\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 08, pp. 1--9.\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-2004/08\hfil Upper and lower solutions for differential inclusions] {The method of upper and lower solutions for Caratheodory n-th order differential inclusions} \author[B. C. Dhage, T. L. Holambe, \& S. K. Ntouyas\hfil EJDE-2004/08\hfilneg] {Bupurao C. Dhage, Tarachand L. Holambe, \& Sotiris K. Ntouyas } % in alphabetical order \address{Bupurao C. Dhage \hfill\break Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India} \email{bcd20012001@yahoo.co.in} \address{Tarachand L. Holambe \hfill\break GMCT's ACS College, Shankarnagar -431 505, Dist: Nanded, Maharashtra, India} \address{Sotiris K. Ntouyas \hfill\break Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{sntouyas@cc.uoi.gr} \date{} \thanks{Submitted November 20, 2003. Published January 2, 2004.} \subjclass[2000]{34A60} \keywords{Differential inclusion, method of upper and lower solutions, \hfill\break\indent existence theorem} \begin{abstract} In this paper, we prove an existence theorem for n-th order differential inclusions under Carath\'eodory conditions. The existence of extremal solutions is also obtained under certain monotonicity condition of the multi-function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Let $\mathbb{R}$ denote the real line and let $J=[0,a]$ be a closed and bounded interval in $\mathbb{R}$. Consider the initial value problem (in short IVP) of $n^{\text{th}}$ order differential inclusion \begin{equation}\label{e1} \begin{gathered} x^{(n)}(t)\in F(t,x(t)) \quad \mbox{a.e. } t\in J,\\ x^{(i)}(0)=x_{i}\in \mathbb{R} \end{gathered} \end{equation} where $F:J\times\mathbb{R}\to 2^{\mathbb{R}}$, $i\in \{0,1,\ldots,n-1\}$ and $2^{\mathbb{R}}$ is the class of all nonempty subsets of $\mathbb{R}$. By a solution of (\ref{e1}) we mean a function $x\in AC^{n-1}(J,\mathbb{R})$ whose $n^{\text{th}}$ derivative $x^{(n)}$ exists and is a member of $L^{1}(J,\mathbb{R})$ in $F(t,x)$, i.e. there exists a $v\in L^{1}(J,\mathbb{R})$ such that $v(t)\in F(t,x(t))$ a.e\,\,\ $t\in J$, and $x^{(i)}(0)=x_{i}\in \mathbb{R}, i=0,1,\ldots, n-1$, where $AC^{n-1}(J,\mathbb{R})$ is the space of all continuous real-valued functions whose $(n-1)$ derivatives exist and are absolutely continuous on $J$. The method of upper and lower solutions has been successfully applied to the problem of nonlinear differential equations and inclusions. For the first direction, we refer to Heikkila and Laksmikantham \cite{HL} and Bernfield and Laksmikantham \cite{BL} and for the second direction we refer to Halidias and Papageorgiou \cite{HP} and Benchohra \cite{B}. In this paper we apply the multi-valued version of Schaefer's fixed point theorem due to Martelli \cite{M} to the initial value problem (\ref{e1}) and prove the existence of solutions between the given lower and upper solutions, using the Carath\'eodory condition on $F$. \section{Preliminaries} Let $X$ be a Banach space and let $2^{X}$ be a class of all non- empty subsets of $X$. A correspondence $T: X\to 2^{X}$ is called a multi-valued map or simply multi and $u\in Tu$ for some $u\in X$, then $u$ is called a fixed point of $T$. A multi $T$ is closed (resp. convex and compact) if $Tx$ is closed (resp. convex and compact) subset of $X$ for each $x\in X$. $T$ is said to be bounded on bounded