\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 50, pp. 1--10.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/50\hfil Existence of viable solutions] {Existence of viable solutions for nonconvex differential inclusions} \author[M. Bounkhel, T. Haddad\hfil EJDE-2005/50\hfilneg] {Messaoud Bounkhel, Tahar Haddad} % in alphabetical order \address{Messaoud Bounkhel \hfill\break King Saud University, College of Science, Department of Mathematics, Riyadh 11451, Saudi Arabia} \email{bounkhel@ksu.edu.sa} \address{Tahar Haddad \hfill\break University of Jijel, Department of Mathematics, B.P. 98, Ouled Aissa, Jijel, Algeria} \email{haddadtr2000@yahoo.fr} \date{} \thanks{Submitted December 26, 2004. Published May 11, 2005.} \subjclass[2000]{34A60, 34G25, 49J52, 49J53} \keywords{Uniformly regular functions; normal cone; \hfill\break\indent nonconvex differential inclusions} \begin{abstract} We show the existence result of viable solutions to the differential inclusion \begin{gather*} \dot x(t)\in G(x(t))+F(t,x(t)) \\ x(t)\in S \quad \mbox{on } [0,T], \end{gather*} where $F: [0,T]\times H\to H$ $(T>0)$ is a continuous set-valued mapping, $G:H\to H$ is a Hausdorff upper semi-continuous set-valued mapping such that $G(x)\subset \partial g(x)$, where $g :H\to \mathbb{R}$ is a regular and locally Lipschitz function and $S$ is a ball, compact subset in a separable Hilbert space $H$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Porposition} \newtheorem{rem}[theorem]{Remark} \newtheorem{coro}[theorem]{Corolloray} \section{Introduction} Let $T>0$. It is well known that the solution set of the differential inclusion \begin{gather*} % (*) \dot x(t)\in G(x(t)) \quad \mbox{ a.e. } \quad [0,T]\\ x(0)=x_0 \in \mathbb{R}^d, \end{gather*} can be empty when the set-valued mapping $G$ is upper semicontinuous with non\-empty nonconvex values. Bressan, Cellina and Colombo \cite{bressancellinacolombo}, proved an existence result fo the above equation by assuming that the set-valued mapping $G$ is included in the subdifferential of a convex lower semicontinuous (l.s.c.) function $g:\mathbb{R}^d \to \mathbb{R}$. This result has been extended in many ways by many authors; see for example \cite{anconacolombo,benabdellah2,benabdellahcastaing,bounkhel1, morchadi,rossi,Trongducha}. The recent extension of the above equation was studied by Bounkhel \cite{bounkhel1}, in which the author proved an existence result of viable solutions in the finite dimensional case for the differential inclusion $$%\label{DI} \begin{gathered} \dot x(t)\in G(x(t))+F(t,x(t))\quad\mbox{ a.e. } [0,T]\\ x(t)\in S, \quad \mbox{on } [0,T]. \end{gathered} \label{2}$$ This extension covers all the other extensions given in the finite dimensional case. In the present paper we extend this result to the infinite dimensional setting. A function $x(\cdot)$ is called a viable solution if it satisfies the differential inclusion and $x(t)\in S$ for all $t\in [0,T]$ and for some closed set $S$. \section{Uniformly regular functions} Let $H$ be a real separable Hilbert space. Let us recall the concept of regularity that will be used in the sequel \cite{bounkhel1}. \begin{definition}[\cite{bounkhel1}] \label{def1} \rm Let $f:H\to \mathbb{R}\cup \{+\infty \}$ be a l.s.c. function and let $O\subset \mathop{\rm dom} f$ be a nonempty open subset. We will say that $f$ is uniformly regular over $O$ if there exists a positive number $\beta \geq 0$ such that for all $x\in O$ and for all $\xi \in \partial ^{P}f(x)$ one has $$\langle \xi ,x'-x\rangle \leq f(x')-f(x)+\beta \Vert x'-x\Vert^{2} \quad \mbox{for all } x'\in O.