}e^{\tau L(s)}\|y_{1}(s)-y_{2}(s)\|\, ds\\ &\leq& \|y_{1}-y_{2}\|_{B}\int_{0}^{t}l(s)e^{\tau L(s)}ds\\ &=& \|y_{1}-y_{2}\|_{B}\frac{1}{\tau}\int_{0}^{t}(e^{\tau L(s)})'ds\\ &\leq& \frac{\|y_{1}-y_{2}\|_{B}}{\tau}e^{\tau L(t)}ds, \end{eqnarray*} where $L(t)=\int_{0}^{t}l(s)\,ds$, $\tau>1$, and $\|\cdot\|_{B}$ is the Bielecki-type norm on $C([0,T],E)$, $$ \|y\|_{B}=\max_{t\in [0,T]}\{\|y(t)\|e^{-\tau L(t)}\}.$$ Then $\|h_{1}-h_{2}\|_{B}\leq \frac{1}{\tau}\|y_{1}-y_{2}\|_{B}$. By the analogous relation, obtained by interchanging the roles of $y_{1}$ and $y_{2}$, it follows that $$ H(N(y_{1}),N(y_{2}))\leq \frac{1}{\tau}\|y_{1}-y_{2}\|_{B}.$$ Therefore, $N$ is a contraction and thus, by Lemma \ref{l1}, it has a fixed point $y$, which is a solution to (\ref{e1})-(\ref{e2}). \hfill$\diamondsuit$\smallskip The next theorem gives an existence result for the boundary-value problem (\ref{e3})--(\ref{e4}). \paragraph{Definition} A function $y: [-r,T]\to E$ is called solution of (\ref{e3})-(\ref{e4}) if $y\in C([-r,0],E)\cap AC^1([0,T],E)$ and satisfies the differential inclusion (\ref{e3}) a.e. on $[0,T]$ and the condition (\ref{e4}). \begin{theorem} \label{t2} Let $F$ satisfy (H1) and (H2). Then (\ref{e3})-(\ref{e4}) has at least one solution on $[-r,T]$. \end{theorem} \paragraph{Proof} As in Theorem \ref{t1} we transform the problem into a fixed point problem. Consider the multi-valued operator, $N_{1}:C([-r,T],E)\to \mathcal{P}(C([-r,T],E))$ defined by \begin{eqnarray*} N_{1}(y)&:=& \Big\{h\in C([-r,T],E)\mbox{ such that }\\ && h(t)=\Big\{\begin{array}{ll} \phi(t) &\mbox{if $t\in [-r,0]$}\\ \phi(0)+t\eta +\int_{0}^{t}(t-s)g(s)\,ds &\mbox{if $t\in [0,T]$,} \end{array} \Big\} \end{eqnarray*} where $$ g\in S_{F,y}=\Bigl\{g\in L^1([0,T],E) : g(t)\in F(t,y_{t}) \quad \hbox{for a.e. } t\in [0,T] \Bigr\}. $$ \textbf{Remark} It is clear that the fixed points of $N_{1}$ are solutions to (\ref{e3})-(\ref{e4}). \smallskip We shall show that $N_{1}$ satisfies the assumptions of Lemma \ref{l1}. Using the same reasoning as in Step 1 of Therem \ref{t1} we can show that $N_{1}(y)\in P_{cl}(C(-r,T],E)$, for each $y\in C([-r,T],E)$. $N_{1}$ is a contraction multi-valued map. Indeed, let $y_{1},y_{2} \in C[-r,T],E)$ and $h_{1}\in N_{1}(y_{1})$. Then there exists $g_{1}(t)\in F(t,y_{t})$ such that $$ h_{1}(t)=\phi(0)+t\eta+\int_{0}^{t}(t-s)g_{1}(s)\,ds, \quad t\in [0,T].$$ From (H2) it follows that $$ H(F(t,y_{1t}), F(t,y_{2t}))\leq l(t)\|y_{1t}-y_{2t}\|.$$ Hence there is $w\in F(t,y_{2t})$ such that $$ \|g_{1}(t)-w\|\leq l(t)\|y_{1t}-y_{2t}\|, \quad t\in [0,T].$$ Consider $U:[0,T]\to \mathcal{P}(E)$, given by $$ U(t)=\{w\in E: \|g_{1}(t)-w\|\leq l(t)\|y_{1t}-y_{2t}\|\}.$$ Since the multi-valued operator $V(t)=U(t)\cap F(t,y_{2t})$ is measurable \cite[Prop. III.4]{CaVa} there exists $g_{2}(t)$ a measurable selection for $V$. So, $g_{2}(t)\in F(t,y_{2t})$ and $$ \|g_{1}(t)-g_{2}(t)\|\leq l(t)\|y_{1}-y_{2}\|, \quad\hbox{for each } t\in J.