sets if $T(B)=\bigcup_{x\in B}T(x)=\bigcup T(B)$ is a bounded subset of $X$ for all bounded sets $B$ in $X$. $T$ is called upper semi-continuous (u.s.c.) if for every open set $N\subset X$, the set $\{x\in X: Tx\subset N\}$ is open in $X$. $T$ is said to be totally bounded if for any bounded subset $B$ of $X$, the set $\cup T(B)$ is totally bounded subset of $X$. Again $T$ is called completely continuous if it is upper semi-continuous and totally bounded on $X$. It is known that if the multi-valued map $T$ is totally bounded with non empty compact values, the $T$ is upper semi-continuous if and only if $T$ has a closed graph (that is $x_n\to x_*, y_n\to y_*, y_n\in Tx_n\Rightarrow y_*\in Tx_*)$. By $KC(X)$ we denote the class of nonempty compact and convex subsets of $X$. We apply the following form of the fixed point theorem of Martelli \cite{M} in the sequel. \begin{theorem}\label{t1} Let $T: X\to KC(X)$ be a completely continuous multi-valued map. If the set $${\mathcal E}=\{u\in X: \lambda u\in Tu\quad \mbox{for some } \lambda>1\}$$ is bounded, then $T$ has a fixed point. \end{theorem} We also need the following definitions in the sequel. \begin{definition} \rm A multi-valued map map $F: J\to KC(\mathbb{R})$ is said to be measurable if for every $y\in {\mathbb R}$, the function $t\to d(y,F(t))=\inf \{\|y-x\|: x\in F(t)\}$ is measurable. \end{definition} \begin{definition} \rm A multi-valued map $F: J\times\mathbb{R}\to 2^{\mathbb{R}}$ is said to be $L^{1}$-Carath\'eodory if \begin{itemize} \item[(i)] $t\to F(t,x)$ is measurable for each $x\in \mathbb{R}$, \item[(ii)] $x\to F(t,x)$ is upper semi-continuous for almost all $t\in J$, and \item[(iii)] for each real number $k>0$, there exists a function $h_{k}\in L^{1}(J,\mathbb{R})$ such that $$\|F(t,x)\|=\sup\{|v|: v\in F(t,x)\}\le h_{k}(t), \quad a.e. \quad t\in J$$ for all $x\in \mathbb{R}$ with $|x|\le k$. \end{itemize} \end{definition} Denote $$S_{F}^{1}(x)=\{v\in L^{1}(J,\mathbb{R}): v(t)\in F(t,x(t))\quad \mbox{a.e. } t\in J\}. $$ Then we have the following lemmas due to Lasota and Opial \cite{LO}. \begin{lemma}\label{l1} If $\dim (X)<\infty$ and $F: J\times X\to KC(X)$ then $S_{F}^{1}(x)\ne \emptyset$ for each $x\in X$. \end{lemma} \begin{lemma}\label{l2} Let $X$ be a Banach space, $F$ an $L^{1}$-Carath\'eodory multi-valued map with $S_{F}^{1}\ne \emptyset$ and $\mathcal{K}:L^{1}(J,X)\to C(J,X)$ be a linear continuous mapping. Then the operator $$ \mathcal{K} \circ S_{F}^{1}:C(J,X)\longrightarrow KC(C(J,X))$$ is a closed graph operator in $C(J,X)\times C(J,X)$. \end{lemma} We define the partial ordering $\le$ in $W^{n,1}(J,\mathbb{R})$ (the Sobolev class of functions $x: J\to \mathbb{R}$ for which $x^{(n-1)}$ are absolutely continuous and $x^{(n)}\in L^{1}(J,\mathbb{R})$) as follows. Let $x,y\in W^{n,1}(J,\mathbb{R})$. Then we define $$ x\le y\Leftrightarrow x(t)\le y(t), \,\, \forall t\in J. $$ If $a,b\in W^{n,1}(J,\mathbb{R})$ and $a\le b$, then we define an order interval $[a,b]$ in $W^{n,1}(J,\mathbb{R})$ by $$ [a,b]=\{x\in W^{n,1}(J,\mathbb{R}): a\le x\le b\}. $$ The following definition appears in Dhage {\em et al.