\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 06, 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-2005/06\hfil BVP with integral boundary conditions] {Existence results for nonlinear boundary-value problems \\ with integral boundary conditions} \author[A. Belarbi, M. Benchohra\hfil EJDE-2005/06\hfilneg] {Abdelkader Belarbi, Mouffak Benchohra} % in alphabetical order \address{Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel Abb\es,\\ BP 89, 22000, Sidi Bel Abb\es, Alg\'erie} \email[A. Belarbi]{aek\_belarbi@yahoo.fr} \email[M. Benchohra]{benchohra@univ-sba.dz} \date{} \thanks{Submitted July 19, 2004. Published January 4, 2005.} \subjclass[2000]{34A60, 34B15} \keywords{Nonlinear boundary-value problem; integral boundary conditions; \hfill\break\indent contraction; fixed point} \begin{abstract} In this paper, we investigate the existence of solutions for a second order nonlinear boundary-value problem with integral boundary conditions. By using suitable fixed point theorems, we study the cases when the right hand side has convex and nonconvex values. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This paper concerns the existence of solutions of a nonlinear boundary-value problem with integral boundary conditions. More precisely, in Section 3, we consider the nonlinear boundary-value problem \begin{gather}\label{e1} x''(t)\in F(t,x(t)), \quad \hbox{a.e. } t\in [0,1], \\ \label{e2} x(0)-k_1x'(0)=\int_0^1 h_1(x(s)) ds, \\ \label{e3} x(1)+k_2x'(1)=\int_0^1 h_2(x(s))ds, \end{gather} where $F: [0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a compact valued multivalued map, $\mathcal{P}(\mathbb{R})$ is the family of all subsets of $\mathbb{R}$, $h_i:\mathbb{R}\to \mathbb{R}$ are continuous functions and $k_i$ are nonnegative constants ($i=1,2$). Boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo \cite{ Gall}, Karakostas and Tsamatos \cite{KaTs}, Lomtatidze and Malaguti \cite{LoMa} and the references therein. Moreover, boundary-value problems with integral boundary conditions have been studied by a number of authors, for instance, Brykalov \cite{Bry}, Denche and Marhoune \cite{DenM}, Jankowskii \cite{Jan} and Krall \cite{Kra} and Rahmat and Bashir \cite{RaBa}. The present paper is motivated by a recent one due to Rahman \cite{Rah} in which the generalized method of quasilinearization was applied to a class of second order boundary-value problem with integral boundary conditions of the form (\ref{e2}) and (\ref{e3}). In this paper, we shall present three existence results for the problem (\ref{e1})-(\ref{e3}) when the right hand side is convex as well as nonconvex valued. The first one relies on the nonlinear alternative of Leray-Schauder type. In the second one, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler, while in the third one, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposables values. These results extend to the multivalued case some ones considered in the literature. \section{Preliminaries} In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper. $C([0,1],\mathbb{R})$ is the Banach space of all continuous functions from $[0,1]$ into $\mathbb{R}$ with the norm $$\|x\|_{\infty}=\sup\{|x(t)|: 0\leq t\leq 1\}.$$ $L^1 ([0,1],\mathbb{R})$ denotes the Banach space of measurable functions $x:[0,1]\to \mathbb{R}$ which are Lebesgue integrable normed by $$\|x\|_{L^1 }=\int_0^1 |x(t)|dt \quad \hbox{for all } x\in L^1 ([0,1],\mathbb{R}).