$$ \end{definition} Here $\partial ^{P}f(x)$ denotes the proximal subdifferential of $f$ at $x$ (for its definition the reader is refereed for instance to \cite{bounkhelthibault3}). We will say that $f$ is uniformly regular over closed set $S$ if there exists an open set $O$ containing $S$ such that $f$ is uniformly regular over $O$. The class of functions that are uniformly regular over sets is so large. For more details and examples we refer the reader to \cite{bounkhel1}. The following proposition summarizes some important properties for uniformly regular locally Lipschitz functions over sets needed in the sequel. For the proof of these results we refer the reader to \cite{bounkhel1,bounkhel3}. \begin{proposition}\label{prop1} Let $f:H\to \mathbb{R}$ be a locally Lipschitz function and $S$ a nonempty closed set. If $f$ is uniformly regular over $S$, then the following hold: \begin{itemize} \item[(i)] The proximal subdifferential of $f$ is closed over $S$, that is, for every $x_{n}\to x\in S$ with $x_{n}\in S$ and every $\xi _{n}\to \xi$ with $\xi _{n}\in \partial ^{P}f(x_{n})$ one has $\xi \in \partial ^{P}f(x)$ \item[(ii)] The proximal subdifferential of $f$ coincides with $\partial ^{C}f(x)$ the Clarke subdifferential for any point $x$ (see for instance \cite{bounkhelthibault3} for the definition of $\partial ^{C}f$) \item[(iii)] The proximal subdifferential of $f$ is upper hemicontinuous over $S$, that is, the support function $x\mapsto \langle v,\partial ^{P}f(x)\rangle$ is u.s.c. over $S$ for every $v\in H$ \item[(iv)] For any absolutely continuous map $x:[0,T]\to S$ one has $$\frac{d}{dt}(f\circ x)(t)=\langle \partial^C f(x(t));\dot x(t) \rangle.$$ \end{itemize} \end{proposition} Now we are in position to state and prove our main result in this paper. \begin{theorem} \label{thm2.1} Let $g:H\to \mathbb{R}$ be a locally Lipschitz function and $\beta$-uniformly regular over $S\subset H$. Assume that \begin{itemize} \item[(i)] $S$ is nonempty ball compact subset in $H$, that is, the set $S\cap r \mathbb{B}$ is compact for any $r>0$; \item[(ii)] $G:H \to H$ is a Hausdorff $u.s.c$ set valued map with compact values satisfying $G(x)\subset \partial ^{C}g(x)$ for all $x\in S$; \item[(iii)] $F:[0,T]\times H\to H$ is a continuous set valued map with compact values; \item[(iv)] For any $(t,x)\in I\times S$, the following tangential condition holds $$\liminf_{h\to 0} \frac{1}{h}e\big(x+h\left[G(x)+F(t,x)\right];S\big)=0\,, \label{3}$$ where $e(A;S):= \sup_{a\in A} d_S(a)$. \end{itemize} Then, for any $x_{0}\in S$ there exists $a\in ]0,T[$ such that the differential inclusion (\ref{DI}) has a viable solution on $[0,a]$. \end{theorem} \begin{proof} Let $\rho >0$ such that $K_{0}:=S\cap (x_{0}+\rho B)$ is compact and $g$ is $L$-Lipschitz on $x_{0}+\rho B$. Since $F$ and $G$ are continuous and the set $I\times K_0$ is compact, there exists a positive scalar $M$ such that $$\| G(x)\| +\| F(t,x)\| \leq M,$$ for all $(t,x)\in I\times K_{0}$. Since $(t_{0},x_{0})\in I\times K_{0}$, then (by $(\ref{3})$) $\liminf_{h\to 0}\frac{1}{h}e\Big(x_{0}+h\big[G(x_{0}) +F(t_{0},x_{0})\big];S\Big)=0.