$$ For $t$ in $J$, let us define $h_{2}(t)=\phi(0)+t\eta+\int_{0}^{t}(t-s)g_{2}(s)\,ds$. Then we have \begin{eqnarray*} \|h_{1}(t)-h_{2}(t)\|&\leq&\int_{0}^{t}(t-s)\|g_{1}(s)-g_{2}(s)\|\, ds \\ &\leq&\int_{0}^{t}(t-s)l(s)\|y_{1}(s)-y_{2}(s)\|ds \\ &=& \int_{0}^{t}(t-s)l(s)e^{-\tau L(s)}e^{\tau L(s)}\|y_{1}(s)-y_{2}(s)\|\, ds\\ &\leq& \|y_{1}-y_{2}\|_{B}\int_{0}^{t}(t-s)l(s)e^{\tau L(s)}ds\\ &\leq& \|y_{1}-y_{2}\|_{B}\frac{T}{\tau}\int_{0}^{t}(e^{\tau L(s)})'ds\\ &\leq& \frac{T\|y_{1}-y_{2}\|_{B}}{\tau}e^{\tau L(t)}ds. \end{eqnarray*} Then $\|h_{1}-h_{2}\|_{B}\leq \frac{T}{\tau}\|y_{1}-y_{2}\|_{B}$. By the analogous relation, obtained by interchanging the roles of $y_{1}$ and $y_{2}$, it follows that $$ H(N_{1}(y_{1}),N_{1}(y_{2}))\leq \frac{T}{\tau}\|y_{1}-y_{2}\|_{B}.$$ Therefore, when $\tau>T$, $N_{1}$ is a contraction, and thus, by Lemma \ref{l1}, it has a fixed point $y$, which is solution to (\ref{e3})-(\ref{e4}). \hfill$\diamondsuit$ \paragraph{Remark} It seems that the reasoning used above can be applied for other boundary value problems for functional differential inclusions such as \begin{eqnarray}\label{e5} &y''\in F(t,y_{t}), \quad\mbox{a.e. } t\in [0,T]&\\ \label{e6} &y(t)=\phi(t) \quad t\in [-r,0],\ y(T)=\eta.& \end{eqnarray} \begin{thebibliography}{99} \frenchspacing \bibitem{CaVa} C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977. \bibitem{CoNa} H. Covitz and S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, {\em Israel J. Math.} {\bf 8} (1970), 5-11. \bibitem{Dei} K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin-New York, 1992. \bibitem{DuGr} J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mat. PWN, Warsaw, 1982. \bibitem{ErQiZh} L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics, Marcel Dekker, 1994. \bibitem{Gor} L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999. \bibitem{Hen} J. Henderson, Boundary Value Problems for Functional Differenial Equations, World Scientific, Singapore, 1995. \bibitem{HuPa} Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dordrecht, 1997. \bibitem{LaLaVa} G. S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston M. A. 1985. \bibitem{LiNi} E. Liz and J. J. Nieto, Periodic BVPs for a class of functional differential equations, {\em J. Math. Anal. Appl.} {\bf 200} (1996), 680-686. \bibitem{NiJiJu} J. J. Nieto, Y. Jiang and Y. Jurang, Monotone iterative method for functional differential equations, {\em Nonlinear Anal.} {\bf 32} (6) (1998), 741-749. \bibitem{Nto} S. K. Ntouyas, Initial and boundary value problems for functional differential equations via the topological transversality method: A survey, {\em Bull. Greek Math. Soc.} {\bf 40} (1998), 3-41. \end{thebibliography} \noindent\textsc{M. Benchohra } \\ Department of Mathematics, University of Sidi Bel Abbes \\ BP 89, 22000 Sidi Bel Abbes, Algeria \\ e-mail: benchohra@yahoo.com \smallskip \noindent\textsc{S. K. Ntouyas }\\ Department of Mathematics, University of Ioannina \\ 451 10 Ioannina, Greece \\ e-mail: sntouyas@cc.uoi.gr\\ http://www.uoi.gr/schools/scmath/math/staff/snt/snt.htm \end{document}