} \cite{DOA}. \begin{definition} \rm A function $\alpha\in W^{n,1}(J,\mathbb{R})$ is called a lower solution of IVP (\ref{e1}) if there exists $v_{1}\in L^{1}(J,\mathbb{R})$ with $v_{1}(t)\in F(t,\alpha(t))$ a.e. $t\in J$ we have that $\alpha^{(n)}(t)\le v_{1}(t)$ a.e. $t\in J$ and $\alpha^{(i)}(0)\le x_{i}, i=0,1,\ldots,n-1$. Similarly a function $\beta\in W^{n,1}(J,\mathbb{R})$ is called an upper solution of IVP (\ref{e1}) if there exists $v_{2}\in L^{1}(J,\mathbb{R})$ with $v_{2}(t)\in F(t,\beta(t))$ a.e. $t\in J$ we have that $\beta^{(n)}(t)\ge v_{2}(t)$ a.e. $t\in J$ and $\beta^{(i)}(0)\ge x_{i}, i=0,1,\ldots,n-1$. \end{definition} Now we are ready to prove in the next section our main existence result for the IVP (\ref{e1}). \section{Existence Result} We consider the following assumptions: \begin{itemize} \item[(H1)] The multi $F(t,x)$ has compact and convex values for each $(t,x)\in J\times \mathbb{R}$. \item[(H2)] $F(t,x)$ is $L^{1}$-Carath\'eodory. \item[(H3)] The IVP (\ref{e1}) has a lower solution $\alpha$ and an upper solution $\beta$ with $\alpha\le \beta$. \end{itemize} \begin{theorem}\label{t2} Assume that (H1)--(H3) hold. Then the IVP (\ref{e1}) has at least one solution $x$ such that $$\alpha(t)\le x(t)\le \beta(t), \quad \mbox{for all}\quad t\in J.$$ \end{theorem} \begin{proof} First we transform (\ref{e1}) into a fixed point inclusion in a suitable Banach space. Consider the IVP \begin{equation}\label{e2} \begin{gathered} x^{(n)}(t)\in F(t,\tau x(t)) \quad \mbox{a.e. } t\in J,\\ x^{(i)}(0)=x_{i}\in \mathbb{R} \end{gathered} \end{equation} for all $i\in \{0,1,\ldots, n-1\}$, where $\tau: C(J,\mathbb{R})\to C(J,\mathbb{R})$ is the truncation operator defined by \begin{equation}\label{e3} (\tau x)(t)=\begin{cases} \alpha(t), & \mbox{if $x(t)<\alpha(t)$}\\ x(t), & \mbox{if $\alpha(t)\le x(t)\le \beta(t)$}\\ \beta(t), & \mbox{if $\beta(t)0$ such that $\|x\|\le r, \forall x\in B$. Now for each $u\in Tx$, there exists a $v\in \overline{S_{F}^{1}}(\tau x)$ such that $$u(t)=\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}+\int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds.$$ Then for each $t\in J$, \begin{align*} |u(t)|&\leq \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!}+\int_{0}^{t}\frac{a^{n-1}}{(n-1)!}|v(s)|ds\\ &\le \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!}+\int_{0}^{t} \frac{a^{n-1}}{(n-1)!}h_{r}(s)ds\\ &= \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!} +\frac{a^{n-1}}{(n-1)!}\|h_{r}\|_{L^{1}}. \end{align*} This further implies that $$ \|u\|_{C}\le \sum_{i=0}^{n-1}\frac{|x_{i}a^{i}}{i!} +\frac{a^{n-1}}{(n-1)!}\|h_{r}\|_{L^{1}} $$ for all $u\in Tx\subset \bigcup T(B)$. Hence $\bigcup T(B)$ is bounded. \smallskip \noindent{\bf Step III.} Next we show that $T$ maps bounded sets into equicontinuous sets. Let $B$ be a bounded set as in step II, and $u\in Tx$ for some $x\in B$. Then there exists $v\in \overline{S_{F}^{1}}(\tau x)$ such that $$u(t)=\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}+\int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds.$$ Then for any $t_1, t_2\in J$ we have \begin{align*} &|u(t_1)-u(t_2)|\\ &\leq \Big|\sum_{i=0}^{n-1}\frac{x_{i}t_{1}^{i}}{i!} -\sum_{i=0}^{n-1}\frac{x_{i}t_{2}^{i}}{i!}\Big| +\Big|\int_{0}^{t_{1}}\frac{(t_{1}-s)^{n-1}} {(n-1)!}v(s)ds -\int_{0}^{t_{2}}\frac{(t_{2}-s)^{n-1}}{(n-1)!}v(s)ds\Big|\\ &\leq |q(t_{1})-q(t_{2})| +\Big|\int_{0}^{t_{1}}\frac{(t_{1}-s)^{n-1}} {(n-1)!}v(s)ds -\int_{0}^{t_{1}}\frac{(t_{2}-s)^{n-1}}{(n-1)!} v(s)ds\Big|\\ &\quad\quad+\Big|\int_{0}^{t_{1}}\frac{(t_{2}-s)^{n-1}} {(n-1)!}v(s)ds -\int_{0}^{t_{2}}\frac{(t_{2}-s)^{n-1}}{(n-1)!