$$ $AC^1((0,1),\mathbb{R})$ is the space of differentiable functions $x:(0,1) \to \mathbb{R}$, whose first derivative, $x'$, is absolutely continuous. Let $(X,|\cdot|)$ be a normed space, $P_{cl}(X)=\{Y\in \mathcal{P}(X): Y$ closed$\}$, $P_{b}(X)=\{Y\in \mathcal{P}(X): Y$ bounded$\}$, $P_{cp}(X)=\{Y\in \mathcal{P}(X): Y$ compact$\}$ and $P_{cp,c}(X)=\{Y\in \mathcal{P}(X): Y$ compact and convex$\}$. A multivalued map $G:X\to P(X)$ is convex (closed) valued if $G(x)$ is convex (closed) for all $x\in X$. $G$ is bounded on bounded sets if $G(B)=\cup_{x\in B}G(x)$ is bounded in $X$ for all $B\in P_{b}(X)$ (i.e. $\sup_{x\in B}\{\sup\{|y|: y\in G(x) \}\}<\infty). \, G$ is called upper semi-continuous (u.s.c.) on $X$ if for each $x_0\in X$ the set $G(x_0)$ is a nonempty closed subset of $X$ and if for each open set $N$ of $X$ containing $G(x_0)$, there exists an open neighbourhood $N_0$ of $x_0$ such that $G(N_0)\subseteq N. \, G$ is said to be completely continuous if $G(\mathcal{B})$ is relatively compact for every $\mathcal{B}\in P_{b}(X)$. If the multivalued map $G$ is completely continuous with nonempty compact values, then $G$ is u.s.c. if and only if $G$ has a closed graph (i.e. $x_{n}\to x_{*}, \ y_{n}\to y_{*}, \ y_{n}\in G(x_{n})$ imply $y_{*}\in G(x_{*})$). $G$ has a fixed point if there is $x\in X$ such that $x\in G(x)$. The fixed point set of the multivalued operator $G$ will be denoted by $Fix G$. A multivalued map $G:[0,1]\to P_{cl}(\mathbb{R})$ is said to be measurable if for every $y\in \mathbb{R}$, the function $t\mapsto d(y,G(t))=\inf\{|y-z|: z\in G(t) \}$ is measurable. For more details on multivalued maps see the books of Aubin and Cellina \cite{AuCe}, Aubin and Frankowska \cite{AuFr}, Deimling \cite{Dei} and Hu and Papageorgiou \cite{HuPa} . \begin{definition} \rm A multivalued map $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is said to be $L^1$-Carath\'eodory if \begin{itemize} \item[(i)] $t\mapsto F(t,x)$ is measurable for each $x\in \mathbb{R}$; \item[(ii)] $x\mapsto F(t,x)$ is upper semicontinuous for almost all $t\in [0,1]$; \item[(iii)] for each $q>0$, there exists $\varphi_{q} \in L^1 ([0,1],\mathbb{R}_{+})$ such that $$\|F(t,x)\|=\sup\{|v|: v\in F(t,x)\}\leq \varphi_{q}(t) \quad \hbox{for all } |x|\leq q \mbox{ and for } a.e. \; t\in [0,1].$$ \end{itemize} For each $x\in C([0,1],\mathbb{R})$, define the set of selections of $F$ by $$S_{F,x}=\{v\in L^1([0,1],\mathbb{R}): v(t)\in F(t,x(t))\, \, a.e. \, \, t\in [0,1]\}.$$ \end{definition} Let $E$ be a Banach space, $X$ a nonempty closed subset of $E$ and $G:X\to \mathcal{P}(E)$ a multivalued operator with nonempty closed values. $G$ is lower semi-continuous (l.s.c.) if the set $\{x\in X: G(x)\cap B\not=\emptyset\}$ is open for any open set $B$ in $E$. Let $A$ be a subset of $[0,1]\times \mathbb{R}$. $A$ is $\mathcal{L}\otimes\mathcal{B}$ measurable if $A$ belongs to the $\sigma$-algebra generated by all sets of the form $\mathcal{J}\times D$, where $\mathcal{J}$ is Lebesgue measurable in $[0,1]$ and $D$ is Borel measurable in $\mathbb{R}$. A subset $A$ of $L^1([0,1],\mathbb{R})$\ is decomposable if for all $u,v\in A$ and $\mathcal{J}\subset [0,1]$ measurable, the function $u\chi_\mathcal{J}+v\chi_{J-\mathcal{J}}\in A$, where $\chi_{\mathcal{J}}$ stands for the characteristic function of $\mathcal{J}$. \begin{definition} \rm Let $Y$ be a separable metric space and