\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 175, pp. 1--8.\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 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/175\hfil Existence of solutions] {Existence of solutions to differential inclusions with delayed arguments} \author[L. Boudjenah\hfil EJDE-2010/175\hfilneg] {Lotfi Boudjenah} \address{Lotfi Boudjenah \newline Department of Computer Science, Faculty of Sciences, University of Oran, BP 1524 Oran 31000, Algeria} \email{lotfi60@yahoo.fr} \thanks{Submitted May 24, 2010. Published December 15, 2010.} \subjclass[2000]{34A60, 49J24, 49K24} \keywords{Delayed argument; differential inclusion; fixed point; set-valued;\hfill\break\indent upper semi-continuity} \begin{abstract} In this work we investigate the existence of solutions to differential inclusions with a delayed argument. We use a fixed point theorem to obtain a solution and then provide an estimate of the solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} This note concerns the existence of solutions to differential inclusions with delayed argument, $x'(t)\in F(t,x_t)$ for $t\geq t_0$ with the initial condition $x(t)=\varphi (t)$ for $t\leq t_0$. The first works on differential inclusions were published in 1934-35 by Marchaud \cite{m1} and Zaremba \cite{z1}. They used the terms contingent and paratingent equations. Later, Wasewski and his collaborators published a series of works and developed the elementary theory of differential inclusions \cite{w1,w2}. Within few years after the first publications, the differential inclusions became a basic tool in optimal control theory. Starting from the pioneering work of Myshkis \cite{m2}, there exists a whole series of papers devoted to paratingent and contingent differential inclusions with delay; see for example Campu \cite{c1,c2} and Kryzowa \cite{k3}. After this, many works appeared on differential inclusions with delay, for example Anan'ev \cite{a1}, Deimling \cite{d1}, Hong \cite{h1} and Zygmunt \cite{z2}. Recent results about differential inclusions in Banach spaces were obtained by Boudjenah \cite{b1}, Syam \cite{s2} and Castaing-Ibrahim \cite{c3}. For more details on differential inclusions see the books by Aubin and Cellina \cite{a2}, Deimling \cite{d1}, Smirnov \cite{s1}, and Kisielewicz \cite{k1}. In this work, we study the existence of solutions to differential inclusions with delayed argument, and we extend a result obtained by Anan'ev \cite{a1}. \section{Preliminaries} Let $\mathbb{R}^n$ denote the $n$ dimensional Euclidean space and $\|\cdot\|$ its norm. Let $B$ be a Banach space with norm $\|\cdot\|_{B}$. If $x\in B^n=B\times B\times \dots\times B$, then $x_i\in B$, $i=1,\dots,n$, and $\|x\|_{B^n}=\big(\sum \|x_i\|_{B}^2\big) ^{1/2}$. Let $(M,d)$ be a metric space, $A\subset M$, and $\epsilon$ a positive number. We denote by $A^{\epsilon }$ the closed $\epsilon$-neighborhood of $A$; i.e., $A^{\epsilon }=\{ x\in M: d(x,a)\leq \epsilon \}$. Let $\overline{A}$ denote the closure of $A$ and $\operatorname{co}A$ the convex hull of $A$. Let $C_{[a,b]}$ be the space of continuous real functions on $[a,b]$ and $L_{p[a,b]}$ the space of real-valued functions whose $p$-power is integrable on $[a,b]$. For $f\in L_{p[a,b ]}$, let $\|f\|_{p}=(\int_a^b |f(x)|^pdx)^{1/p}$. Let $\operatorname{Conv}\mathbb{R}^n$ denote the set of all compact, convex and nonempty subsets of $\mathbb{R}^n$. Fix $t_0\in \mathbb{R}$, let $h(t)$ be a continuous and positive function for $t\geq t_0$ and $F$ be a set-valued map: $[t_0,+\infty [ \times C_{ [-h(t),0]}^n\to \operatorname{Conv}\mathbb{R}^n$ such that: $(t,[x]_t)\to F(t,x_t)\in \operatorname{Conv}\mathbb{R}^n$, for $t\geq t_0$ and $x_t\in C_{[-h(t), 0]}^n$, where $x_t(\zeta )=x(t+\zeta )$. For $-h(t)\leq \zeta \leq 0$, $x_t(\cdot)$ represents the history of the state from time $t-h(t)$ to time $t$. For fixed $t$, the map $F(t,.):C_{[-h(t), 0]}^n\to \operatorname{Conv}\mathbb{R}^n$ is called upper semi-contin\-uous, u.s.c for short, if: for all $\epsilon >0$ there exists $\delta >0$ such that $\|x_t-y_t\|_{C^n}\leq \delta$ implies $F(t,y_t)\subset F^{\epsilon }(t,x_t)$ where $x_t,y_t\in C_{[ -h(t), 0 ]}^n$ (See \cite{a2}). The map $F(.,x_{.})$ is called Lebesgue-measurable on $[t_0,\gamma ]$, if the set $Z=\{ t\in [ t_{0},\gamma]: F(t,x_t)\cap K\neq \emptyset \}$ is Lebesgue-measurable for any closed set $K\subset \mathbb{R}^n$ (See \cite{a2}). Let $F$ be a set valued map: $[t_0,+\infty ]\times C_{[-h(t), 0]}^n\to \operatorname{Conv}\mathbb{R}^n$. A relation of the form $$x'(t)\in F(t,x_t)\quad \text{for }t\geq t_0. \label{e1}$$ is called a differential inclusion with delayed argument. The generalized Cauchy problem consists of searching a solution of the differential inclusion \eqref{e1} which satisfies the initial condition $$x(t)=\varphi (t)\quad \text{for }t\leq t_0\,. \label{e2}$$ A function $x$ is called solution of \eqref{e1}-\eqref{e2} if $x$ is absolutely continuous on $[t_0,\gamma ]$ and satisfies the differential inclusion \eqref{e1} a.e, (almost everywhere) on $[t_0,\gamma ]$ and the initial condition $x(t)=\varphi (t)$ for $t\leq t_0$. For the proof of our main theorem we need some lemmas including Opial's theorem wich is presented next. \begin{lemma}[\cite{l1}] \label{lem1} Let $w(t,y)$ be a continuous function from $\mathbb{R}^{+}\times \mathbb{R}^{+}$ to $\mathbb{R}^{+}$, increasing in $y$ and $M(t)$ a maximal solution of the ordinary differential equation $y'=w(t,y)$, with the initial condition $y(t_0)=y_0$, on the interval $[t_0,T]$, where $T>t_0$ an arbitrary positive number. Let $m(t)$ be a continuous function increasing on $[t_0,T]$ and such that $m'(t)\leq w(t,m(t))$ a.e. on $[t_0,T]$. If $m(t_0)\leq y_0$, then $m(t)\leq M(t)$ for all $t\in [ t_0,T]$. \end{lemma} \begin{lemma}[\cite{c4}] \label{lem2} Let $\Gamma$ be an upper semicontinuous set-valued map defined on a metric space $T$ with compact and nonempty value in a metric space $U$ and $\{\Theta _n\}$ a sequence of elements of $T$ converging to $\Theta _0$. Then we have $\emptyset \neq \cap_{k=1}^{\infty }\overline{\rm co} (\cup_{n=k}^{\infty }\Gamma (\Theta _n))\subset \Gamma (\Theta _0).