\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 11, 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} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2011/11\hfil Existence of positive solutions] {Existence of positive solutions for self-adjoint boundary-value problems with integral boundary conditions at resonance} \author[A. Yang, B. Sun, W. Ge\hfil EJDE-2011/11\hfilneg] {Aijun Yang, Bo Sun, Weigao Ge} % not in alphabetical order \address{Aijun Yang \newline College of Science, Zhejiang University of Technology, Hangzhou, Zhejiang, 310032, China} \email{yangaij2004@163.com} \address{Bo Sun \newline School of Applied Mathematics, Central University of Finance and Economics, Beijing, 100081, China} \email{sunbo19830328@163.com} \address{Weigao Ge \newline Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, China} \email{gew@bit.edu.cn} \thanks{Submitted September 29, 2010. Published January 20, 2011.} \thanks{Supported by grant 11071014 from NNSF of China, by the Youth PhD Development Fund \hfill\break\indent of CUFE 121 Talent Cultivation Project} \subjclass[2000]{34B10, 34B15, 34B45} \keywords{Boundary value problem; resonance; cone; positive solution; \hfill\break\indent coincidence} \begin{abstract} In this article, we study the self-adjoint second-order boundary-value problem with integral boundary conditions, \begin{gather*} (p(t)x'(t))'+ f(t,x(t))=0,\quad t\in (0,1),\\ p(0)x'(0)=p(1)x'(1),\quad x(1)=\int_0^1x(s)g(s)ds, \end{gather*} which involves an integral boundary condition. We prove the existence of positive solutions using a new tool: the Leggett-Williams norm-type theorem for coincidences. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} This paper concerns the existence of positive solutions to the following boundary value problem at resonance: \begin{gather} (p(t)x'(t))'+ f(t,x(t))=0,\quad t\in (0,1), \label{e1.1}\\ p(0)x'(0)=p(1)x'(1),\quad x(1)=\int_0^1x(s)g(s)ds,\label{e1.2} \end{gather} where $g\in L^1[0,1]$ with $g(t)\geq0$ on $[0,1]$, $\int_0^1g(s)ds=1$, $p\in C[0,1]\cap C^1(0,1)$, $p(t)>0$ on $[0,1]$. Recently much attention has been paid to the study of certain nonlocal boundary value problems (BVPs). The methodology for dealing with such problems varies. For example, Kosmatov \cite{k1} applied a coincidence degree theorem due to Mawhin and obtained the existence of at least one solution of the BVP at resonance \begin{gather*} u''(t)=f(t,u(t),u'(t)),~~t\in (0,1),\\ u'(0)=u'(\eta),\quad \sum_{i=1}^{n}\alpha_iu(\eta_i)=u(1), \end{gather*} under the assumptions $\sum_{i=1}^{n}\alpha_i=1$ and $\sum_{i=1}^{n}\alpha_i\eta_i=1$. Han \cite{h1} studied the three-point BVP at resonance \begin{gather*} x''(t)=f(t,x(t)),\quad t\in (0,1),\\ x'(0)=0,\quad x(\eta)=x(1). \end{gather*} The author rewrote the original BVP as an equivalent problem, and then used the Krasnolsel'skii-Gue fixed point theorem. Although the existing literature on solutions of BVPs is quite wide, to the best of our knowledge, only a few papers deal with the existence of positive solutions to multi-point BVPs at resonance. In particular, there