\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 98, pp. 1--8.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/98\hfil Positive solutions and continuous branches] {Positive solutions and continuous branches for boundary-value problems of differential inclusions} \author[N. T. Hoai, N. V. Loi\hfil EJDE-2007/98\hfilneg] {Nguyen Thi Hoai, Nguyen Van Loi} % in alphabetical order \address{Nguyen Thi Hoai \newline Faculty of mathematics \\ Voronezh State Pedagogical University, Russia} \email{nthoai0682@yahoo.com} \address{Nguyen Van Loi \newline Faculty of mathematics \\ Voronezh State Pedagogical University, Russia} \email{loitroc@yahoo.com} \thanks{Submitted February 16, 2007. Published July 13, 2007.} \subjclass[2000]{34B16, 34A60, 34B18, 47H04} \keywords{Boundary value problems; positive solutions; multivalued map; \hfill\break\indent differential inclusions} \begin{abstract} In this paper, we consider second order differential inclusions with periodic boundary conditions. We obtain the existence of positive solutions and of continuous branches of positive solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \section{Introduction} Consider the boundary-value problem \label{e1.1} \begin{gathered} Lu\in\lambda{F(t,u)},\quad 00$,$q\geq{0}$on$[0,1]$,$\alpha,\beta,\gamma,\delta\quad \geq{0}$with$\alpha\delta+\alpha\gamma+\beta\gamma\quad >0$,$F\colon [0,1]\times{[0,+\infty)\to {P([0,+\infty))}}$, and$\lambda$is a positive parameter. When$F$is a continuous map, the existence of positive solutions of \eqref{e1.1} was studied in \cite{Dang}. In this paper, the results in \cite{Dang,Loi} will be used to prove the existence of positive solutions of \eqref{e1.1}. First, we recall the following notion (see, e.g. \cite{BGMO,KOZ}). Let$X,Y$be two Banach spaces. Let$P(Y)$,$K(Y)$,$Kv(Y)$,$C(Y)$,$Cv(Y)$denote the collections of all nonempty, nonempty compact, nonempty convex compact, nonempty closed, nonempty convex closed subsets of$Y$, respectively. A multimap$F\colon{X}\to {P(Y)}$is said to be upper semicontinuous (u.s.c.) [lower semicontinuous (l.s.c.)] if the set$F_{+}^{-1}(V)=\{x\in{X}: {F(x)\subset{V}}\}$is open [respectively, closed] for every open [respectively, closed] subset$V\subset{Y}$.$F$is said to be compact if the set$F(X)$is relatively compact in$Y$. Let$A\subset{K(Y)}$and the max-normal and min-normal be $$\|A\|=\max\{\|x\| : x\in{A}\}\quad \text{and}\quad \|A\|_{0}=\min\{\|z\| : z\in{A}\}.$$ Let$C_{+}[0,1]\,(L_{+}^{1}[0,1])$denote the cone of all positive continuous (respectively, integrable) functions on$[0,1]$. We will consider the cone$C_{+}[0,1]\quad (L_{+}^{1}[0,1])$as subspace of the space$C[0,1]$(respectively,$L^{1}[0,1]$) with induced topology. The nonempty subset$M\subset{L_{+}^{1}[0,1]}$is said to be decomposable provided for every$f,g\in{M}$and each Lebesgue measurable subset$m\subset{[0,1]}$, $$f\chi_{m}+g\chi_{[0,1]\setminus{m}}\in{M},$$ where$\chi_{m}$is the characteristic function of the set$m$. \section{ Existence of positive solutions} Let$G(t,s)$be the Green's function for \eqref{e1.1}. Then$u$is a solution of \eqref{e1.1} if and only if $$u(t)\in\lambda\int_{0}^{1}G(t,s)F(s,u(s))ds.