\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 168, 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} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/168\hfil Existence of solutions] {Existence of solutions for convex sweeping processes in $p$-uniformly smooth and $q$-uniformly convex Banach spaces} \author[M. Bounkhel \hfil EJDE-2012/168\hfilneg] {Messaoud Bounkhel} \address{Messaoud Bounkhel \newline King Saud University, Department of Mathematics, P.O. Box 2455, Riyadh 11451, Riyadh, Saudi-Arabia} \email{bounkhel@ksu.edu.sa, bounkhel\_messaoud@yahoo.fr} \thanks{Submitted February 14, 2012. Published October 4, 2012.} \subjclass[2000]{34A60, 49J53} \keywords{Uniformly smooth and uniformly convex Banach spaces; \hfill\break\indent state dependent sweeping process; generalized projection; duality mapping} \begin{abstract} We show the existence of at least one Lipschitz solution for extensions of convex sweeping processes in reflexive smooth Banach spaces. Our result is proved under a weaker assumption on the moving set than those in \cite{bounkhel_rabab}, and using a different discretization. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Main result} Bounkhel and Al-yusof \cite{bounkhel_rabab} studied the following extension of the convex sweeping processes from Hilbert spaces $H$ to reflexive smooth Banach spaces $X$: \begin{itemize} \item[(SP)] Find $u:[0,T]\to X$ such that $u(t)=u_0+\int_0^t \dot u(s) ds$, $$ -\frac{d}{dt}(J(u(t))) \in N(C(t);u(t)) \text{ a.e. in } [0,T] \text{ and } u(t)\in C(t), \forall t\in [0,T], $$ \end{itemize} where $J:X\to X^*$ is the duality mapping defined from $X$ into $X^*$ (see Section 2 for the definition). Clearly, (SP) coincides with the well known convex sweeping process introduced and studied in \cite{moreau} in the Hilbert space setting in which $J$ is the identity mapping. The authors in \cite{bounkhel_rabab} proved the following theorem. \begin{theorem} \label{thm2.1} Let $p,q >1$, $X$ be a $p$-uniformly convex and $q$-uniformly smooth Banach space, $T>0$, $I=[0,T]$ and $C:I\rightrightarrows X$ be a set-valued mapping closed convex values satisfying for any $t, t' \in I$ and any $x\in X$ \begin{equation}\label{dVcond} |(d^V_{C(t')})^{1/q'}(\psi)-(d^V_{C(t)})^{1/q'}(\phi)|\le \lambda|t'-t|+ \gamma \|\psi-\phi\|, \end{equation} where $\lambda, \gamma>0$, and $q'=\frac{q}{q-1}$. Assume that \begin{equation}\label{compac_cond} J(C(t))\subset K, \forall t\in I \text{ for some convex compact set } K \text{ in } X^*. \end{equation} Then {\rm (SP)} has at least one Lipschitz solution \end{theorem} They proved the existence of solutions under the Lipschitz continuity of the function $(t,\psi)\mapsto (d^V_{C(t)})^{1/q'}(\psi)$ defined on $I\times X^*$, and under the compactness assumption \eqref{compac_cond}. Using a different discretization we prove the previous theorem under the boundedness of $C$ and the compactness of their values which is clearly weaker than the compactness assumption \eqref{compac_cond}, and under the Lipschitz continuity of the usual distance function $t\mapsto (d_{C(t)})^{1/q'}(u)$, for all $u\in X$, defined on $I$ which is easier to handle with, than the function used in \eqref{dVcond}. Although, both Lipschitz assumptions coincide in the Hilbert space setting, in the case of