$ Put $\alpha:=\min\{T, \frac{\rho}{M+1},1\}$. Hence for every $m \ge 1$ there exists $0<\xi< \frac{\alpha}{2}$ such that $$\label{eq:6} e\Big(x_{0}+\xi \left[G(x_{0})+F(t_{0},x_{0})\right];S\Big)<\frac{\xi}{m}.$$ Let $b_{0}\in G(x_{0})+F(t_{0},x_{0})$ and put $$\lambda _{0}^{m}:=\max\big\{\xi\in (0,\frac{\alpha}{2}]: \xi \le T-t_0 \mbox{ and } d_{S}(x_{0}+\xi b_{0})<\frac{\xi}{m} \big\}.$$ Since $x_0\in S$, we have $d_{S}(x_{0}+\lambda _{0}^{m}b_{0})\leq \lambda _{0}^{m}\Vert b_{0}\Vert \le \lambda _{0}^{m} M < M.$ So, there exists $\Psi_{0}^{m}\in S \cap {\mathbb{B}} (x_{0}+\lambda _{0}^{m}b_{0},M+1)$ such that $\Vert \Psi _{0}^{m}-x_{0}-\lambda_{0}^{m}b_{0}\Vert =d_{S}(x_{0} +\lambda _{0}^{m}b_{0}),$ and so $\| \frac{1}{\lambda _{0}^{m}}[\Psi _{0}^{m}-x_{0}]-b_{0}\| =\frac{1}{\lambda _{0}^{m}}d_{S}(x_{0}+\lambda _{0}^{m}b_{0}) < \frac{1}{m}$ by (\ref{eq:6}) and the definition of $\lambda _{0}^{m}$. Let $w_{0}^{m}:=\frac{\Psi^{m}_{0}-x_{0}}{\lambda _{0}^{m}}$ and $x_1^m:=x_{0}+\lambda _{0}^{m}w_{0}^{m}\in S$. Thus, we obtain $$\begin{gathered} w_{0}^{m}\in G(x_{0})+F(t_{0},x_{0})+\frac{1}{m}B, \\ \| x_{1}^{m}-x_{0}\| =\lambda _{0}^{m}\| w_{0}^{m}\| <\lambda _{0}^{m}(M+\frac{1}{m})<\lambda _{0}^{m}(M+1). \end{gathered} \label{eq:resume1}$$ We can choose, a priori, $a<\alpha$ and find $\lambda _{0}^{m} < a$ such that 0<\lambda _{0}^{m} t^m_{i+1}-t^m_{i}=\lambda^m_{i}. Thus, there exists some \lambda > \lambda^m_i such that 0<\lambda < T-\bar t\le T-t^m_i (for all i\ge i_0) and d_S(x^m_i+\lambda b_i)\le \frac{\lambda}{2m}<\frac{\lambda}{m}. This contradicts the definition of \lambda^m_{i}. Therefore, there is an integer \nu_m \ge 1 such that t_{\nu_m}\leq aj), we have \begin{align*} \Vert x_{m}(t_{i}^{m})-x_{m}(t_{j}^{m}) \Vert &\le \sum_{k=j+1}^{i} \| x_{m}(t_{k}^{m})-x_{m}(t_{k-1}^{m})\| \\ &\le (M+1) \sum_{k=j+1}^{i} (t_{k}^{m}-t_{k-1}^{m})= (M+1)|t_{i}^{m}-t_{j}^{m}|. \end{align*} Also, we have by construction for a.e. t\in [t_{i}^{m},t_{i+1}^{m}] and for all i\in \{0,\dots ,\nu _{m}\} $$\Vert \dot x_{m}(t) \Vert = \Vert w_{i}^{m} \Vert \le M+1.\label{9}$$ \subsection*{Convergence of approximate solutions} We note that the sequence f_{m} can be constructed with the relative compactness property in the space of bounded functions (see \cite{Trongducha}). Therefore, without loss of generality we can suppose that there is a bounded function f such that $$\lim_{m\to \infty} \sup_{t\in [0,a]} \Vert f_{m}(t)-f(t) \Vert =0\,.$$ Now, we prove that the approximate solutions x_{m}(.) converge to a viable solution of \eqref{2}. It is clear by construction that \{x_{m}\}_m are Lipschitz continuous with constant M+1 and% $x_{m}(t) = x_{i}^{m}+(t-t_{i}^{m})w_{i}^{m} =x_{i}^{m}+(\frac{t-t_{i}^{m}}{\lambda _{i}^{m}})(\Psi _{i}^{m}-x_{i}^{m}).$ On the other hand, we have 0 \le t - t_{i}^{m} \le t_{i+1}^{m}-t_{i}^{m}=\lambda_{i}^{m} and so 0\le \frac{t-t_{i}^{m}}{\lambda _{i}^{m}}\leq 1, and hence we get $$(\frac{t-t_{i}^{m}}{\lambda _{i}^{m}})(\Psi _{i}^{m}-x_{i}^{m})\in \overline{% co}[\{0\}\cup (K_{1}-K_{0})].