$$ Recall that $G(t,s)= \begin{cases} c^{-1}\phi(t)\psi(s) & \text{if t\leq{s}}\\ c^{-1}\phi(s)\psi(t) & \text{if s\leq{t}}, \end{cases}$ where$\phi$and$\psi$satisfy \begin{gather*} L{\phi}=0,\quad \phi(0)=\beta,\quad \phi'(0)=\alpha, \\ L{\psi}=0,\quad \psi(1)=\delta,\quad \psi'(1)=-\gamma \end{gather*} and$c=r(t)(\phi'(t)\psi(t)-\psi'(t)\phi(t))>0$. Note that$\phi'>0$on$(0,1]$and$\psi'<0$on$[0,1)$. Let$G=\max\{G(t,s):0\leq{t,s}\leq{1}\}$. We shall make the following assumptions: \begin{itemize} \item[(H1)] For every$x\in{[0,+\infty)}$the multifunction$F(\cdot ,x)\colon{[0,1]}\to {Kv([0,+\infty))}$has a measurable selection, i.e., there exists a measurable function$f$such that$f(t)\in{F(t,x)}$for a.e.$t\in{[0,1]}$; \item[(H2)] For a.e.$t\in{[0,1]}$the multimap$F(t,\cdot)\colon{[0,+\infty)}\to {Kv([0,+\infty))}$is u.s.c.; \item[(H3)] There exists a positive function$\omega\in{L^{1}[0,1]}$such that $$\|F(t,x)\|\leq\omega(s)(1+x),$$ for all$x\in{[0,+\infty)}$and a.e.$t\in{[0,1]}$; \item[(H4)] The multioperator$F\colon{[0,1]\times{[0,+\infty)}}\to {K([0,\infty))}$is almost lower semicontinuous; i.e., there exists a sequence of disjoint compact sets$\{I_m\}, {I_m}\subset{{[0,1]}}$such that: \begin{itemize} \item[(i)]$\mathop{\rm meas}([0,1]\setminus\bigcup_{m}I_{m})=0$; \item[(ii)] the restriction of$F$on each set${J_m}={I_m}\times{[0,\infty)}$is l.s.c.; \end{itemize} \end{itemize} We will use the method in \cite{Loi} to prove the following results. \begin{theorem} \label{thm1} Let (H1)--(H3) hold. If \eqref{e1.1} has no zero solution, then for each$0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$, \eqref{e1.1} has a positive solution. \end{theorem} \begin{theorem} \label{thm2} Let (H3)-(H4) hold. If \eqref{e1.1} has no zero solution, then for each$0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$, \eqref{e1.1} has a positive solution. \end{theorem} \begin{proof}[Proof of Theorem \ref{thm1}] From (H1)--(H3) it follows easily that the multioperator superposition \begin{gather*} \wp_{F}\colon{C_{+}[0,1]}\to {Cv(L_{+}^{1}[0,1])},\\ \wp_{F}(u)=\{f\in{L_{+}^{1}[0,1]}:f(s)\in{F(s,u(s))}\text{ for a.e. } s\in{[0,1]}\}.\notag \end{gather*} is defined and closed (see, e.g. \cite{BGMO}). Consider a completely continuous operator $Q_{\lambda}\colon{L_{+}^{1}[0,1]}\to {C_{+}[0,1]},\quad Q_{\lambda}(f)(t)=\lambda\int_{0}^{1}G(t,s)f(s)ds,$ Let$\Gamma_{\lambda}=Q_{\lambda}\circ\wp_{F}$. From \cite[Theorem 1.5.30]{BGMO} it follows that the multioperator$\Gamma_{\lambda}$is closed. We can easily prove that for every bounded subset$U\subset{C_{+}[0,1]}$, the set$\Gamma_{\lambda}(U)$is relatively compact in$C_{+}[0,1]$. Hence applying \cite[Theorem 1.2.48]{BGMO}, we have that the Hammerstein's multioperator \begin{gather*} \Gamma_{\lambda}\colon{C_{+}[0,1]}\to {Kv(C_{+}[0,1])},\\ \Gamma_{\lambda}(u)=\lambda\int_{0}^{1}G(t,s)F(s,u(s))ds. \end{gather*} is upper semicontinuous. Let$T_{+}=\{u\in{C_{+}[0,1]}:\|u\|_{C}\leq\rho, \text{ where }\rho>0\}$For$u$in${T_{+}}$we have $$\big\|\Gamma_{\lambda}(u)\big\|_{C} =\max \big\{\big\|\lambda\int_{0}^{1}G(t,s)f(s)ds \big\|_{C}:f\in{\wp_{F}(u)}\big\},$$ where $$\big\|\int_{0}^{1}G(t,s)f(s)ds\big\|_{C} =\sup_{t\in{[0,1]}} \big\{\int_{0}^{1}G(t,s)f(s)ds\big\}.