let $N: Y\to \mathcal{P}(L^1([0,1],\mathbb{R}))$ be a multivalued operator. We say $N$ has property (BC) if \begin{itemize} \item[1)] $N$ is lower semi-continuous (l.s.c.); \item[2)] $N$ has nonempty closed and decomposable values. \end{itemize} \end{definition} Let $F: [0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ be a multivalued map with nonempty compact values. Assign to $F$ the multivalued operator $$\mathcal{F}: C([0,1],\mathbb{R})\to \mathcal{P}(L^1([0,1],\mathbb{R}))$$ by letting $$\mathcal{F}(x)=\{w\in L^1([0,1],\mathbb{R}): w(t)\in F(t, x(t)) \quad \hbox{for a.e. } t\in[0,1]\}.$$ The operator $\mathcal{F}$ is called the Nymetzki operator associated with $F$. \begin{definition} \rm Let $F: [0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ be a multivalued function with nonempty compact values. We say $F$ is of lower semi-continuous type (l.s.c. type) if its associated Nymetzki operator $\mathcal{F}$ is lower semi-continuous and has nonempty closed and decomposable values. \end{definition} Let $(X,d)$ be a metric space induced from the normed space $(X,|\cdot |)$. Consider $H_{d}:\mathcal{P}(X)\times \mathcal{P}(X)\to\mathbb{R}_{+}\cup\{\infty\}$ given by $$H_{d}(A,B)=\max\{\sup_{a\in A}d(a,B),\sup_{b\in B}d(A,b)\},$$ where $d(A,b)=\inf_{a\in A}d(a,b), \ d(a,B)=\inf_{b\in B}d(a,b)$. Then $( P_{b,cl}(X),H_{d})$ is a metric space and $(P_{cl}(X),H_{d})$ is a generalized metric space (see \cite{Kis}). \begin{definition}\rm A multivalued operator $N:X\to P_{cl}(X)$ is called \begin{itemize} \item[a)] $\gamma$-Lipschitz if and only if there exists $\gamma>0$ such that $$H_d(N(x),N(y))\leq \gamma d(x,y), \quad \hbox{for each} \ x,\ y\in X,$$ \item[b)] a contraction if and only if it is $\gamma$-Lipschitz with $\gamma<1$. \end{itemize} \end{definition} \begin{lemma}[\cite{LaOp}] \label{l1} Let $X$ be a Banach space. Let $F:[0,1]\times X\to P_{cp,c}(X)$ be an $L^1$-Carath\'eodory multivalued map and let $\Gamma$ be a linear continuous mapping from $L^1 ([0,1],X)$ to $C([0,1],X)$, then the operator \begin{gather*} \Gamma \circ S_{F}:C([0,1],X) \to P_{cp,c}(C([0,1],X)),\\ x \mapsto (\Gamma \circ S_{F})(x):=\Gamma(S_{F,x}) \end{gather*} is a closed graph operator in $C([0,1],X)\times C([0,1],X)$. \end{lemma} \begin{lemma}[\cite{BrCo}]\label{l2} Let $Y$ be a separable metric space and $N: Y\to \mathcal{P}(L^1([0,1],\mathbb{R}))$ be a multivalued operator which has property (BC). Then $N$ has a continuous selection; i.e., there exists a continuous function (single-valued) $g:Y\to L^1([0,1],\mathbb{R})$ such that $g(x)\in N(x)$\ for every $x\in Y$. \end{lemma} \begin{lemma}[\cite{CoNa}]\label{l4} Let $(X,d)$ be a complete metric space. If $N: X\to P_{cl}(X)$ is a contraction, then $Fix N \not= \emptyset$. \end{lemma} \section{Main Results} In this section, we are concerned with the existence of solutions for the problem (\ref{e1})-(\ref{e3}) when the right hand side has convex as well as nonconvex values. Initially, we assume that $F$ is a compact and convex valued multivalued map. \begin{definition} \rm A function $x\in AC^1((0,1),\mathbb{R})$ is said to be a solution of (\ref{e1})-(\ref{e3}) if there exists a function $v\in L^1([0,1],\mathbb{R})$ with $v(t)\in F(t,x(t))$ for a.e. $t\in [0,1]$ such that $x''(t)= v(t)$ a.e. on $[0,1]$ and the function $x$ satisfies the conditions (\ref{e2}) and (\ref{e3}). \end{definition} We need the following auxiliary