$$ Thus, $$x_{m}(t) \in K:=K_{0}+\overline{co}[\{0\}\cup (K_{1}-K_{0})].$$ Therefore, since the set K is compact ( because K_0 and K_1 are compact), then the assumptions of the Arzela-Ascoli theorem are satisfied. Hence a subsequence of x_{m} my be extracted (still denoted x_{m}) that converges to an absolutely continuous mapping x:[0,a]\to H such that $$\begin{gathered} \lim_{m\to \infty} \max_{t\in [0,a]} \Vert x_{m}(t)-x(t) \Vert =0 \\ \dot x_{m}(.)\rightharpoonup \dot x(.)\mbox{ in the weak topology of } L^{2}([0,a],H). \end{gathered} \label{10}$$ Recall now that f_{m} converges pointwise a.e. on [0,a] to f. Then the continuity of the set-valued mapping F and the closedness of the set F(t,x(t)) entail f(t)\in F(t,x(t)) . Now, it remains to prove that $$\begin{gathered} x(t)\in S; \\ -f(t)+x'(t)\in G(x(t))\quad \mbox{a.e. on } \quad [0,a]. \end{gathered}$$ By construction we have x_{i}^{m}\in K_{0} (for all i \in \{0,\dots ,\nu _{m}-1\}). This ensures $d_{K_{0}}(x(t)) \leq \| x_{m}(t)-x_{i}^{m}\| +\| x_{m}(t)-x(t)\| \leq \frac{1+M}{m}+\| x_{m}(t)-x(t)\|$ which approaches 0 as m approaches \infty. The closedness of K_0 yields d_{K_{0}}(x(t))=0 and so x(t)\in K_{0}\subset S. By construction, we have for a.e. t\in [0,a] $$\label{15} \dot x_{m}(t)-f_{m}(t)-c_{m}(t)\in G(x_{m}(\theta _{m}(t)))\subset \partial ^{C}g(x_{m}(\theta _{m}(t)))=\partial ^{P}g(x_{m}(\theta _{m}(t))),$$ where the above equality follows from the uniform regularity of g over C and the part (ii) in Proposition \ref{prop1}. We can thus apply Castaing techniques (see for example \cite{castaing}). The weak convergence (by (\ref{10})) in L^2([0,a],H) of \dot x_m(\cdot) to \dot x(\cdot) and Mazur's Lemma entail \dot x(t)\in \bigcap_m\overline{co}\{\dot x_k(t):\, k\geq m\}, \quad \mbox{ for a.e. on } [0,a]. $$Fix any such t and consider any \xi\in H. Then, the last relation above yields$$ \big<\xi,\dot x(t)\big>\leq \inf_m\sup_{k\geq m}\big<\xi,\dot x_m(t)\big> and hence Proposition \ref{prop1} part (iii) and (\ref{15}) yield \begin{align*} \langle \xi ,\dot x(t) \rangle &\leq \lim_m \sup \sigma (\xi ,\partial ^{P}g(x_{m}(\theta _{m}(t))) +f_{m}(t)+c_{m}(t)) \\ &\leq \sigma (\xi ,\partial ^{P}g(x(t))+f(t))\quad \mbox{for any } \xi \in H, \end{align*} So, the convexity and the closedness of the set \partial ^{P}g(x(t)) ensure $$-f(t)+\dot x (t)\in \partial ^{P}g(x(t)).$$ Now, since g is uniformly regular over C and x:[0,a] \to C we have \begin{align*} \frac{d}{dt}(g\circ x)(t) &=\langle \partial^{P}g(x(t)),\dot x (t)\rangle\\ &=\langle -f(t)+\dot x(t),\dot x (t)\rangle \\ &=\| \dot x (t)\| ^{2}-\langle f(t), \dot x (t)\rangle. \end{align*} Consequently, $$g(x(a))-g(x_{0})=\int_{0}^{a}\| \dot x (s)\| ^{2}ds-\int_{0}^{a}\left\langle f(s),\dot x (s)\right\rangle ds \label{11}$$% On the other hand, by (\ref{15}) and Definition \ref{def1} we have for all i\in \{0,\dots \nu _{m}-1\} \begin{align*} g(x_{i+1}^{m})-g(x_{i}^{m}) &\geq \langle \dot x_{m} (t)-f_{i}^{m}-c_{i}^{m},x_{i+1}^{m}-x_{i}^{m}\rangle -\beta \| x_{i+1}^{m}-x_{i}^{m}\| ^{2} \\ &=\big\langle \dot x_{m} (t)-f_{m}(t)-c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m (s)ds\big\rangle -\beta \| x_{i+1}^{m}-x_{i}^{m}\| ^{2}\\ &\geq \int_{t_{i}^{m}}^{t_{i+1}^{m}}\| \dot