\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 141, 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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/141\hfil Existence of solutions] {Existence of solutions for nonconvex functional differential inclusions} \author[Vasile Lupulescu\hfil EJDE-2004/141\hfilneg] {Vasile Lupulescu} % in alphabetical order \address{Vasile Lupulescu \hfill\break Constantin Br\^{a}ncu\c{s}i'' University of T\^{a}rgu-Jiu\\ Bulevardul Republicii, nr. 1 \\ 1400 T\^{a}rgu-Jiu, Romania} \email{vasile@utgjiu.ro} \date{} \thanks{Submitted September 24, 2004. Published November 29, 2004.} \subjclass[2000]{34A60, 34K05, 34K25} \keywords{Functional differential inclusions; existence result} \begin{abstract} We prove the existence of solutions for the functional differential inclusion $x'\in F(T(t)x)$, where $F$ is upper semi-continuous, compact-valued multifunction such that $F(T(t)x)\subset \partial V(x(t))$ on $[0,T]$, $V$ is a proper convex and lower semicontinuous function, and $(T(t)x)(s)=x(t+s)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction} Let $\mathbb{R}^{m}$ be the $m$-dimensional Euclidean space with the norm $\|\cdot \|$ and the scalar product $\langle \cdot ,\cdot \rangle$. When $I$ is a segment in $\mathbb{R}$, we denote by $\mathcal{C}(I,\mathbb{R}^{m})$ the Banach space of continuous functions from $I$ to $\mathbb{R}^{m}$ with the norm $\|x(.)\|_{\infty }:=\sup \{\|x(t)\|;t\in I\}$. When $\sigma$ is a positive number, we put $\mathcal{C}_{\sigma }:=\mathcal{C}([-\sigma ,0],\mathbb{R}^{m})$ and for any $t\in [ 0,T]$, $T>0$, we define the operator $T(t)$ from $\mathcal{C}([-\sigma ,T],\mathbb{R}^{m})$ to $\mathcal{C}_{\sigma }$ as $(T(t)x)(s):=x(t+s)$, $s\in [-\sigma ,0]$. Let $\Omega$ be a nonempty subset in $\mathcal{C}_{\sigma }$. For a given multifunction $F:\Omega \to 2^{\mathbb{R}^{m}}$ we consider the following functional differential inclusion: $$x'\in F(T(t)x). \label{e1.1}$$ We recall that a continuous function $x(.):[-\sigma ,T]\to \mathbb{R}^{m}$ is said to be a solution of \eqref{e1.1} if $x(.)$ is absolutely continuous on $[0,T]$, $T(t)x\in \Omega$ for all $t\in [ 0,T]$ and $x'(t)\in F(T(t)x)$ for almost all $t\in [0,T]$; see \cite{h1}. The functional differential equation \eqref{e1.1} with $F$ single-valued, has been studied by many authors; for results, references, and applications, see for example \cite{h2,l1}. The existence of solutions for the functional differential inclusion \eqref{e1.1} was proved by Haddad \cite{h1} when $F$ is upper semicontinuous with convex compact values. The nonconvex case in Banach space has been studied by Benchohra and Ntouyas \cite{b1}. The case when $F$ is lower semicontinuous with compact value has been studied by Fryszkowski \cite{f1}. In this paper we prove the existence of solutions for functional differential inclusion \eqref{e1.1} when $F$ is upper semicontinuous, compact valued multifunction such that $F(\psi )\subset\partial V(\psi (0))$ for every $\psi \in \Omega$ and $V$ is a proper convex and lower semicontinuous function. Our existence result contains Peano's existence theorem as a particular case. On the other hand, our result may be considered as an