$ \end{lemma} \begin{lemma}[\cite{d2}] \label{lem3} If $X$ is a Banach space and $\{x_n\}$ a sequence of elements of $X$ weakly convergent to $x$, then there exists a sequence of convex combinations of the elements $\{x_n\}$ which converges strongly to $x$, in the sense of the norm. \end{lemma} We will recall the fixed point theorem for multivalued mappings due to Borisovich et al; see \cite{b2}. \begin{lemma}[\cite{b2}] \label{lem4} Let $X$ be a normed space, $C$ be a convex subset of $X$ and $\Gamma : C\to 2^{C}$ be an upper semicontinuous set-valued map. Suppose that for all $x\in C$, $\Gamma (x)\in \operatorname{Conv}C$, then $\Gamma$ has at least one fixed point in $C$. \end{lemma} \section{Existence result} First we study the existence of solutions to \eqref{e1}-\eqref{e2} on the interval $[t_0,\gamma ]$, where $\gamma >t_0$ ($\gamma$ an arbitrary fixed reel number). Let us consider the interval $[t_{\gamma },\gamma ]$, where $t_{\gamma }=\min \{ t-h(t), t\in [t_0,\text{ }\gamma ]\} \}$, then $x_t$ denote the restriction of the function $x\in C_{[t_0,\gamma ]}^n$ to the interval $[t-h(t),t]$ where $t\in [ t_{0},\gamma ]$. For $x\in C_{[t_\gamma ,\gamma]}^n$, we denote the norm of $x$ by $\|x\|_{c^n}=\max \{\|x(s)\|_{\mathbb{R} ^n}, s\in [ t-h(t),t], t\in [t_\gamma ,\gamma ]\}.$ We use the following hypotheses: \begin{itemize} \item[(H1)] For $t\geq t_0$ the set-valued map $F(t,.):C_{[-h(t), 0 ]}^n\to \operatorname{Conv}\mathbb{R}^n$ is upper semi-continuous. \item[(H2)] For each fixed function $x\in C_{[t_\gamma ,\gamma ]}^n$, the set-valued map $F(.,x_{.}):[t_0,\gamma ]\to \operatorname{Conv}\mathbb{R}^n$, is Lebesgue-measurable on the interval $[t_0,\gamma ]$. \item[(H3)] For any bounded set $Q\subset C_{[t_\gamma ,\gamma ]}^n$, there exists a function $m:[t_0,\gamma]\to [ 0,+\infty [$ Lebesgue integrable such that for each measurable function $y:[t_0,\gamma]\to \mathbb{R}^{n}$ verifying the condition: $y(t)\in F_{Q(t)}=\cup \{ F(t,x_t):x\in Q\}$, almost everywhere on $[t_0,\gamma ]$, we have the inequality $\|y(t)\|\leq m(t)$ a.e. on $[t_0,\gamma ]$. \item[(H4)] For each fixed function $x\in C_{[t_\gamma ,\gamma ]}^n$ and a vector $y\in F(t,x_t)$, we have the inequality: $x'(t).y\leq \Phi (t,\|x_t\|_{c^n}^2)$ where $\Phi (t,z)$ is a continuous function on $[t_0,\gamma]\times \mathbb{R}^{+}\to \mathbb{R}^{+}$, positive, increasing in $z$ and such that the ordinary differential equation $z'=2\Phi (t,z)$ with the initial condition $z(t_0)=A$ ($A$ an arbitrary positive number) has a maximal solution on all $[t_0,\gamma ]$. \item[(H5)] The initial function $\varphi$ is continuous on $[t_\gamma ,t_0]$. \end{itemize} Now we are able to state and prove our existence result. \begin{theorem} \label{thm1} Under hypothesis {\rm (H1)--(H5)}, for each $\varphi \in C_{[t_{\gamma},t_0]}^n$, problem \eqref{e1}-\eqref{e2} has at least one solution on the interval $[t_0,\gamma ]$. \end{theorem} \begin{proof} First we give an estimate of the solution of the differential inclusion \eqref{e1}. Suppose that \eqref{e1} has a solution on $[t_0,\gamma ]$ and let $g(t)=1+\|x(t)\|_{\mathbb{R}^n}^2$. Then $g'(t)=2(x_1(t)x'_1(t)+x_2(t)x_2'(t)+\dots +x_n(t)x_n'(t))=2x(t)x'(t).