result. Its proof uses a standard argument. \begin{lemma}\label{l5} For any $\sigma (t)$, $\rho_1(t)$, $\rho_2(t)\in C([0,1],\mathbb{R})$, the nonhomogeneous linear problem \begin{gather*} x''(t)=\sigma(t), \quad \hbox{a.e. } t\in [0,1],\\ x(0)-k_1x'(0)= \int_0^1 \rho_1(s)ds, \\ x(1)+k_2x'(1)=\int_0^1 \rho_2(s)ds, \end{gather*} has a unique solution $x\in AC^1((0,1),\mathbb{R})$ given by $$x(t)=P(t)+\int_0^1 G(t,s)\sigma(s)ds,$$ where $$P(t)=\frac{1}{1+k_1+k_2}\{(1-t+k_2)\int_0^1 \rho_1(s)ds +(k_1+t)\int_0^1 \rho_2(s)ds\}$$ is the unique solution of the problem \begin{gather*} x''(t)=0,\quad \hbox{a.e. } t\in [0,1],\\ x(0)-k_1x'(0)=\int_0^1 \rho_1(s)ds, \\ x(1)+k_2x'(1)=\int_0^1 \rho_2(s)ds, \end{gather*} and $$G(t,s)=\frac{-1}{k_1+k_2+1}\begin{cases} (k_1+t)(1-s+k_2),& 0\leq t0 such that$$ \frac{M}{\frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}+\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\psi(M)\int_0^1 p(s)ds }>1. $$\end{itemize} \begin{theorem} \label{t1} Suppose hypotheses (H1)--(H4) are satisfied. Then the boundary-value problem (\ref{e1})-(\ref{e3}) has at least one solution. \end{theorem} \begin{proof} We transform (\ref{e1})-(\ref{e3}) into a fixed point problem. Consider the operator N:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R})) defined by$$ N(x)=\{h\in C([0,1],\mathbb{R}): h(t)=P(t)+\int_0^1G(t,s)v(s)ds, \; v\in S_{F,x}\}, $$where$$ P(t)=\frac{1}{1+k_1+k_2}\{(1-t+k_2)\int_0^1 h_1(x(s))ds+(k_1+t) \int_0^1 h_2(x(s))ds\}. $$\begin{remark} \rm Clearly, from Lemma \ref{l5}, the fixed points of N are solutions to (\ref{e1})-(\ref{e3}). \end{remark} We shall show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps. \smallskip \noindent {\bf Step 1:} N(x) is convex for each x\in C([0,1],\mathbb{R}). Indeed, if h_{1},\ h_{2} belong to N(x), then there exist v_{1}, v_{2}\in S_{F,x} such that for each t\in [0,1] we have$$ h_{i}(t)=P(t)+\int_0^1 G(t,s)v_i(s)ds, \, (i=1,2). $$Let 0\leq d\leq 1. Then, for each t\in [0,1] we have$$ (dh_{1}+(1-d)h_{2})(t)=P(t)+\int_0^1G(t,s)[dv_{1}(s)+(1-d)v_{2}(s)]ds. $$Since S_{F,x} is convex (because F has convex values), then$$ dh_{1}+(1-d)h_{2}\in N(x). $$\noindent {\bf Step 2}: {\em N maps bounded sets into bounded sets in C([0,1],\mathbb{R}).} Let B_{q}=\{x\in C([0,1],\mathbb{R}): \|x\|_{\infty}\leq q \} be a bounded set in C([0,1],\mathbb{R}) and x\in B_{q}, then for each h\in N(x), there exists v\in S_{F,x} such that$$ h(t)=P(t)+\int_0^1G(t,s)v(s)ds. From (H2) and (H3) we have \begin{align*} & |h(t)|\\ &\leq |P(t)|+\int_0^1|G(t,s)||v(s)|ds \\ &\leq |P(t)|+\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\int_0^1|v(s)|ds \\ &\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\} +\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\psi(q)\int_0^1 p(s)ds \\ &\leq \frac{1}{1+k_1+k_2}\{(1+k_2)c_1+(k_1+1)c_2\}+\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\psi(q)\|p\|_{L^1}. \end{align*} {\bf Step 3}: {\em N maps bounded sets into equicontinuous sets of C([0,1],\mathbb{R}).