x_{m} (s)\| ^{2}ds-\int_{t_{i}^{m}}^{t_{i+1}^{m}}\langle \dot x_{m} (s),f_{m}(s)\rangle ds \\ &\quad-\langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m (s)ds\rangle -\beta (M+1)^{2}(t_{i+1}^{m}-t_{i}^{m})^{2} \\ &\geq \int_{t_{i}^{m}}^{t_{i+1}^{m}}\| \dot x_{m} (s)\| ^{2}ds-\int_{t_{i}^{m}}^{t_{i+1}^{m}}\langle \dot x_{m} (s),f_{m}(s)\rangle ds \\ &\quad-\langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m (s)ds\rangle -\frac{\beta (M+1)^{2}}{m}(t_{i+1}^{m}-t_{i}^{m}). \end{align*} By adding, we obtain $$g(x_{m}(t_{\nu _{m}}^{m}))-g(x_{0})\geq \int_{0}^{t_{\nu _{m}}^{m}}\| \dot x_{m} (s)\| ^{2}ds-\int_{0}^{t_{\nu _{m}}^{m}}\left\langle f_{m}(s),\dot x_{m} (s)\right\rangle ds-\varepsilon _{1,m} \label{12}$$% with\varepsilon_{1,m}= \sum_{i=0}^{\nu _{m}-1} % \langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m (s)ds \rangle +\frac{\beta (M+1)^{2}t_{\nu _{m}}^{m}}{m} $$and $$g(x_{m}(a))-g((t_{\nu _{m}}^{m}))\geq \int_{t_{\nu _{m}}^{m}}^{a} \Vert \dot x_{m} (s) \Vert ^{2}ds-\int_{t_{\nu _{m}}^{m}}^{a} \left\langle \dot x_{m} (s),f_{m}(s) \right\rangle ds-\varepsilon _{2,m} \label{13}$$ with$$ \varepsilon _{2,m}= \langle c_{\nu _{m}}^{m},\int_{t_{\nu _{m}}^{m}}^{a}\dot x (s)ds \rangle +\frac{\beta (M+1)^{2}(a-t_{\nu _{m}}^{m})}{m}. $$Therefore, we get $$\label{16} g(x_{m}(a))-g((x_{0}))\geq \int_{^{0}}^{a} \Vert \dot x_{m}(s) \Vert ^{2}ds-\int_{0}^{^{a}} \langle f_{m}(s),\dot x_{m}(s) \rangle ds-\varepsilon_{m}$$ where$$ \varepsilon_m =\varepsilon_{1,m}+\varepsilon_{2,m} = \sum_{i=0}^{\nu _{m}-1} \langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x (s)ds \rangle + \langle c_{\nu _{m}}^{m},\int_{t_{\nu _{m}}^{m}}^{a}\dot x(s)ds \rangle +\frac{\beta a(M+1)^{2}}{m}. Using our construction we get \begin{align*} \vert \varepsilon _{m} \vert &\leq \sum_{i=0}^{\nu _{m}-1} \Vert c_{i}^{m} \Vert \int_{t_{i}^{m}}^{t_{i+1}^{m}} \Vert \dot x (s) \Vert ds+ \Vert c_{\nu _{m}}^{m} \Vert \int_{t_{\nu _{m}}^{m}}^{a} \Vert \dot x (s) \Vert ds +\frac{\beta (M+1)^{2}a}{m} \\ &\leq \sum_{i=0}^{\nu _{m}-1}\frac{1}{m} (t_{i+1}^{m}-t_{i}^{m})(M+1)+\frac{1}{m}(a-t_{\nu _{m}}^{m})(M+1)+\frac{ \beta (M+1)^{2}a}{m} \\ &=\frac{(M+1)}{m}\big[ \sum_{i=0}^{\nu _{m}-1} (t_{i+1}^{m}-t_{i}^{m}) +(a-t_{\nu_{m}}^{m}) \big] +\frac{\beta (M+1)^{2}a }{m}\\ &=\frac{(M+1)a}{m}+\frac{\beta (M+1)^{2}a}{m} \to 0 \quad \mbox{ as } m\to \infty. \end{align*} We have also $\lim_{m\to \infty} \int_{0}^{a} \langle f_{m}(s),\dot x_{m}(s) \rangle ds=\int_{0}^{a} \langle f(s),\dot x(s) \rangle ds.$ Taking the limit superior in (\ref{16}) when m\to \infty we obtain $$g(x(a))-g(x_{0})\geq \limsup_m \int_{0}^{a} \Vert \dot x_{m} (s) \Vert ^{2}ds-\int_{0}^{a} \langle f(s),\dot x (s) \rangle ds.$$% This inequality compared with (\ref{11}) yields $\int_{0}^{a}\| \dot x (s)\| ^{2}ds\geq \limsup_m \int_{0}^{a}\|\dot x_{m} (s)\| ^{2}ds,$ that is, $$\| \dot x \| _{L^{2}([0,a],H)}^{2}\geq \limsup_m \| \dot x_{m} \| _{L^{2}([0,a],H)}^{2}.$$% On the other hand the weak l.s.c of the norm ensures $\| \dot x \| _{L^{2}([0,a],H)}^{2}\leq \liminf_m \| \dot x_{m} \| _{L^{2}([0,a],H)}^{2}$ Consequently, we get $\| \dot x \| _{L^{2}([0,a],H)}= \lim_{m} \| \dot x_{m} \| _{L^{2}([0,a],H)}.