} v(s)ds\Big|\\ &\leq |q(t_{1})-q(t_{2})| +\int_{0}^{t_{1}}\left|\frac{(t_{1}-s)^{n-1}}{(n-1)!} -\frac{(t_{2}-s)^{n-1}}{(n-1)!} \right|\,|v(s)|ds\\ &\quad\quad+\Big|\int_{t_{1}}^{t_{2}}\left|\frac{(t_{2}-s)^{n-1}}{(n-1)!}\right||v(s)|ds \Big|\\ &\leq |q(t_{1})-q(t_{2})|+|p(t_{1})-p(t_{2})|\\ &\quad\quad+\frac{1}{(n-1)!}\int_{0}^{t_{1}}\left|(t_{1}-s)^{n-1}-(t_{2}-s)^{n-1} \right|\,\|F(s,u(s))\|\,\,ds\\ &\leq |q(t_{1})-q(t_{2})|+|p(t_{1})-p(t_{2})|\\ &\quad\quad +\frac{1}{(n-1)!}\int_{0}^{a}\left|(t_{1}-s)^{n-1}-(t_{2}-s)^{n-1} \right|h_{r}(s)\,ds \end{align*} where $$ q(t)=\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}\quad\mbox{and}\quad p(t)=\int_{0}^{t}\frac{(a-s)^{n-1}}{(n-1)!}h_{r}(s)ds. $$ Now the functions $p$ and $q$ are continuous on the compact interval $J$, hence they are uniformly continuous on $J$. Hence we have $$|u(t_1)-u(t_2)|\to 0\quad \mbox{as } t_1\to t_2.$$ As a result $\bigcup T(B)$ is an equicontinuous set in $C(J,\mathbb{R})$. Now an application of Arzel\'a-Ascoli theorem yields that the multi $T$ is totally bounded on $C(J,\mathbb{R})$. \smallskip \noindent{\bf Step IV.} Next we prove that $T$ has a closed graph. Let $\{x_n\}\subset C(J,\mathbb{R})$ be a sequence such that $x_n\to x_*$ and let $\{y_n\}$ be a sequence defined by $y_n\in Tx_n$ for each $n\in \mathbb{N}$ such that $y_n\to y_*$. We just show that $y_*\in Tx_*$. Since $y_n\in Tx_n$, there exists a $v_n\in \overline{S_{F}^{1}}(\tau x_n)$ such that $$ y_n(t)=\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!} +\int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v_n(s)ds. $$ Consider the linear and continuous operator $\mathcal{K}: L^{1}(J,\mathbb{R})\to C(J,\mathbb{R})$ defined by $$ \mathcal{K}v(t)=\int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds. $$ Now \begin{align*} \Big|y_n(t)-\sum_{i=0}^{n-1}\frac{|x_{i}|t^{i}}{i!}&-y_*(t) -\sum_{i=0}^{n-1}\frac{|x_{i}|t^{i}}{i!}\Big|\\ &\leq |y_n(t)-y_*(t)|\\ &\leq \|y_n-y_*\|_C\to 0\quad \mbox{as}\quad n\to \infty. \end{align*} From Lemma \ref{l2} it follows that $(\mathcal{K}\circ \overline{S_{F}^{1}})$ is a closed graph operator and from the definition of $\mathcal{K}$ one has $$ y_n(t)-\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}\in (\mathcal{K}\circ \overline{S_{F}^{1}}(\tau x_n)). $$ As $x_n\to x_*$ and $y_n\to y_*$, there is a $v_*\in \overline{S_{F}^{1}}(\tau x_*)$ such that $$ y_*=\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}+\int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1)!}v_*(s)ds. $$ Hence the multi $T$ is an upper semi-continuous operator on $C(J,\mathbb{R})$. \smallskip \noindent {\bf Step V.} Finally we show that the set $$ {\mathcal E}=\{x\in C(J,\mathbb{R}): \lambda x\in Tx\quad \mbox{for some } \lambda>1\} $$ is bounded. Let $u\in {\mathcal E}$ be any element. Then there exists a $v\in \overline{S_{F}^{1}}(\tau x)$ such that $$u(t)=\lambda^{-1}\sum_{i=0}^{n-1}\frac{x_{i}t^{i}}{i!}+ \lambda^{-1}\int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds.$$ Then $$|u(t)|\le \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!}+ \int_{0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}|v(s)|ds.