has been no work done for the BVP \eqref{e1.1}-\eqref{e1.2}. Moreover, Our main approach is different from the ones existing and our main ingredient is the Leggett-Williams norm-type theorem for coincidences obtained by O'Regan and Zima \cite{o1}. \section{Related Lemmas} For the convenience of the reader, we review some standard facts on Fredholm operators and cones in Banach spaces. Let $X$, $Y$ be real Banach spaces. Consider a linear mapping $L:\operatorname{dom}L\subset X\to Y$ and a nonlinear operator $N:X\to Y$. Assume that \begin{itemize} \item[(A1)] $L$ is a Fredholm operator of index zero; that is, $\operatorname{Im}L$ is closed and\\ $\operatorname{dim}\ker L=\operatorname{codim\, Im}L<\infty$. \end{itemize} This assumption implies that there exist continuous projections $P:X\to X$ and $Q:Y\to Y$ such that $\operatorname{Im}P=\ker L$ and $\ker Q=\operatorname{Im}L$. Moreover, since $\text{dim Im}Q=\operatorname{codim\ Im}L$, there exists an isomorphism $J:\operatorname{Im}Q\to \ker L$. Denote by $L_p$ the restriction of $L$ to $\ker P\cap \operatorname{dom}L$. Clearly, $L_p$ is an isomorphism from $\ker P\cap \operatorname{dom}L$ to $\operatorname{Im}L$, we denote its inverse by $K_p:\operatorname{Im}L\to \ker P\cap \operatorname{dom}L$. It is known (see \cite{m1}) that the coincidence equation $Lx=Nx$ is equivalent to $$x=(P+JQN)x+K_P(I-Q)Nx.$$ Let $C$ be a cone in $X$ such that \begin{itemize} \item[(i)] $\mu x\in C$ for all $x\in C$ and $\mu\geq 0$, \item[(ii)] $x,-x\in C$ implies $x=\theta$. \end{itemize} It is well known that $C$ induces a partial order in $X$ by $$x\preceq y \quad \text{if and only if}\quad y-x\in C.$$ The following property is valid for every cone in a Banach space $X$. \begin{lemma}[\cite{p1}] \label{lem2.1} Let $C$ be a cone in $X$. Then for every $u\in C\setminus\{0\}$ there exists a positive number $\sigma(u)$ such that $$\|x+u\|\geq \sigma(u)\|u\| \quad\text{for all }x\in C.$$ \end{lemma} Let $\gamma:X\to C$ be a retraction; that is, a continuous mapping such that $\gamma (x)=x$ for all $x\in C$. Set $$\Psi:=P+JQN+K_p(I-Q)N\quad \text{and}\quad \Psi_\gamma:=\Psi\circ\gamma.$$ We use the following result due to O'Regan and Zima, with the following assumptions: \begin{itemize} \item[(A2)] $QN:X\to Y$ is continuous and bounded and $K_p(I-Q)N:X\to X$ be compact on every bounded subset of $X$, \item[(A3)] $Lx\neq \lambda Nx$ for all $x\in C\cap\partial\Omega_2\cap \emph{Im}L$ and $\lambda\in (0,1)$, \item[(A4)] $\gamma$ maps subsets of $\overline{\Omega}_2$ into bounded subsets of $C$, \item[(A5)] $\deg\{[I-(P+JQN)\gamma]|_{\ker L}, \ker L\cap\Omega_2,0\}\neq 0$, \item[(A6)] there exists $u_0\in C\setminus\{0\}$ such that $\|x\|\leq \sigma(u_0)\|\Psi x\|$ for $x\in C(u_0)\cap\partial\Omega_1$, where $C(u_0)=\{x\in C:\mu u_0\preceq x ~for~some ~\mu>0 \}$ and $\sigma(u_0)$ such that $\|x+u_0\|\geq \sigma(u_0)\|x\|$ for every $x\in C$, \item[(A7)] $(P+JQN)\gamma(\partial\Omega_2)\subset C$, \item[(A8)] $\Psi_\gamma(\overline{\Omega}_2\setminus\Omega_1)\subset C$. \end{itemize} \begin{theorem}[\cite{o1}] \label{thm2.1} Let $C$ be a cone in $X$ and let $\Omega_1$, $\Omega_2$ be open bounded subsets of $X$ with $\overline{\Omega}_1\subset\Omega_2$ and $C\cap(\overline{\Omega}_2\setminus\Omega_1)\neq \emptyset$. Assume that {\rm (A1)--(A8)} hold. Then the equation $Lx=Nx$ has a solution in the set $C\cap(\overline{\Omega}_2\setminus\Omega_1)$. \end{theorem} For simplicity of notation, we set $$\begin{gathered} \omega:=\int_0^1(\int_{s}^1\frac{1}{p(\tau)}d\tau)g(s)ds,\\ l(s):=\int_{s}^1\Big(\int_{\tau}^1\frac{1}{p(r)}dr\Big)g(\tau)d\tau + \int_{s}^1\frac{1}{p(\tau)}d\tau\int_0^{s}g(\tau)d\tau, \end{gathered}\label{e2.1}$$ and $G(t,s) =\begin{cases} \frac{1}{\omega}\big[\int_0^{s}(\int_{s}^1\frac{1}{p(r)}dr-\int_{\tau}^1\frac{r}{p(r)}dr)g(\tau)d\tau +\int_{s}^1\int_{\tau}^1\frac{1-r}{p(r)}drg(\tau)d\tau\big] \\ \times\big[\int_0^1\frac{\tau}{p(\tau)}d\tau -\int_{t}^1\frac{1}{p(\tau)}d\tau\big] +1+\int_0^1\frac{\tau^{2}}{p(\tau)}d\tau +\int_{t}^1\frac{1-\tau}{p(\tau)}d\tau -\int_{s}^1\frac{\tau}{p(\tau)}d\tau,\\ \quad \text{if }0\leq s < t\leq1,\\[4pt] \frac{1}{\omega}\big[\int_0^{s}(\int_{s}^1\frac{1}{p(r)}dr -\int_{\tau}^1\frac{r}{p(r)}dr)g(\tau)d\tau +\int_{s}^1\int_{\tau}^1\frac{1-r}{p(r)}drg(\tau)d\tau\big]\\ \times\big[\int_0^1\frac{\tau}{p(\tau)}d\tau -\int_{t}^1\frac{1}{p(\tau)}d\tau\big] +1+\int_0^1\frac{\tau^{2}}{p(\tau)}d\tau +\int_{s}^1\frac{1-\tau}{p(\tau)}d\tau -\int_{t}^1\frac{\tau}{p(\tau)}d\tau ,\\ \quad \text{if }0\leq t\leq s\leq1. \end{cases}$ Note that $G(t,s)\geq0$ for $t,s\in [0,1]$, and set $$\kappa:=\min\big\{1,\;\frac{1}{\max_{t,s\in[0,1]}G(t,s)}\big\}. \label{e2.2}$$ \section{Main result} To prove the existence result, we present here a definition. \begin{definition} \label{def3.1} \rm We say that the function $f:[0,1]\times\mathbb{R}\to \mathbb{R}$ satisfies the $L^1$-Carath\'eodory conditions, if \begin{itemize} \item[(i)] for each $u\in\mathbb{R}$, the mapping $t\mapsto f(t,u)$ is Lebesgue measurable on $[0,1]$, \item[(ii)] for a.e. $t\in [0,1]$, the mapping $u\mapsto f(t,u)$ is continuous on $\mathbb{R}$, \item[(iii)] for each $r>0$, there exists $\alpha_{r}\in L^1[0,1]$ satisfying $\alpha_{r}(t)>0$ on $[0,1]$ such that $$|u|\leq r\text{ implies }|f(t,u)|\leq \alpha_{r}(t).$$ \end{itemize} \end{definition} Now, we state our result on the existence of positive solutions for \eqref{e1.1}-\eqref{e1.2}. under the following assumptions: \begin{itemize} \item[(H1)] $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ satisfies the $L^1$-Carath\'eodory conditions, \item[(H2)] there exist positive constants $b_1, b_2, b_3, c_1,c_2, B$ with $$B>\frac{c_2}{c_1}+3(\frac{b_2c_2}{b_1c_1}+\frac{b_3}{b_1}) \int_0^1\frac{1+s}{p(s)}ds, \label{e3.1}$$ such that $-\kappa x\leq f(t,x),\quad f(t,x)\leq -c_1x+c_2,\quad f(t,x)\leq -b_1|f(t,x)|+b_2x+b_3$ for $t\in[0,1]$, $x\in[0,B]$, \item[(H3)] there exist $b\in (0,B)$, $t_0\in [0,1]$, $\rho\in (0,1]$, $\delta\in (0,1)$ and $q\in L^1[0,1]$, $q(t)\geq 0$ on $[0,1]$, $h\in C([0,1]\times(0,b],\mathbb{R}^{+})$ such that $f(t,x)\geq q(t)h(t,x)$ for $t\in[0,1]$ and $x\in(0,b]$. For each $t\in[0,1]$, $\frac{h(t,x)}{x^{\rho}}$ is non-increasing on $x\in(0,b]$ with $$\int_0^1G(t_0,s)q(s)\frac{h(s,b)}{b}ds\geq \frac{1-\delta}{\delta^{\rho}}.