$ In view of (H4), \begin{align*} g'(t)&\leq 2\Phi (t,\|x\|_{c^n}^2) =2\Phi (t,\max \{\|x(s)\|_{\mathbb{R}^n}^2, t-h(t)\leq s\leq t\})\\ &=\bigskip 2\Phi (t,\max \{ g(s), s\in [t-h(t),t]\})\\ &\leq 2\Phi (t,\max \{ g(s), \quad s\in [ t_{\gamma },t]\}). \end{align*} For $t_00$ such that $$\int_{t_0}^{t_1}m(t)dt\leq b_1 \label{e4}$$ Now we show that \eqref{e1}-\eqref{e2} has at least one solution on $[t_0,t_1]$. Let us consider the set $X$ of functions $x$ of the space $C_{[t_0,t_1]}^n$ satisfying the following three conditions: $x$ is absolutely continuous on $[t_0,t_1]$, $x=\varphi \in C_{[t_{\gamma },t_0]}^n$ for $t\in [ t_\gamma ,t_0]$, and $$\|x'(t)\|\leq m(t)\text{ a.e. on }[t_0,\ t_1] \label{e5}$$ We claim that $X$ is compact. For this purpose we show that $X$ is uniformly bounded and equi-continuous. For $x\in X$, we have $\|x(t)-\varphi (t_0)\|=\|x(t)-x(t_0)\| =\|\int_{t_0}^{t_1}x'(s)ds\|.$ Furthermore, from \eqref{e4}, we have $\int_{t_0}^{t_1}m(t)dt\leq b_1$. Then, using \eqref{e5}, we obtain $\|\int_{t_0}^{t_1}x'(s)ds\|\leq \int_{t_0}^{t_1}\|x'(s)ds\|\leq \int_{t_0}^{t_1}m(s)ds\leq b_1\quad\text{a.e. on }[t_0,t_1].$ This implies $$\|x(t)-\varphi (t_0)\|\leq b_1\quad\text{a.e. on }[t_0,\ t_1]\,. \label{e6}$$ This shows that $X$ is uniformly bounded. Let $t_0\leq t\leq t'\leq t_1$, then we have $\|x(t)-x(t')\| \leq \int_t^{t'}\|x'(s)ds\|\leq \int_t^{t'}m(s)ds\,.$ As the Lebesgue integral is absolutely continuous, for each $\varepsilon >0$, there exists $\delta>0$, such that $|t-t' |<\delta \Rightarrow \|x(t)-x(t')\|<\varepsilon ,$ which shows that $X$ is equi-continuous. Since $X$ is uniformly bounded and equi-continuous, it is compact by Arzela's theorem. It is easy to show that $X$ is convex. Indeed, let $x,y\in X$ and $\lambda \in [ 0,1]$, we have $\|\frac{d}{dt}(\lambda x(t)+(1-\lambda )y(t))\| \leq \lambda \|x'(t)\|+(1-\lambda )\|y'(t)\|\leq m(t)$ a.e. on $[t_0,t_1]$. Then $\lambda x+(1-\lambda )y\in X\quad\text{for }\lambda \in [0,1].$ Let us fix $x\in X$, and with this function we consider a function $y$ such that $$y'(t)\in F(t,x_t)\quad \text{a.e. on }[t_0,\text{ }t_1] \label{e7}$$ We denote by $G$ the set of pairs $(x,y)\in X\times X$ such that $(x,y)$ fulfills the above relation \eqref{e7}; i.e., $G=\{(x,y)\in X\times X:y'(t)\in F(t,x_t)\text{ a.e. on } [t_0,t_1]\}\,.$ Now we show that $G$ is nonempty and closed. Let $x\in X$. In view of Hypothesis (H2) and selector's theorem (see[13]), there is a measurable function $\psi :[t_0,t_1]\to \mathbb{R}$ such that $\psi (t)\in F(t,x_t)$ a.e. on $[t_0,t_1]$. Define the function $\xi$ as $\xi (t)=\varphi (t_0)+\int_{t_0}^{t}\psi (s)\,ds\quad \text{for }t\in [t_0,t_1].