} Let r_{1}, r_{2}\in [0,1], r_{1}From (H6) it follows that H_d(F(t,x(t)), F(t,\overline x(t)))\leq l(t)|x(t)-\overline x(t)|. $$Hence, there exists w\in F(t,\overline x(t)) such that$$ |v_{1}(t)-w|\leq l(t)|x(t)-\overline x(t)|, \ t\in [0,1]. $$Consider U:[0,1]\to \mathcal{P}(\mathbb{R}) given by$$ U(t)=\{w\in \mathbb{R}: |v_{1}(t)-w|\leq l(t)|x(t)-\overline x(t)|\}. $$Since the multivalued operator V(t)=U(t)\cap F(t,\overline x(t)) is measurable \cite[Proposition III.4]{CaVa}, there exists a function v_{2}(t) which is a measurable selection for V. So, v_{2}(t)\in F(t,\overline x(t)) and for each t\in [0,1]$$ |v_{1}(t)-v_{2}(t)|\leq l(t)|x(t)-\overline x(t)|. $$Let us define for each t\in [0,1]$$ h_{2}(t)=\bar P(t)+\int_0^1G(t,s)v_2(s)ds, $$where$$ \bar P(t)=\frac{1}{1+k_1+k_2}[(1-t+k_2)\int_0^1h_1(\overline x(s))ds +(1+k_1)\int_0^1h_2(\overline x(s))ds]. We have \begin{align*} &|h_{1}(t)-h_{2}(t)|\\ &\leq |P(t)-\bar P(t)|+\int_0^1 |G(t,s)||v_1(s)-v_2(s)|ds \\ &\leq\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2]\|x-\bar x\|_{\infty} +\int_0^1 |G(t,s)|l(s)|x(s)-\overline x(s)|ds. \end{align*} Thus, \begin{align*} &\|h_{1}-h_{2}\|_{\infty}\\ &\leq \Big(\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2] +\sup_{(t,s)\in [0,1]\times [0,1] }|G(t,s)|\|l\|_{L^1}\Big)\|x-\overline x\|_{\infty}. \end{align*} By an analogous relation, obtained by interchanging the roles of x and \overline x, it follows that \begin{align*} &H_d(N(x),N(\overline x))\\ &\leq \Big(\frac{1}{1+k_1+k_2}[(1+k_1)d_1+(1+k_2)d_2] +\sup_{(t,s)\in [0,1]\times [0,1]}|G(t,s)|\|l\|_{L^1}\Big) \|x-\overline x\|_{\infty}. \end{align*} So, N is a contraction and thus, by Lemma \ref{l4}, N has a fixed point x which is solution to (\ref{e1})-(\ref{e3}). \end{proof} In this part, by using the nonlinear alternative of Leray Schauder type combined with the selection theorem of Bresssan and Colombo for semi-continuous maps with decomposable values, we shall establish an existence result for the problem (\ref{e1})-(\ref{e3}). We need the following hypothesis: \begin{itemize} \item[(H8)] F:[0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R}) is a nonempty compact-valued multivalued map such that: \\ a) (t,x)\mapsto F(t,x) is \mathcal{L}\otimes\mathcal{ B} measurable; \\ b)x\mapsto F(t,x) is lower semi-continuous for each t\in [0,1]; \end{itemize} The following lemma is of great importance in the proof of our next result. \begin{lemma}[\cite{FrGr1}] \label{l6} Let F: [0,1]\times \mathbb{R}\to\mathcal{P}(\mathbb{R}) be a multivalued map with nonempty compact values. Assume (H3) and (H8) hold. Then F is of l.s.c. type. \end{lemma} \begin{theorem}\label{t3} Assume that (H2), (H3), (H4) and (H8) hold. Then the BVP (\ref{e1})-(\ref{e3}) has at least one solution. \end{theorem} \begin{proof} Note that (H3), (H8) and Lemma \ref{l6} imply that F is of l.s.c. type. Then from Lemma \ref{l2}, there exists a continuous function f: C([0,1],\mathbb{R})\to L^1([0,1],\mathbb{R}) such that f(x)\in \mathcal{F}(x) for all x\in C([0,1],\mathbb{R}). Consider the problem \begin{gather}\label{eq1} x''(t)= f(x(t)), \quad \hbox{a.e. } t\in [0,1], \\ \label{eq2} x(0)-k_1x'(0)=\int_0^1 h_1(x(s)) ds, \\ \label{eq3} x(1)+k_2x'(1)=\int_0^1 h_2(x(s))ds. \end{gather} It is clear that if x\in AC^1 ((0,1),\mathbb{R}) is a solution of (\ref{eq1})-(\ref{eq3}), then x is a solution to the problem (\ref{e1})-(\ref{e3}). Transform the problem (\ref{eq1})-(\ref{eq3}) into a fixed point theorem. Consider the operator \bar N defined by \bar N(x)(t)=P(t)+\int_0^1G(t,s)f(x(s))ds.  We can easily show that $\bar N$ is continuous and completely continuous. The remaning of the proof is similar to that of Theorem \ref{t1}. \end{proof} \begin{thebibliography}{99} \bibitem{AuCe} J. P. Aubin and A. 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