extension of the previous result of Bressan, Cellina and Colombo \cite{b2}. \section{Preliminaries and statement of the main result} For $x\in \mathbb{R}^{m}$ and $r>0$ let $B(x,r):=\{y\in \mathbb{R}^{m};\|y-x\|0$ there exists $\delta >0$ such that \begin{equation*} F(\psi )\subset F(\varphi )+B(0,\varepsilon ),\quad \forall \psi \in \Omega \cap B_{\sigma }(\varphi ,\delta ). \end{equation*} The definition of the upper semicontinuous multifunctions is the same as \cite[Definition 1.2]{d1}. For a multifunction $F:\Omega \to 2^{\mathbb{R}^{m}}$ we consider the functional differential inclusion \eqref{e1.1} under the following assumptions: \begin{itemize} \item[(H1)] $\Omega \subset \mathcal{C}_{\sigma }$ is an open set and $F$ is upper semicontinuous with compact values; \item[(H2)] There exists a a proper convex and lower semicontinuous function $V:\mathbb{R}^{m}\to \mathbb{R}$ such that $$F(\psi )\subset \partial V(\psi (0))\text{ for every }\psi \in \Omega . \label{e2.2}$$ \end{itemize} \noindent\textbf{Remark.} A convex function $V:\mathbb{R}^{m}\to \mathbb{R}$ is continuous in the whole space $\mathbb{R}^{m}$ \cite[Corollary 10.1.1]{r1} and almost everywhere differentiable \cite[Theorem 25.5]{r1}. Therefore, (H2) restricts strongly the multivaluedness of $F$. Our main result is the following: \begin{theorem} \label{thm2.1} If $F:\Omega \to 2^{\mathbb{R}^{m}}$ and $V:\mathbb{R}^{m}\to \mathbb{R}$ satisfy assumptions (H1) and (H2) then for every $\varphi \in \Omega$ there exists $T>0$ and $x(.):[-\sigma ,T]\to \mathbb{R}^{m}$ a solution of the functional differential inclusion \eqref{e1.1} such that $T(0)x=\varphi$ on $[-\sigma ,0]$. \end{theorem} \section{Proof of the main result} Let $\varphi \in \Omega$ be arbitrarily fixed. Since the multifunction $% x\to \partial V(x)$ is locally bounded \cite[Proposition 2.9]{b3}, there exists $r>0$ and $M>0$ such that $V$ is Lipschitz continuous with constant $M$ on $B(\varphi (0),r)$. Since $\Omega$ is an open set we can choose $r$ such that $\overline{B}_{\sigma }(\varphi ,r)\subset \Omega$. Moreover, by \cite[Proposition 1.1.3]{a1}, $F$ is locally bounded; therefore, we can assume that $$\sup \{\|y\|: y\in F(\psi ),\,\psi \in B(\varphi ,r)\}\leq M\,. \label{e3.1}$$ Since $\varphi$ is continuous on $[-\sigma ,0]$ we can choose $\eta >0$ such that \|\varphi (t)-\varphi (s)\|0. \end{equation*} Then we have: \begin{align*} \|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty } &= \sup_{-\sigma\leq s\leq 0} \|x_{n}(\theta _{n}(t)+s)-x_{n}(t+s)\| \\ &\leq \omega (x_{n},[-\sigma ,T],\frac{T}{n}) \\ &\leq \omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\omega (x_{n},[0,T], \frac{T}{n}) \\ &\leq \omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\frac{T}{n}M; \end{align*} hence $$\|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty }\leq \delta _{n}\quad \text{for every }t\in [ 0,T], \label{e3.14}$$ where $\delta _{n}:=\omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\frac{T}{n}M$. Thus, by continuity of $\varphi$, we have $\delta _{n}\to 0$ as $n\to \infty$ and hence \begin{equation*} \|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty }\to 0\text{ as } n\to \infty . \end{equation*} Therefore, since