Banach spaces the Lipschitz continuity of the distance function is easier to be checked than \eqref{dVcond}. Before proving our main result in Theorem \ref{th2}, we recall from \cite{bounkhel_rabab} some needed concepts and results and for more details we refer the reader to \cite{bounkhel_rabab} and the references therein. \section{Preliminaries} Let $X$ be a Banach space with topological dual space $X^*$. We denote by $d_S$ the usual distance function to $S$; i.e., $d_S(x):=\inf_{u\in S}\|x-u\|$. Let $S$ be a nonempty closed convex set of $X$ and $\bar x$ be a point in $S$. The convex normal cone of $S$ at $\bar x$ is defined by (see for instance \cite{clarke1}) \begin{equation}\label{eq2.1} N(S;\bar x)=\{\varphi\in X^*: \langle \varphi, x-\bar x\rangle \le 0\text{ for all } x\in S\}. \end{equation} The normalized duality mapping $J:X\rightrightarrows X^*$ is defined by $$ J(x)=\{j(x)\in X^*:\langle j(x),x\rangle =\|x\|^2=\|j(x)\|^2\}. $$ Many properties of the normalized duality mapping $J$ have been studied. For the details, one may see the books \cite{yakov,takahashi,Vainberg}. Let $V:X^*\times X \to \mathbb{R}$ be defined by $$ V(\varphi,x)=\|\varphi\|^2-2\langle\varphi,x\rangle+\|x\|^2, \text{ for any } \varphi \in X^* \text{ and } x\in X. $$ Based on the functional $V$, a set $\pi_S(\varphi)$ of generalized projections of $\varphi\in X^*$ onto $S$ is defined as follows (see \cite{alber1}). \begin{definition} \label{def0.2.1} \rm Let $S$ be a nonempty subset of $X$ and $\varphi \in X^*$. If there exists a point $\bar x \in S$ satisfying $$ V(\varphi,\bar x)=\inf_{ x\in S} V(\varphi,x), $$ then $\bar x$ is called a generalized projection of $\varphi $ onto $S$. The set of all such points is denoted by $\pi_S(\varphi)$. When the space $X$ is not reflexive $\pi_S(\varphi)$ may be empty for some elements $\varphi\in X^*$ even when $S$ is closed and convex (see \cite[Example 1.4]{Li}). \end{definition} The two following propositions are needed in the proof of the main theorem. For their proofs we refer the reader to \cite{bounkhelthibault,penot} respectively. \begin{proposition}\label{prop2} Let $S$ be a nonempty closed convex subset of $X$ and $x\in S$. Then $$ \partial d_S(x)=N_S(x)\cap \mathbf{B}. $$ \end{proposition} \begin{proposition}\label{prop1} For a nonempty closed convex subset $S$ of a reflexive smooth Banach space $X$ and $u\in S$, the following assertions are equivalent: \begin{itemize} \item[(i)] $\bar x\in S$ is a projection of $u$ onto $S$, that is $\bar x\in P_S(u)$; \item[(ii)] $\langle J(u-\bar x),x-\bar x\rangle\le 0$ for all $x\in S$; \item[(iii)] $J(u-\bar x)\in N(S;\bar x)$. \end{itemize} \end{proposition} Assume now that $X$ is $p$-uniformly convex and $q$-uniformly smooth Banach space and let $S$ be closed nonempty set in $X$. Recall the definition of the function $d^V_S: X^* \to [0,\infty[ $, given by $d^V_S(\varphi)=\inf_{x\in S}V(\varphi,x)$. Clearly, in Hilbert spaces, $d^V_S$ coincides with $d_S^2$. We need the two following lemmas proved in \cite{bounkhel_rabab}. \begin{lemma} \label{lem5.1} Let $p,q>1$, $X$ be a $p$-uniformly convex and $q$-uniformly smooth Banach space, and let $S$ be a bounded set. Then there exist two constants $\alpha >0$ and $\beta >0$ so that $\alpha\|x-y\|^p \le V(J(x),y)\le \beta \|x-y\|^q, \text{ for all } x,y \in S$. \end{lemma} \begin{proposition}\label{propo5.0} If $S$ is a bounded set in $X$, then $d^V_S(\varphi)\le \beta (d_S(J^*(\varphi)))^q$, where $\beta$ depends on the bound of $S$ and on $\varphi$. As a consequence, for sets $S_1$ and $S_2$ in $X$ and $X^*$ bounded by $l_1$ and $l_2$ respectively, we have $d^V_S(\varphi)\le \beta (d_S(J^*(\varphi)))^q, \text{ for all } \varphi \in S_2$, where $\beta$ depends on $l_1$ and $l_2$. \end{proposition} The following proposition is taken from \cite{yakov}. \begin{proposition}\label{prop5.2} Let $p\ge 2$ and let $X$ be a $p$-uniformly convex and $q$-uniformly smooth Banach space. The duality mapping $J:X\to X^*$ is Lipschitz on bounded sets; that is, $$ \|J(x)-J(y)\|\le C(R)\|x-y\|, \text{ for all } \|x\\\le R, \|y\|\le R. $$ Here $C(R):=32Lc_2^2(q-1)^{-1}$ and $c_2=\max\{1,R\}$ and $10$ such that $\bar x\in \pi_S(J(\bar x)+\alpha\varphi)$. \end{itemize} \end{proposition} \section{Main result} Now, we are ready to prove our main result in the following theorem. \begin{theorem}\label{th2} Instead of \eqref{dVcond} and \eqref{compac_cond} in Theorem \ref{thm2.1}, assume that $C$ is bounded with compact values and that \begin{equation}\label{distcond} |(d_{C(t')})^{p/q}(u)-(d_{C(t)})^{p/q}(u)|\le \lambda|t'-t|. \end{equation} Then {\rm (SP)} has at least one Lipschitz solution. \end{theorem} \begin{proof} Assume that $T=1$. Consider $\forall n\in N$ the following partition of $I$ $$ I_{n,i}=(t_{n,i},t_{n,i+1}], \quad t_{n,i}=\frac{i}{n}, \quad 0\leq i\leq n-1,\quad I_{n,0}=\{0\}. $$ Put $\mu_n=1/n$. Fix $n\ge 2$. Define by induction \begin{gather*} u_{n,0}= {u_0}\in C(0); \\ {u_{n,{i+1}}} \in \pi(C(t_{n,{i+1}});u_{n,i}), \quad\text{for } 0\leq i\leq n-1, \end{gather*} and \begin{gather*} u_n(t):= J^*(u^*_n(t))\\ u^*_n(t):= J(u_{n,i})+\frac{(t-t_{n,i})}{\mu_{n}}(J(u_{n,i+1})-J(u_{n,i})), \quad\text{for all } t\in I_{n,i} \end{gather*} and $u^*_n(0)=J(u_0)$. The construction is well defined since the generalized projection $\pi$ exits by Proposition \ref{prop0.0.6}. Clearly $u^*_n$ and $u_n$ are continuous on all $I$ and $u^*_n$ is differentiable on $I\setminus \{t_{n,i}\}$ and $\dot u^*_n(t)=\frac{J(u_{n,i+1})-J(u_{n,i})} {\mu_{n}}$, for all $t\in I\setminus \{t_{n,i}\}$. Let us find an upper bound estimate for the expression $\|J(u_{n,i+1})-J(u_{n,i})\|$. First, we have to point out that the sequence $u_i^n$ is bounded by some $l$ because the set-valued mapping $C$ is bounded. Now, since $X$ is $q$-uniformly smooth and $p$-uniformly convex and the sequence $u_i^n$ is bounded by $l$, there exist some constants $\alpha$ and $\beta$ depending on $l$ such that \[ \alpha\|u_{n,i+1}-u_{n,i}\|^p \le V(J(u_{n,i}),u_{n,i+1}) \le \beta\|u_{n,i+1}-u_{n,i}\|^q, \] and so by the construction of the sequence $u_i^n$ and Proposition \ref{propo5.0} we obtain \[ \alpha\|u_{n,i+1})-u_{n,i}\|^p \le d^V_{C(t_{n,i+1})}(J(u_{n,i})) \le \beta d^q_{C(t_{n,i+1})}(u_{n,i}) \] and so by the Lipschitz continuity in \eqref{distcond} we obtain \begin{align*} (\frac{\alpha}{\beta})^{\frac{1}{p}}\|u_{n,i+1})-u_{n,i}\| &\le d^{q/p}_{C(t_{n,i+1})}(u_{n,i})- d^{q/p}_{C(t_{n,i})}(u_{n,i})\\ &\leq \lambda|t_{n,i+1}-t_{n,i}|=\lambda \mu_n, \end{align*} and so \[ \|u_{n,i+1})-u_{n,i}\| \le \bar{\lambda}\mu_n, \] where $ \bar{\lambda}= (\frac{\beta}{\alpha})^{\frac{1}{p}}\lambda$. Using now the Lipschitz property of the duality mapping $J$ in Proposition \ref{prop5.2}, we can write $$ \|J(u_{n,i+1})-J(u_{n,i})\|\le C(l)\|u_{n,i+1}-u_{n,i}\|\le C(l)\bar{\lambda}\mu_n. $$ This inequality ensures the Lipschitz continuity of $u^*_n$ on all $I$ with ratio $\delta:=C(l)\bar{\lambda}$. Using the characterization of the normal cone, in terms of the generalized projection $\pi$ projection operator stated in Proposition \ref{prop0.0.6}, we can write for a.e. $t\in I$ \[ J(u_{n,i+1})-J(u_{n,i})\in -N (C(t_{n,i+1});u_{n,i+1}), \] which ensures together with Proposition \ref{prop2} that \[ -\frac{J(u_{n,i+1})-J(u_{n,i})}{\mu_n} \in \delta \partial d_{C(t_{n,i+1})}(u_{n,i+1}). \] Define now on $I_{n,i}$ the functions $\theta_n :I\rightarrow I$ by $\theta_n(0)=0$, and $$ \theta_n(t)=t_{n,i+1}, \quad \text{for all } t \in I_{n,i}. $$ Then the above inclusion becomes \begin{eqnarray}\label{eqn;2biss} -\dot u^*_n(t) \in \delta \partial d_{C(\theta_n(t))}(u_n(\theta_n(t))). \end{eqnarray} Now, let us prove that the sequence $(u_n)$ has a convergent subsequence. Clearly, we have $B=\{u_n; n\ge 2\}$ is equi-Lipschitz and bounded. So it remains to prove that $B(t)=\{u_n(t); n\ge 2\}$ is relatively compact in $X$, for all $t\in I$. By construction we have \begin{equation} \label{eqn:3} u_n(\theta_n(t))\in C(\theta_n(t)), \quad \forall t\in I \text{ and all }n\ge 2, \end{equation} and hence by the Lipschitz property of $d^{p/q}_C$ and the equi-Lipschitz property of $u_n$ we can write \begin{align*} d^{p/q}_{C(t)}(u_n(t)) &= d^{p/q}_{C(t)}(u_n(t))-d^{p/q}_{C(\theta_n(t))}( u_n(\theta_n(t))\le \lambda\mu_n+\|u_n(\theta_n(t))-u_n(t)\| \\ &\leq (\lambda+\delta)\mu_n. \end{align*} Assume by contradiction that $B(t_0)$ is not relatively compact in $X$ for some $t_0\in I$. So, $\gamma (B(t_0))\ge 2\bar \delta >0$, for some $\bar\delta\in(0,1]$. Fix now $n_0\in \mathbb{N}$ such that $\mu_n \le \mu_{n_0} < \frac{(\frac{\bar\delta}{2})^{p/q}}{\lambda+\delta}$, for all $n\ge n_0$. So \[ u_n(t)\in C(t)+ (\lambda+\delta)^{q/p}\mu_{n_0}^{q/p} \mathbb{B}, \quad \text{for all } n\ge n_0 \text{ and all } t\in I, \] which implies \[ B(t) \subset C(t)+ (\lambda+\delta)^{q/p}\mu_{n_0}^{q/p} \mathbb{B}, \quad \text{ for all } t\in I. \] Then the properties of $\gamma$ and the compactness of the values of $C$ imply \begin{align*} \gamma(B(t_0)) &= \gamma(\{u_n(t_0):n\ge n_0\})\le \gamma((C(t_0)) +\gamma((\lambda+\delta)^{q/p}\mu_{n_0}^{q/p}\mathbb{B}) \\ &\leq 2(\lambda+\delta)^{q/p}\mu_{n_0}^{q/p} < \bar\delta, \end{align*} which is a contradiction. Therefore, the set $B(t)$ is relatively compact in $X$ for any $t\in I$. Thus, Arzela-Ascoli theorem concludes that $(u_n)$ has a subsequence (still denoted $u_n$) converging uniformly to some $u$. Since $\lim_n \theta_n(t)=t$, we can write $\lim_n u_n(\theta_n(t)) =\lim_n u_n(t)=u(t)$ uniformly on $I$. So the sequence $u^*_n=J(u_n)$ will converge uniformly to $u^*=J(u)$ on $I$, since $J$ is uniformly continuous on bounded sets. We also have $(\dot u ^*_n)$ converges weakly star in $L^\infty(I,X^*)$ to some $w$. So, by the reflexivity and the separability of the space $X$, we can write $$ u^*(t)=J(u(t))=\lim_n u^*_n(t) =\lim_n\left(u^*_n(0)+\int^t_0 \dot u^*_n(s) ds\right) = u_0+\int^t_0 