\label{e3.2}$$ \end{itemize} \begin{theorem} \label{thm3.1} Under assumptions {\rm (H1)--(H3)}, The problem \eqref{e1.1}-\eqref{e1.2} has at least one positive solution on $[0,1]$. \end{theorem} \begin{proof} Consider the Banach spaces $X=C[0,1]$ with the supremum norm $\|x\|=\max_{t\in[0,1]}|x(t)|$ and $Y=L^1[0,1]$ with the usual integral norm $\|y\|=\int_0^1|y(t)|dt$. Define $L: \operatorname{dom}L\subset X\to Y$ and $N:X\to Y$ with \begin{align*} \operatorname{dom}L=\big\{&x\in X: p(0)x'(0)=p(1)x'(1),\; x(1)=\int_0^1x(s)g(s)ds,\\\ & x,px'\in AC[0,1],\; (px')'\in L^1[0,1]\big\} \end{align*} with $Lx(t)=-(p(t)x'(t))'$ and $Nx(t)=f(t,x(t))$, $t\in[0,1]$. Then \begin{gather*} \ker L=\{x\in \operatorname{dom}L: x(t)\equiv c\text{ on }[0,1]\},\\ \operatorname{Im}L=\{y\in Y:\int_0^1y(s)ds=0\}. \end{gather*} Next, we define the projections $P:X\to X$ by $(Px)(t)=\int_0^1x(s)ds$ and $Q:Y\to Y$ by $$(Qy)(t)=\int _0^1y(s)ds.$$ Clearly, $\operatorname{Im}P= \ker L$ and $\ker Q=\operatorname{Im}L$. So $\operatorname{dim\,ker}L=1=\operatorname{dim\, Im}Q =\operatorname{codim\, Im}L$. Notice that $\operatorname{Im}L$ is closed, $L$ is a Fredholm operator of index zero; i.e. (A1) holds. Note that the inverse $K_p:\operatorname{Im}L\to \operatorname{dom}L\cap \ker P$ of $L_p$ is given by $$(K_py)(t)=\int_0^1k(t,s)y(s)ds,$$ where $$k(t,s):=\begin{cases} -\int_{s}^1\frac{\tau}{p(\tau)}d\tau+\frac{1}{\omega}l(s) \big[\int_0^1\frac{\tau}{p(\tau)}d\tau -\int_{t}^1\frac{1}{p(\tau)}d\tau\big]\\ +\int_{t}^1\frac{1}{p(\tau)}d\tau, &0\leq s\leq t\leq1,\\[4pt] -\int_{s}^1\frac{\tau}{p(\tau)}d\tau+\frac{1}{\omega}l(s) \big[\int_0^1\frac{\tau}{p(\tau)}d\tau -\int_{t}^1\frac{1}{p(\tau)}d\tau\big]\\ +\int_{s}^1\frac{1}{p(\tau)}d\tau, &0\leq t< s\leq1, \end{cases} \label{e3.3}$$ It is easy to see that $|k(t,s)|\leq 3\int_0^1\frac{1+s}{p(s)}ds$. Since $f$ satisfies the $L^1$-Carath\'eodory conditions, (A2) holds. Consider the cone $$C=\{x\in X: x(t)\geq 0\text{ on }[0,1]\}.$$ Let \begin{gather*} \Omega_1=\{x\in X: \delta\|x\|<|x(t)|0\text{ on }[0,1]\}$and we can take$\sigma(u_0)=1$. Let$x\in C(u_0)\cap \partial\Omega_1$. Then$x(t)>0$on$[0,1]$,$0<\|x\|\leq b$and$x(t)\geq \delta \|x\|$on$[0,1]$. For every$x\in C(u_0)\cap \partial\Omega_1, by (H3), we have \begin{align*} (\Psi x)(t_0) &= \int_0^1x(s)ds+\int_0^1G(t_0,s)f(s,x(s))ds\\ &\geq \delta\|x\|+\int_0^1G(t_0,s)q(s)h(s,x(s))ds\\ &= \delta\|x\|+\int_0^1G(t_0,s)q(s)\frac{h(s,x(s))}{x^{\rho}(s)} x^{\rho}(s)ds\\ &\geq \delta\|x\|+\delta^{\rho}\|x\|^{\rho}\int_0^1G(t_0,s) q(s)\frac{h(s,b)}{b^{\rho}}ds\\ &= \delta\|x\|+\delta^{\rho}\|x\|\cdot \frac{b^{1-\rho}}{\|x\|^{1-\rho}}\int_0^1G(t_0,s)q(s) \frac{h(s,b)}{b}ds\\ &\geq \delta\|x\|+\delta^{\rho}\|x\|\int_0^1G(t_0,s)q(s) \frac{h(s,b)}{b}ds \geq \|x\|. \end{align*} Thus,\|x\|\leq\sigma(u_0)\|\Psi x\|$for all$x\in C(u_0)\cap \partial\Omega_1$. By Theorem \ref{thm2.1}, the BVP \eqref{e1.1}-\eqref{e1.2} has a positive solution$x^{*}$on$[0,1]$with$b\leq\|x^{*}\|\leq