$$ Since$f(s)\in{F(s,u(s))}$for a.e.$s\in{[0,1]}$and (H3), for a.e.$s\in{[0,1]}$we have $$f(s)\leq{\|F(s,u(s))\|}\leq\omega(s)(1+u(s))\leq\omega(s)(1+\|u\|_{C}) \leq{\omega(s)(1+\rho)}.$$ Therefore, $$\int_{0}^{1}G(t,s)f(s)ds\leq{G(1+\rho)\int_{0}^{1}\omega(s)ds},$$ and hence $$\big\|\int_{0}^{1}G(t,s)f(s)ds\big\|_{C}\leq {G(1+\rho)\int_{0}^{1}\omega(s)ds}.$$ Because the last inequality holds for all$f\in{\wp_{F}(u)}$, $${\|\Gamma_{\lambda}(u)\|}_{C}\leq\lambda{G(1+\rho)\int_{0}^{1}\omega(s)ds}.$$ Choose$\rho\geq\frac{\lambda{G\int_{0}^{1}\omega(s)ds}}{1-\lambda{G\int_{0}^{1} \omega(s)ds}}$then$\|\Gamma_{\lambda}(u)\|_{C}\leq\rho$, i.e.,$\Gamma_{\lambda}$maps the set$T_{+}$in to itself. The existence of positive solution of the problem \eqref{e1.1} can be easily follow from the Bohnenblust-Karlin fixed point theorem \end{proof} For the proof of Theorem \ref{thm2} we need the following result proved in \cite{Deim,HuPa}. \begin{lemma} \label{lem1} Let$X$be a separable metric space;$E$be a Banach space. Then every l.s.c. multimap$\tilde{F}\colon{X}\to {P(L^{1}([0,1],E))}$with closed decomposable values has a continuous selection. \end{lemma} \begin{proof}[Proof of theorem \ref{thm2}] From conditions (H3)--(H4) it follows that $$\wp_{F}\colon{C_{+}[0,1]}\to {C(L_{+}^{1}[0,1])}$$ is a l.s.c. multioperator with closed decomposable values (see, e.g. \cite{BGMO,KOZ}). Consider again the Hammerstein's multioperator$\Gamma_{\lambda}=Q_{\lambda}\circ\wp_{F}$. By Lemma \ref{lem1}, the multioperator superposition$\wp_{F}$has a continuous selection $\ell\colon{C_{+}[0,1]}\to {L_{+}^{1}[0,1]},\quad \ell(u)\in\wp_{F}(u).$ Hence the operator $\gamma_{\lambda}\colon{C_{+}[0,1]}\to {C_{+}[0,1]},\quad \gamma_{\lambda}(u)(t)=\lambda\int_{0}^{1}G(t,s)\ell(u)(s)ds,$ is a completely continuous selection of the multioperator$\Gamma_{\lambda}$. As shown above, for each$0<\lambda<\frac{1}{G\int_{0}^{1}\omega(s)ds}$, we can choose$\rho>0$such that the multioperator$\Gamma_{\lambda}$maps the set$T_{+}$in to itself. From the Schauder fixed theorem it follows that the operator$\gamma_{\lambda}$has a fixed point in$T_{+}$, i.e., \eqref{e1.1} has a positive solution \end{proof} Now we use the result in \cite{Dang} to prove the existence and multiplicity of positive solutions for \eqref{e1.1}, when$F$is lower semicontinuous. Assume that \begin{itemize} \item[(F1)]$F\colon{(0,1)\times{[0,+\infty)}}\to {Kv([0,+\infty))}$is l.s.c.; \item[(F2)] For each$M>0$, there exists a continuous function$g_{M}$on$(0,1)$such that$\|F(t,x)\|\leq{g_{M}(t)}$for$t\in{(0,1)}\text{,\quad }x\in{[0,M]}$, and $$\int_{0}^{1}G(s,s)g_{M}(s)ds<\infty.