$ Hence there exists a subsequence of \{\dot x_{m}\}_{_m} (still denoted \{\dot x_{m}\}_{_m}) converges poitwisely a.e on [0.a] to \dot x. \vskip2mm \noindent Since (x_{m}(t),\dot x_{m} (t)-f_{m}(t)-c_{m}(t))\in gphG, \quad \mbox{a.e. on } [0.a], $$and as G has a closed graph, we obtain$$ (x(t),\dot x (t)-f(t))\in gphG \quad \mbox{a.e. on } [0.a], $$and so $\dot x (t)\in G(x(t))+F(t,x(t))\quad \mbox{a.e. on } [0.a]$ The proof is complete. \end{proof} \begin{rem}\label{r1} {\rm An inspection of the proof of Theorem \ref{thm2.1} shows that the uniformity of the constant \beta was needed only over the set K_0 and so it was not necessary over all the set S. Indeed, it suffices to take the uniform regularity of g locally over S, that is, for every point \bar x\in S there exist \beta\geq 0 and a neighborhood V of x_0 such that g is uniformly regular over S\cap V.} \end{rem} We conclude the paper with two corollaries of our main result in Theorem \ref{thm2.1}. \begin{coro} Let K\subset H be a nonempty uniformly prox-regular closed subset of a finite dimensional space H and F: [0,T]\times H\to H be a continuous set-valued mapping with compact values. Then, for any x_0\in K there exists a\in ]0,T[ such that the following differential inclusion \begin{gather*} \dot x(t)\in -\partial^C d_K(x(t)) + F(t,x(t)) \quad \mbox{\rm a.e. on }[0,a]\\ x(0)=x_0 \in K, \end{gather*} has at least one absolutely continuous solution on [0,a]. \end{coro} \begin{proof} In \cite[Theorem 3.4]{bounkhelthibault1} (see also \cite[theorem 4.1]{bounkhel1}) it is shown that the function g:=d_K is uniformly regular over K and so it is uniformly regular over some neighborhood V of x_0\in K. Thus, by Remark \ref{r1}, we apply Theorem \ref{thm2.1} with S=H (hence the tangential condition (\ref{3}) is satisfied), K_0:=V\cap S=V, and the set-valued mapping G:=\partial^C d_K which satisfies the hypothesis of Theorem \ref{thm2.1}. \end{proof} Our second corollary concerns the following differential inclusion $$\label{DI} \begin{gathered} \dot x(t)\in -N^C(S;x(t)) + F(t,x(t)) \quad \mbox{a.e. }\\ x(t)\in S, \quad \mbox{ for all t and } x(0)=x_0 \in S. \end{gathered}$$ This type of differential inclusion has been introduced in \cite{henry} for studying some economic problems. \begin{coro} Let H be a separable Hilbert space. Assume that \begin{enumerate} \item F: [0,T]\times H\to H is a continuous set-valued mapping with compact values; \item S is a nonempty uniformly prox-regular closed subset in H; \item For any (t,x)\in I\times S the tangential condition$$ \liminf_{h\downarrow 0} h^{-1}e\big(x+h(\partial^Cd_S (x)+F(t,x));S\big)=0,$for any$(t,x)\in I\times S$holds. \end{enumerate} Then, for any$x_0\in S$, there exists$a\in ]0,T[$such that the differential inclusion \eqref{DI} has at lease one absolutely continuous solution on$[0,a]\$. \end{coro} \subsection*{Acknowledgement} The authors would like to thank the anonymous referee for his careful reading of the paper and for his pertinent suggestions and remarks. \begin{thebibliography}{9} \bibitem{anconacolombo} F. Ancona and G. Colombo; {\it Existence of solutions for a class of non convex differential inclusions}, Rend. Sem. Mat. Univ. Padova, Vol. {\bf 83} (1990), pp. 71-76. \bibitem{benabdellah2} H. 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