\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 85, 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} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/85\hfil Second order differential inclusions] {Second-order differential inclusions with Lipschitz right-hand sides} \author[D. Azzam-Laouir, F. Bounama\hfil EJDE-2010/85\hfilneg] {Dalila Azzam-Laouir, Fatiha Bounama} % in alphabetical order \address{Dalila Azzam-Laouir \newline Laboratoire de Math\'ematiques Pures et Appliqu\'ees, D\'epartement de Math\'ematiques, Universit\'e de Jijel, Alg\'erie} \email{laouir.dalila@gmail.com} \address{Fatiha Bounama \newline Laboratoire de Math\'ematiques Pures et Appliqu\'ees, D\'epartement de Math\'ematiques, Universit\'e de Jijel, Alg\'erie} \email{bounamaf@yahoo.fr} \thanks{Submitted March 29, 2010. Published June 18, 2010.} \subjclass[2000]{34A60, 34B15, 47H10} \keywords{Differential inclusion; Lipschitz multifunction} \begin{abstract} We study the existence of solutions of a three-point boundary-value problem for a second-order differential inclusion, \begin{gather*} \ddot u(t)\in F(t,u(t),\dot u(t)),\quad\text{a.e. }t\in [0,1],\\ u(0)=0, \quad u(\theta)=u(1). \end{gather*} Here $F$ is a set-valued mapping from $[0,1]\times E\times E$ to $E$ with nonempty closed values satisfying a standard Lipschitz condition, and $E$ is a separable Banach space. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} We study the existence of solutions to the second-order differential inclusion $$\label{PF} \begin{gathered} \ddot u(t)\in F(t,u(t),\dot u(t)),\quad \text{a.e. }t\in [0,1],\\ u(0)=0, \quad u(\theta)=u(1), \end{gathered}$$ where $F:[0,1]\times E\times E \to E$ is a nonempty closed valued multifunction and $\theta$ is a given number in $[0,1]$. Existence of solutions for \eqref{PF} has been investigated by many authors \cite{ACT,AL,ALT,IG} under the assumption that $F$ is a convex bounded-valued multifunction upper semicontinuous on $E\times E$ and integrably compact. The aim of this article is to provide existence of solutions for \eqref{PF} under the standard Lipschitz condition for the multifunction $F$, when it is nonconvex. After some preliminaries in section 3, we present our main result which is the existence of $\mathbf{W}^{2,1}_{E}([0,1])$-solutions for \eqref{PF}. We suppose that $F$ is a closed valued multifunction satisfying the Lipschitz condition $$\mathcal{H}(F(t,x_1,y_1),F(t,x_2,y_2)) \leq k_1(t)\| x_1-y_1\|+k_2(t)\| x_2-y_2\|$$ where $\mathcal{H}(\cdot,\cdot)$ stands for the Hausdorff distance. For first-order differential inclusions satisfying the standard Lipschitz condition we refer the reader to \cite{CV, F,G,W} and the references therein. \section{Notation and preliminaries} In this article, $(E, \|\cdot\|)$ is a separable Banach space and $E'$ is its topological dual, $\overline{\mathbf{B}}_E$ is the closed unit ball of $E$, $\mathcal{L}([0,1])$ is the $\sigma$-algebra of Lebesgue-measurable sets of $[0,1]$, $\lambda=dt$ is the Lebesgue measure on $[0,1]$, and $\mathcal{B}(E)$ is the $\sigma$-algebra of Borel subsets of $E$. By $L^{1}_{E}([0,1])$, we denote the space of all Lebesgue-Bochner integrable E-valued mappings defined on $[0,1]$. Let $\mathbf{C}_{E}([0,1])$ be the Banach space of all continuous mappings $u:[0,1] \to E$, endowed with the supremum norm, and let $\mathbf{C}^{1}_{E}([0,1])$ be the Banach space of all continuous mappings $u : [0,1] \to E$ with continuous derivative, equipped with the norm $$\| u\|_{\mathbf{C}^{1}}=\max\{\max_{t\in [0,1]} \| u(t)\|, \max_{t\in [0,1]}\| \dot u(t)\|\}.