$$ \item[(F3)] There exist an interval$I\subset{(0,1)}$and a non-zero function$m\in{L^{1}(I)}$with$m\geq{0}$such that for every$b>0$, there exists$r_{b}>0$such that $$\|F(t,x)\|_{0}\geq bm(t)x \quad \text{for } t\in{I}, \; x\in{(0,r_{b})};$$ \item[(F4)] There exist an interval$I_{1}\subset{(0,1)}$and a non-zero function$m_{1}\in{L^{1}(I_{1})}$with$m_{1}\geq{0}$such that for every$c>0$, there exists$R_{c}>0$such that $$\|F(t,x)\|_{0}\geq c\,m_{1}(t)x \quad \text{for } t\in{I_{1}},\; x\geq{R_{c}};$$ \end{itemize} \begin{theorem} \label{thm3} Let (F1)--(F3) hold. Then there exists$\lambda_{0}>0$such that \eqref{e1.1} has a positive solution for$0<\lambda<\lambda_{0}$. If, in addition, (F4) holds, then \eqref{e1.1} has at least two positive solutions for$0<\lambda<\lambda_{0}$\end{theorem} For the proof of this we need the following result (see, e.g. \cite{BGMO,Mic}). \begin{lemma} \label{lem2} Let$X$be a metric space;$Y$be a Banach space. Then every l.s.c. multi-map$W\colon{X}\to {Cv(Y)}$has a continuous selection. \end{lemma} \begin{proof}[Proof of Theorem \ref{thm3}] Let$f\colon{(0,1)\times{[0,+\infty)}}\to {[0,+\infty)}$be a continuous selection of$F$, i.e., $$f(t,x)\in{F(t,x)}\quad \text{for all } (t,x)\in{(0,1)\times{[0,+\infty)}}.$$ It is easy to see that for all$(t,x)\in{(0,1)\times{[0,+\infty)}}$the following inequality holds $$\|F(t,x)\|_{0}\leq{f(t,x)}\leq{\|F(t,x)\|}.$$ Consider now the problem \label{e1.2} Lu=\lambda{f(t,u)},\quad 00$, there exists a continuous function $g_{M}$ on $(0,1)$ such that $f(t,x)\leq{g_{M}(t)}$ for $t\in{(0,1)}$, $0\leq{x}\leq{M}$ and $\int_{0}^{1}G(s,s)g_{M}(s)ds<\infty.$ \item[(f3)] There exist an interval $I\subset{(0,1)}$ and a non-zero function $m\in{L^{1}(I)}$ with $m\geq{0}$ such that for every $b>0$, there exists $r_{b}>0$ such that $f(t,x)\geq{bm(t)x},\quad \text{for } t\in{I},\; x\in{(0,r_{b})};$ \end{itemize} If $(F4)$ holds then we have \begin{itemize} \item[(f4)] There exist an interval $I_{1}\subset{(0,1)}$ and a non-zero function $m_{1}\in{L^{1}(I_{1})}$ with $m_{1}\geq{0}$ such that for every $c>0$, there exists $R_{c}>0$ such that $f(t,x)\geq{c\,m_{1}(t)x},\text{\quad for } t\in{I_{1}},\quad x\geq{R_{c}};$ \end{itemize} From \cite[Theorem 1.1]{Dang} it follows that if (f1)--(f3) hold then there exists $\lambda_{0}>0$ such that \eqref{e1.2} has a positive solution for $0<\lambda<\lambda_{0}$. If, in addition, $(f4)$ holds then \eqref{e1.2} has at least two positive solutions for $0<\lambda<\lambda_{0}$. Hence we obtain our result \end{proof} \section{Continuous branch of positive solutions} A sphere and a ball with center at the point $0$ (the zero function) and radius $r$ in the cone $C_{+}[0,1]$ will be denoted respectively by \begin{gather*} S_{+}(0,r)=\{u\in{C_{+}[0,1]} :\|u\|_{C}=r\},\\ T_{+}(0,r)=\{u\in{C_{+}[0,1]} :\|u\|_{C}\leq{r}\}. \end{gather*} Recall the following notion (see, \cite{Bakh1,Bakh2,Kras2}). \noindent\textbf{Definition} A set $V$ of positive solutions of \eqref{e1.1} is said to form a continuous branch connecting the spheres $S_{+}(0,r)$ and $S_{+}(0,R)$, with $0\leq{r}0. $$Then the positive solutions of the equation$$ Ax=\mu{x},\quad x\in{\mathbf{K}\setminus\{0\}} $$form a continuous branch with infinite length. \end{lemma} Let