$$ Since $\tau x\in [\alpha,\beta], \forall x\in C(J,\mathbb{R})$, we have $$ \|\tau x\|_{C}\le \|\alpha\|_{C}+\|\beta\|_{C}:=l. $$ By (H2) there is a function $h_{l}\in L^{1}(J,\mathbb{R})$ such that $$ \|F(t,\tau x)\|=\sup\{|u|: u\in F(t,\tau x)\}\le h_{l}(t)\quad \mbox{a.e. } t\in J $$ for all $x\in C(J,\mathbb{R})$. Therefore \[ \|u\|_{C} \leq \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!}+\frac{a^{n-1}}{(n-1)!} \int_{0}^{a}h_{l}\,ds = \sum_{i=0}^{n-1}\frac{|x_{i}|a^{i}}{i!}+ \frac{a^{n-1}}{(n-1)!}\|h_{l}\|_{L^{1}} \] and so, the set ${\mathcal E}$ is bounded in $C(J,\mathbb{R})$. Thus $T$ satisfies all the conditions of Theorem \ref{t1} and so an application of this theorem yields that the multi $T$ has a fixed point. Consequently (\ref{e3}) has a solution $u$ on $J$. Next we show that $u$ is also a solution of (\ref{e1}) on $J$. First we show that $u\in [\alpha,\beta]$. Suppose not. Then either $\alpha \not\leq u$ or $u\not\leq \beta$ on some subinterval$J'$ of $J$. If $u\not\geq \alpha$, then there exist $t_0, t_1 \in J, t_0u(t)$ for all $t\in (t_0,t_1)\subset J$. From the definition of the operator $\tau$ it follows that $$ u^{(n)}(t)\in F(t, \alpha(t))\quad \mbox{a.e. } t\in J. $$ Then there exists a $v(t)\in F(t, \alpha(t))$ such that $v(t)\ge v_{1}(t), \forall t\in J$ with $$ u^{(n)}(t)=v(t)\quad \mbox{a.e. } t\in J. $$ Integrating from $t_0$ to $t$ $n$ times yields $$ u(t)-\sum_{i=0}^{n-1}\frac{u_{i}(0)(t-t_0)^{i}}{i!} =\int_{t_0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds. $$ Since $\alpha$ is a lower solution of (\ref{e1}), we have \begin{align*} u(t)&= \sum_{i=0}^{n-1}\frac{u_{i}(0)(t-t_0)^{i}}{i!} +\int_{t_0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}v(s)ds\\ &\ge \sum_{i=0}^{n-1}\frac{\alpha_{i}(0)(t-t_0)^{i}}{i!} +\int_{t_0}^{t}\frac{(t-s)^{n-1}}{(n-1)!}\alpha(s)ds\\ &= \alpha(t) \end{align*} for all $t\in (t_0,t_1)$. This is a contradiction. Similarly if $u\not\leq \beta$ on some subinterval of $J$, then also we get a contradiction. Hence $\alpha\le u\le \beta$ on $J$. As a result (\ref{e3}) has a solution $u$ in $[\alpha,\beta]$. Finally since $\tau x=x, \forall x\in [\alpha,\beta]$, $u$ is a required solution of (\ref{e1}) on $J$. This completes the proof. \end{proof} \section{Existence of Extremal Solutions} In this section we establish the existence of extremal solutions to (\ref{e1}) when the multi-map $F(t,x)$ is isotone increasing in $x$. Here our technique involves combining method of upper and lower solutions with an algebraic fixed point theorem of Dhage \cite{D} on ordered Banach spaces. Define a cone $K$ in $C(J,\mathbb{R})$ by \begin{equation}\label{e6} K=\{x\in C(J,\mathbb{R}): x(t)\ge 0, \forall t\in J\}. \end{equation} Then the cone $K$ defines an order relation, $\le$, in $C(J,\mathbb{R})$ by \begin{equation}\label{e7} x\le y\quad \mbox{iff}\quad x(t)\le y(t), \quad \forall t\in J. \end{equation} It is known that the cone $K$ is normal in $C(J,\mathbb{R})$. See Heikkila and Laksmikantham \cite{HL} and the references therein. For any $A, B\in 2^{C(J,\mathbb{R})}$ we define the order relation, $\le$, in $2^{C(J,\mathbb{R})}$ by \begin{equation}\label{e8} A\le B\quad \mbox{iff}\quad a\le b, \quad \forall a\in A\quad \mbox{and}\quad \forall b\in B. \end{equation} In particular, $a\le B$ implies that $a\le b,\quad \forall b\in B$ and if $A\le A$, then it follows that $A$ is a singleton set. \begin{definition} \rm A multi-map $T: C(J,\mathbb{R})\to 2^{C(J,\mathbb{R})}$ is said to be isotone increasing if for any $x,y\in C(J,\mathbb{R})$ with $x