w(s) ds. $$ Hence $\dot u^*(t)=\frac{d}{dt}J(u(t))=w(t)$ a.e. on $I$. Let us prove that $u$ is the solution of our problem. First, we have to prove that $u(t)\in C(t)$, for all $t\in I$. Using now the Lipschitz property of the function $t\mapsto d^{q/p}_{C(t)}$ to write for all $t\in I$ \begin{align*} d^{q/p}_{C(t)}(u_n(\theta_n(t)) &= d^{q/p}_{C(t)}(u_n(\theta_n(t))- d^{q/p}_{C(\theta_n(t))}(u_n(\theta_n(t))\\ &\leq \lambda|\theta_n(t)-t| \le \lambda\mu_n, \end{align*} and so \begin{align*} d_{C(t)}(u(t)) &= d_{C(t)}(u_n(\theta_n(t))+\|u_n(\theta_n(t)-u(t)\| \\ &\leq (\lambda\mu_n)^{p/q}+\|u_n(\theta_n(t))-u(t)\| \to 0, \end{align*} as $n\to \infty$, by the fact that $\lim_n u_n(\theta_n(t))=u(t)$ uniformly on $I$. So the closedness of the set $C(t)$ ensures $u(t)\in C(t)$, for all $t\in I$. Going back to \eqref{eqn;2biss} we have \[ -\dot u^*_n(t)) \in N(C(\theta_n(t));u_n(\theta_n(t))), \quad \text{a.e. on } I. \] So, Proposition \ref{prop2} ensures for a.e. $t\in I$, \begin{equation} \label{eqn:5bis} \langle -\dot u^*_n(t));x-u_n(\theta_n(t))\rangle \le 0, \quad \forall x\in C(\theta_n(t)). \end{equation} Using the fact that $\dot u^*_n$ converges to $\frac{d}{dt}J(u(\cdot))$ in the weak star topology of $L^\infty(I,X^*)$, we can pass to the limit in \eqref{eqn:5bis} to obtain \begin{equation} \label{eqn:6} \langle -\frac{d}{dt}J(u(t));x-u(t)\rangle \le 0, \quad \forall x\in C(t), \text{ a.e. on } I. \end{equation} Indeed, fix $t\in I$, for which $\dot u^*_n(t)$ exists and converges weakly to $\frac{d}{dt}J(u(t))$, and let $x$ be any element in $C(t)$. Then, we have $$ x\in C(\theta_n(t))+(\lambda \mu_n)^{q/p} \mathbb{B}; $$ that is, $x=y_n(t)+(\lambda \mu_n)^{q/p}b_n$, with $b_n\in \mathbb{B}$ and $y_n(t)\in C(\theta_n(t))$. Hence \eqref{eqn:5bis} yields \begin{align*} &\langle -\frac{d}{dt}J(u(t)),x-u(t)\rangle \\ &= \langle -\frac{d}{dt}J(u(t))+\dot u^*_n(t)),x-u(t)\rangle+ \langle -\dot u^*_n(t)),x-u(t)\rangle \\ &= \langle -\frac{d}{dt}J(u(t))+\dot u^*_n(t)),x-u(t)\rangle+\langle -\dot u^*_n(t)), u_n(\theta_n(t))-u(t)\rangle\\ &\quad + \langle -\dot u^*_n(t),y_n(t)-u_n(\theta_n(t))\rangle+ \langle -\dot u^*_n(t)), (\lambda \mu_n)^{q/p}b_n\rangle \\ &\leq \langle \dot u^*_n(t))-\frac{d}{dt}J(u(t)),x-u(t)\rangle+ \lambda(\lambda \mu_n)^{q/p} + \lambda\| u_n\big(\theta_n(t)-u(t)\big)\| \to 0 \end{align*} as $n\to \infty$. So, \begin{eqnarray} \langle -\frac{d}{dt}J(u(t)),x-u(t)\rangle \le 0, \quad \text{for all } x\in C(t), \end{eqnarray} which by Proposition \ref{prop2} gives \begin{equation} \label{eqn:7ter} -\frac{d}{dt}J(u(t))\in N(C(t);u(t)), \quad \text{ a.e. on } I \end{equation} and hence the proof is complete. \end{proof} \subsection*{Acknowledgments} The author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No. RGP-VPP-024. \begin{thebibliography}{00} \bibitem{yakov} Y. Alber, I. Ryazantseva; \emph{Nonlinear ill-posed problems of monotone type}. Springer, Dordrecht, 2006. \bibitem{alber1} Y. Alber; \emph{Generalized Projection Operators in Banach Spaces: Properties and Applications,} Funct. Different. Equations 1 (1), 1-21 (1994). \bibitem{bounkhel_rabab} M. Bounkhel, R. Al-yusof; \emph{First and second order Convex Sweeping Processes in Reflexive smooth Banach spaces}, Set-Valued Var. 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