a be a positive constant. Consider now the problem \eqref{e1.1} with the multimap $F\colon{[0,1]\times{[0,+\infty)}}\to {K([a,+\infty))}$ satisfying the following assumptions: \begin{itemize} \item[(A1)] F is almost lower semicontinuous; \item[(A2)] For every nonempty bounded subset \Omega\subset{[0,+\infty)} there exists a function \vartheta_{\Omega}\in{L_{+}^{1}[0,1]} such that$$ \|F(t,x)\|\leq\vartheta_{\Omega}(t), $$for all x\in\Omega and a.e. t\in{[0,1]}; \item[(A3)] There exists q>0 such that the Green's function satisfies G(t,s)\geq{q}, for all 0\leq{t,s}\leq{1}; \end{itemize} \begin{theorem} \label{thm4} Let (A1)--(A3) hold. Then the positive solutions of \eqref{e1.1} form a continuous branch with infinite length. \end{theorem} \begin{proof} Note that the condition (H3) is special case of the condition (A2). As is shown above, from (A1)--(A2) the multioperator \Gamma_{\lambda} has a completely continuous selection \gamma_{\lambda} on the cone C_{+}[0,1]. Let \Xi\ni{0} be an open bounded subset of C_{+}[0,1]. For all u\in\Xi, since \ell(u)(s)\in{F(s,u(s))} for a.e. s\in{[0,1]} we have$$ \gamma_{\lambda}(u)(t)=\lambda\int_{0}^{1}G(t,s)\ell(u)(s)ds \geq{\lambda{aq}}>0. $$Hence$$ \inf_{u\in\partial\Xi}\|l(u)\|_{C}\geq{aq}>0,\quad \text{where } l=\frac{\gamma_{\lambda}}{\lambda}. $$On the cone C_{+}[0,1] consider the equation $$\label{e1.3} l(u)=\frac{1}{\lambda}{u}$$ By Lemma \ref{lem3}, the positive solutions of \eqref{e1.3} form a continuous branch with infinite length. And hence we obtain our result \end{proof} \section{Examples} \begin{example} \label{exa1} \rm Let D\subset{[0,1]} be a nonmeasurable set;$$ F\colon{[0,1]\times{[0,+\infty)}}\to {Kv([0,+\infty))} $$be the multimap $F(t,x)=\begin{cases} [0,x+1] & \text{if x=t and t\in{[0,1]\setminus{D}}}\\ [0,x+1] & \text{if x=t+1 and t\in{D}}\\ x+1 & \text{otherwise.} \end{cases}$ Consider the differential inclusion \label{e1.4} \begin{gathered} -u''(t)\in{\lambda\,F(t,u(t))},\quad \lambda>0,\quad 00, let g_{M}(t)=(M^{2}+\frac{1}{\varepsilon})(t+1). We have $\|F(t,x)\|\leq(t+1)(x^{2}+\frac{1}{x+\varepsilon})\leq{g_{M}(t)},$ for 00$$ \|F(t,x)\|_{0}=t(x^{2}+\frac{1}{1+x})\geq{b\,m(t)x}\quad \text{for } t\in{I},\; x\in{(0,r_{b})}, $$where r_{b}=\min\{\frac{-b+(b^{2}+4b)^{1/2}}{2b},1\}. The condition (F3) holds. For every c>0$$ \|F(t,x)\|_{0}\geq{t(x^{2}+\frac{1}{1+x})}\geq{c\,m(t)x},\quad \text{for }t\in{I},\;x\geq{c}.$$The condition (F4) holds. By Theorem \ref{thm3}, there exists$\lambda_{0}>0$such that \eqref{e1.5} has at least two positive solutions for$0<\lambda<\lambda_{0}$\end{example} \begin{example} \label{exa3} \rm Let$F\colon{[0,1]\times{[0,+\infty)}}\to {K([1,+\infty))}\$ be the multimap $F(t,x)= \begin{cases} (t^{2}+2)(x^{2}+\frac{1}{x+1}) & \text{if 0\leq{t}\leq{1}, 0\leq{x}\leq{1}}\\ (t+2)(x^{2}+\frac{1}{x+1}) & \text{if 0\leq{t}\leq{1}, 2\leq{x}\leq{3}}\\ [(t^{2}+2)(x^{2}+\frac{1}{1+x}),\,(t+2)(x^{2}+\frac{1}{x+1})] & \text{otherwise.} \end{cases}$ Consider the problem \label{e1.6} \begin{gathered} -(1+e^{t})u''-e^{t}u'\in{\lambda{F(t,u)}},\quad 0