the uniform convergence of $x_{n}$ to $x$ on $[-\sigma ,T]$ implies $$T(t)x_{n}\to T(t)x\quad \text{uniformly on }[-\sigma ,0], \label{e3.15}$$ we deduce that $$T(\theta _{n}(t))x_{n}\to T(t)x\quad\text{in }\mathcal{C}_{\sigma }\,. \label{e3.16}$$ Moreover, by \eqref{e3.13} and \eqref{e3.16}, we have that $T(t)x\in \overline{B}_{\sigma }(\varphi ,r)\subset \Omega$. Also, by \eqref{e3.11} and \eqref{e3.14}, we have $$d((T(t)x_{n},x_{n}'(t)),\mathop{\rm graph}(F))\leq \delta _{n}\quad \text{ for every }t\in [ 0.T]. \label{e3.17}$$ By (H2), (ii), \eqref{e3.16} and \cite[Theorem 1.4.1]{a1}, we obtain $$x'(t)\in coF(T(t)x)\subset \partial V(x(t))\quad \text{a.e. on }[0,T], \label{e3.18}$$ where $co$ stands for the closed convex hull. Since the functions $t\to x(t)$ and $t\to V(x(t))$ are absolutely continuous, we obtain from \cite[Lemma 3.3]{b3} and \eqref{e3.18} that \begin{equation*} \frac{d}{dt}V(x(t))=\|x'(t)\|^{2}\quad\text{a.e. on }[0,T]; \end{equation*} therefore, $$V(x(T))-V(x(0))=\int_{0}^{T}\|x'(t)\|^{2}dt. \label{e3.19}$$ On the other hand, since \begin{equation*} x_{n}'(t)=y_{n}^{j}\in F(T(t_{n}^{j})x_{n})\subset \partial V(x_{n}(t_{n}^{j})) \end{equation*} for every $t\in [ t_{n}^{j},t_{n}^{j+1}]$ and for every $j\in\{0,1,\dots ,n-1\}$, it follows that \begin{align*} V(x_{n}(t_{n}^{j+1}))-V(x_{n}(t_{n}^{j})) &\geq \langle x_{n}'(t),x_{n}(t_{n}^{j+1})-x_{n}(t_{n}^{j})\rangle \\ &=\langle x_{n}'(t),\int_{t_{n}^{j}}^{t_{n}^{j+1}}x_{n}'(t)dt\rangle =\int_{t_{n}^{j}}^{t_{n}^{j+1}}\|x'(t)\|^{2}dt. \end{align*} By adding the $n$ inequalities above, we obtain \begin{equation*} V(x_{n}(T))-V(x(0))\geq \int_{0}^{T}\|x_{n}'(t)\|^{2}dt \end{equation*} and passing to the limit as $n\to \infty$, we obtain $$V(x(T))-V(x(0))\geq \underset{n\to \infty }{\lim \sup } \int_{0}^{T}\|x_{n}'(t)\|^{2}dt. \label{e3.20}$$ Therefore, by b\eqref{e3.19} and \eqref{e3.20}, \begin{equation*} \int_{0}^{T}\|x'(t)\|^{2}dt\geq \underset{n\to \infty }{% \lim \sup }\int_{0}^{T}\|x_{n}'(t)\|^{2}dt \end{equation*} and, since $(x_{n}')$ converges weakly in $L^{2}([0,T],\mathbb{R}^{m})$ to $x'$, by applying \cite[Proposition III.30]{b4}, we obtain that $(x_{n}')$ converges strongly in $L^{2}([0,T],\mathbb{R}^{m})$% . Hence there exists a subsequence, still denote by $(x_{n}')$, which converges pointwiese a.e. to $x'$. Since, by (H1), the graph of $F$ is closed \cite[Proposition 1.1.2]{a1}, by \eqref{e3.17}, \begin{equation*} \lim_{n\to \infty } d((T(t)x_{n},x_{n}'(t)),\mathop{\rm graph}(F))=0, \end{equation*} we obtain \begin{equation*} x'(t)\in F(T(t)x)\quad \text{a.e. on }[0,T]\,. \end{equation*} Therefore, the functional differential inclusion \eqref{e1.1} has solutions. \begin{thebibliography}{99} \bibitem{a1} J. P. Aubin and A. Cellina; \emph{Differential inclusions}, Berlin, Springer-Verlag, 1984. \bibitem{b1} M. Benchohra and S. K. Ntouyas; \emph{Existence results for functional differential inclusions}, Electron. J. Diff. Eqns., Vol. 2001(2001), No. 41, pp. 1-8. \bibitem{b2} A. Bressan, A. Cellina and G. Colombo; \emph{Upper semicontinuous differential inclusions without convexity}, Proc. Amer. Math. Soc., 106(1984), 771-775. \bibitem{b3} H. 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