$$ Recall that a mapping $v:[0,1]\to E$ is said to be scalarly derivable when there exists some mapping $\dot v:[0,1]\to E$ (called the weak derivative of $v$) such that, for every $x'\in E'$, the scalar function $\langle x',v(\cdot)\rangle$ is derivable and its derivative is equal to $\langle x',\dot v(\cdot)\rangle$. The weak derivative $\ddot v$ of $\dot v$ when it exists is the weak second derivative. By $\mathbf{W}^{2,1}_{E}([0,1])$ we denote the space of all continuous mappings $u\in \mathbf{C}_{E}([0,1])$ such that their first usual derivatives are continuous and scalarly derivable and such that $\ddot u\in L^{1}_{E}([0,1])$. For closed subsets $A$ and $B$ of $E$, the Hausdorff distance between $A$ and $B$ is defined by $$\mathcal{H}(A,B)=\sup(e(A,B)), e(B,A))$$ where $$e(A,B) =\sup_{a\in A}d(a,B)=\sup_{a\in A}(\inf_{b\in B}\| a-b\|)$$ stands for the excess of $A$ over $B$. \section{Existence results under Lipschitz condition} We begin with a proposition that summarizes some properties of some Green type function (see \cite{Az,ACT,H,IG}). It will use it in the study of our boundary value problems. \begin{proposition} \label{prop3.1} Let $E$ be a separable Banach space and let $G:[0,1]\times [0,1]\to \mathbb{R}$ be the function defined by \[ G(t,s) =\begin{cases} -s &\text{if } 0\leq s\leq t,\\ -t &\text{if } t0$satisfying $$(1+\alpha)\|k_1+k_2\|_{ L^1_{\mathbb{R}}}<1,\quad (1+\alpha)\|\eta\|_{ L^1_{\mathbb{R}}}<[1-(1+\alpha)\| k_1+k_2\|_{ L^1_{\mathbb{R}}}]r .\label{e3.5}$$ We will define a sequence of mappings$f_n$,$n\in \mathbb{N}$, of$ L^1_E([0,1])$such that the following conditions are fulfilled (see \eqref{e3.2} for the definition of$u_f$). \begin{gather} f_n\in L^1_E([0,1]), \quad f_{n}(t)\in F(t,u_{f_{n-1}}(t),\dot u_{f_{n-1}}(t)),\quad \text{a.e. }t\in [0,1];\label{e3.6} \\ \| f_{n}(t)-f_{n-1}(t)\|\leq(1+\alpha)d(f_{n-1}(t),F(t,u_{f_{n-1}}(t), \dot u_{f_{n-1}}(t))),\quad \forall t\in[0,1];\label{e3.7} \\ \mathop{\rm gph}(u_{f_n}(\cdot),\dot u_{f_n}(\cdot)) =\{(u_{f_n}(t),\dot u_{f_n}(t)):\;t\in[0,1]\} \subset \mathbf{X}_r.\label{e3.8} \end{gather} We put$f_0=g$and$u_{f_0}(t)=\int_0^1G(t,s)f_0(s)ds=u_g(t)$, for all$t\in [0,1]$. Let us consider the multifunction$H_0:[0,1]\to E$defined by $$H_0(t)=\{v\in F(t,u_{f_0}(t),\dot u_{f_0}(t)):\| v-f_0(t)\| \leq(1+\alpha)d(f_0(t),F(t,u_{f_0}(t),\dot u_{f_0}(t)))\}.$$ Observe first that$H_0(t)\neq \emptyset$for any$t\in [0,1]$. Since$F(\cdot,u_{f_0}(\cdot),\dot u_{f_0}(\cdot))$is measurable, the multifunction$H_0$is also measurable with nonempty closed values. In view of the existence theorem of measurable selections (see \cite{CV}), there is a measurable mapping$f_1:[0,1]\to E$such that$f_1(t)\in H_0(t)$, for all$t\in [0,1]$. This yields, for all$t\in [0,1]$,$f_1(t)\in F(t,u_{f_0}(t),\dot u_{f_0}(t))$and$\|f_1(t)-f_0(t)\|\leq(1+\alpha)d(f_0(t),F(t,u_{f_0}(t),\dot u_{f_0}(t)))$, and hence according to the assumption (ii), $$\| f_1(t)-f_0(t)\|\leq (1+\alpha)\eta(t).$$ So, we have $$\| f_1(t)\| \leq \| f_1(t)-f_0(t)\| +\| f_0(t)\| \leq(1+\alpha)\eta(t) +\| f_0(t)\|\label{e3.9}.$$ Since$\eta\in L^1_{\mathbb{R}}([0,1])$and$f_0\in L^1_{E}([0,1])$, the last inequality shows that$f_1\in L^1_{E}([0,1])$. Then we define the mapping$u_{f_1}:[0,1]\to E\$ by $$u_{f_1}(t)=\int_0^1 G(t,s)f_1(s)ds,\quad\forall t\in [0,1],$$ and by relation \eqref{e3.3} in Proposition \ref{prop3.1} $$\dot u_{f_1}(t)=\int_0^1 \frac{\partial G}{\partial t}(t,s)f_1(s)ds, \quad \forall t\in [0,1].$$ On the other hand, \begin{align*} \| u_{f_1}(t)-u_{f_0}(t)\| &=\|\int_0^1 G(t,s)(f_1(s)-f_0(s))ds\|\\ &\leq \int_0^1 \| f_1(s)-f_0(s)\| ds\\ &\leq (1+\alpha)\int_0^1 d(f_0(s),F(t,u_{f_0}(s),\dot u_{f_0}(s)))ds\\ &\leq (1+\alpha)\|\eta\|_{ L^1_{\mathbb{R}}}\\ &<[1-(1+\alpha)\| k_1+k_2\|_{ L^1_{E}}]r