$ Then $\xi '(t)\in F(t,x_t)$ a.e. on $[t_0,t_1]$. In view of \eqref{e5}, we have $\|\xi '(t)\|\leq m(t)\quad\text{a.e. on }[t_0,t_1],$ which implies that $G$ is nonempty. $G$ is closed. Indeed, let $(x_{k}$, $y_{k})$ a sequence of elements of $G$ converging to $(x,y)$, we will show that the sequence of derivatives $\{y_{k}'\}$ is bounded with the norm of $L_{1[t_0,t_1]}^n$. We have: \begin{align*} \|y_{k}'\|_{L_1^n}^2 &=\sum_{i=1}^n\|[{y'_{k}}^i]\|_{L_1^n}^2 =\sum_{i=1}^n\Big(\int _{t_0}^{t_1}|{y'_{k}}^{i}(s)|ds\Big)^2\\ &= \sum_{i=1}^n\Big(\int_{t_0}^{t_1}m(s)\Big)^2d = \sum_{i=1}^nb_1^2 \leq nb_1. \end{align*} We will prove that the sequence $\{y_{k}'\}$ satisfies the condition: $\lim \int_{E_i}y_{k}'(s)ds=0$ uniformly, and for each decreasing sequence $\{E_i\}$ of measurable sets $E_1\supset E_2\supset \dots\supset E_n\supset \dots$ such that $\cap_{i=1}^{\infty }E_i=\emptyset$. We have \begin{align*} |\int_{E_i}y_{k}'(s)ds| &\leq \int_{E_i}|y_{k}'(s)|ds =\int_{t_0}^{t_1}\chi _{E_i}(s)|y_{k}'(s)|ds\\ &\leq \int_{t_0}^{t_1}\chi _{E_i}(s)m(s)ds =\int_{E_i}m(s)ds, \end{align*} where $\chi _{E_i}$ denotes the characteristic function of the set $E_i$. For $E_i\subset [ t_0,t_1]$, the integral exists and $\lim \mu (E_i)=\mu (\cap_{i=1}^{\infty }E_i)=\mu (\emptyset)=0,$ where $\mu$ denote the Lesbegue's measure. From the absolute continuity of the integral, we obtain: For each $\epsilon>0$, there exists $j$ such that $$i>j \Rightarrow \int_{E_i}\chi _{E_i}(s)m(s)ds<\epsilon .$$ Applying the weak criterion of compactness (see \cite{d2}), we show that the sequence $\{$ $y_{k}\}$ is weakly compact in the sequential sense. Therefore, there is a subsequence, also denoted by $\{y_{k}\}$, weakly convergent to a function $z\in L_{1[t_0,t_1]}^n$. Thus, for $t\in [ t_0,t_1]$ we have $y(t)=\lim y_{k}(t)=\lim (\varphi (t_0)+\int_{t_0}^{t}y_{k}'(s)ds)=\varphi (t_0)+\int_{t_0}^{t}z(s)ds.$ This implies $y'(t)=z(t)$. Weak convergence in $L_{1[t_0,t_1]}^n$ is equivalent to the convergence of the integrals and applying Lemma \ref{lem3}, we prove the existence of a sequence of convex combinations $z_{j}=\{y_{j}'$, $y_{j+1}',\dots\}$ strongly convergent to $z\in L_{1[t_0,t_1]}^n$. As $L_{1[t_0,t_1]}^n$ is a complete space, from any strongly convergent sequence we can extract a subsequence which converges almost everywhere. Then from the sequence $\{z_{j}\}$ we can extract a subsequence, also denoted by $\{z_{j}\}$, which converges a.e. to $z$. Thus we have $\lim z_{j}(t)=z(t)\quad\text{a.e. on } [t_0,t_1].$ We claim that $y'(t)=z(t)\in F(t,x_t)$ a.e. on $[t_0,t_1]$. So, we will show that $z(t)\in \cap_{j=1}^{\infty} \overline{\rm co}(\cup _{n=j}^{\infty }y_n'(t)).$ Let $\{[z_{j}]\}$ the sequence of the convex combinations of the functions $\{y_{j}'$, $y_{j+1}',\dots\}=\cup_{n=j}^{\infty }y_n'$. We have $z_{j}=\sum_{j=1}^{n_{j}}a_iy_i', \quad i\in \{j,j+1,\dots\},\quad a_i>0, \quad \sum_{j=1}^{n_{j}}a_i=1.$ Then $z_{j}(t)\in \operatorname{co}(\cup_{n=j}^{\infty }y_n'(t))$ for $j$ fixed. As $\lim$ $z_{j}(t)=z(t)$ a.e. on $[t_0,t_1]$, we have that implication: For each neighborhood $U_{z(t)}$ of $z(t)$, there exists $N_0$ such that $z_{j}(t)\in U_{z(t)}$ for all $j>N_0$. Therefore $U_{z(t)}\cap \operatorname{co}(\cup_{n=j}^{\infty }y_n'(t))\neq \emptyset$ and hence $z(t)\in \overline{\rm co}(\cup_{n=j}^{\infty }y_n'(t)).$ We have $\overline{\rm co} (\cup_{n=j}^{\infty }y_n'(t))=\operatorname{co}\overline{ (\cup_{n=j}^{\infty }y_n'(t))}.$ Whence we obtain $z(t)\in \operatorname{co}\overline{ (\cup_{n=j}^{\infty }y_n'(t))},$ so that $z(t)\in \cap_{j=1}^{\infty}\overline{\rm co}(\cup_{n=j}^{\infty }y_n'(t)).$ From the definition of $G$, we have $y_n'(t)\in F(t,(x_n)_t)$ a.e. on $[t_0,t_1]$. This implies $\cap_{j=1}^{\infty}\overline{\rm co}(\cup_{n=j}^{\infty }y_n'(t))\subset \cap_{j=1}^{\infty}\overline{\rm co}(\cup_{n=jn}^{\infty }(F(t,(x_n)_t)$ a.e. on $[t_0,t_1]$. Using Lemma \ref{lem2}, we obtain $\cap_{j=1}^{\infty}\overline{\rm co}(\cup_{n=jn}^{\infty }(F(t,(x_n)_t) \subset F(t,x_t).$ From which it follows $z(t)\in F(t,x_t)$; i.e., $y'(t)\in F(t,x_t)$ a.e. on $[t_0,t_1]$. So we conclude that $G$ is closed. Let us define the set-valued map $\Gamma :X\to 2^{X}$ such that $\Gamma (x)=\{y:y'(t)\in F(t,x_t)\text{ a.e. on }[t_0,t_1]\}.$ The set $G=\{(x,y)\}\subset X\times X\}$ is the graph of $\Gamma$. Since $G$ is closed, the application $\Gamma$ is upper semi-continuous \cite{a2}. Let us show that $\Gamma (x)$ is compact. As $\Gamma (x)\subset X$ and $X$ is compact, then $\Gamma (x)$ is uniformly bounded and we prove the equicontinuity of $\Gamma (x)$ in the same way as we did for $X$. It is also easy to show that $\Gamma (x)$ is convex. Using Lemma \ref{lem4}, we show that the map $\Gamma$ has at least one fixed point. Therefore, there is a function $x\in X$ such that $x(t)\in F(t,x_t)$ a.e. on $[t_0,t_1]$, then $x$ is a solution of \eqref{e1}-\eqref{e2} on $[t_0,$ $t_1]$. To complete the proof, we extend the solution on $[t_1,\gamma ]$. For $t_1<\gamma$, we have the implication: $\|x(t_1)\|0$ such that $\{x\in \mathbb{R}^n:\|x(t)-x(t_1)\|\leq b_2\}\subset \{x\in \mathbb{R}^n:\|x\|\leq L\}.$ Thus, there exist $t_2>t_1$ such that $\int _{t_1}^{t_2}m(t)dt\leq b_2$ and we extend the solution on $[t_{1,}t_2]$. We can choose all $b_i$'s such that $b_i\geq \epsilon >0$, hence the sequence $\{b_i\}$ does not converge to $0$. After a finite number of steps we can extend the solution to the entire interval $[t_0,\gamma ]$. \end{proof} \subsection*{Remark} Anan'ev \cite{a1} assumed that $y.x'(t)\leq K(1+\|x_t\|_{c^n}^2)$ with $K>0$. Our hypothesis (H4) is more general than that the one by Anan'ev. \subsection*{Acknowledgement} The author would like to thank an anonymous referee for his/her helpful suggestions for improving the original manuscript. \begin{thebibliography}{00} \bibitem{a1} B. I. 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