B$. This completes the proof. \end{proof} \begin{remark} \label{rmk3.1} \rm Note that with the projection$P(x)=x(0)$, conditions (A7) and (A8) of Theorem \ref{thm2.1} are no longer satisfied. \end{remark} To illustrate how our main result can be used in practice, we present here an example. \subsection*{Example} Consider the problem $$\begin{gathered} (e^{54t}(1+t)x'(t))'+f(t,x(t))=0,\quad t\in (0,1),\\ x'(0)=2e^{54}x'(1),\quad x(1)=\int_0^12sx(s)ds. \end{gathered} \label{e3.7}$$ Corresponding to \eqref{e1.1}-\eqref{e1.2}, we have \begin{gather*} p(t)=e^{54t}(1+t),\quad g(t)=2t, \\ f(t,x)=\begin{cases} \sin (\pi x/2), &(t,x)\in [0,1]\times(-\infty,3),\\ 2-x, &(t,x)\in [0,1]\times[3,+\infty). \end{cases} \end{gather*} When$\kappa=1/2$, choose$c_1=1$,$c_2=3$,$b_1=1/2$,$b_2=3/2$,$b_3=9/2$,$B=4$and$b=1/2$,$t_0=0$,$\rho=1$,$\delta=1/2$,$q(t)=1-t$,$h(t,x)=\sin (\pi x/2)$. We can check that all the conditions of Theorem \ref{thm3.1} are satisfied, then the BVP \eqref{e3.7} has a positive solution on$[0,1]$. \begin{thebibliography}{00} \bibitem{d1} K. Deimling; \emph{Nonlinear Functional Analysis}. New York, 1985. \bibitem{g1} R. E. Gaines and J. Santanilla; \emph{A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations}. Rocky Mountain. J. Math., 12 (1982) 669-678. \bibitem{g2} W. Ge; \emph{Boundary value problems for ordinary nonlinear differential equations}, Science Press, Beijing, 2007. \bibitem{g3} D. Guo and V. Lakshmikantham; \emph{Nonlinear Problems in Abstract Cones}. New York, 1988. \bibitem{h1} X. Han; \emph{Positive solutions for a three-point boundary value problem at resonance}, J. Math. Anal. Appl., 336 (2007), 556-568. \bibitem{i1} G. Infante and M. Zima; \emph{Positive solutions of multi-point boundary value problems at resonance}, Nonlinear Analysis, 69 (2008), No. 8, 2458-2465. \bibitem{k1} N. Kosmatov; \emph{A multi-point boundary value problem with two critical conditions}, Nonlinear Anal., 65 (2006), 622-633. \bibitem{m1} J. Mawhin; \emph{Topological degree methods in nonlinear boundary value problems}, in NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1979. \bibitem{o1} D. O'Regan and M. Zima; \emph{Leggett-Williams norm-type theorems for coincidences}, Arch. Math., 87 (2006), 233-244. \bibitem{p1} W. V. Petryshyn; \emph{On the solvability of$x \in T x+\lambda Fx$in quasinormal cones with$T$and$Fk$-set contractive}, Nonlinear Anal., 5 (1981), 585-591. \bibitem{y1} A. J. Yang; \emph{An extension of Leggett-Williams norm-type theorem for coincidences and its applications} Topological Methods in Nonlinear Analysis, in press. \bibitem{y2} A. Yang and W. Ge; \emph{Positive solutions for boundary value problems of$N$-dimension nonlinear fractional differential system}, Boundary Value Problems, 2008, 437-453. \bibitem{y3} A. Yang and H. Wang; \emph{Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition}, E. J. Qualitative Theory of Diff. Equ., 1 (2011), 1-15. \end{thebibliography} \section*{Addendum posted on March 14, 2011} In response to comments from a reader, we want to make the following corrections: Page 2, Line 9: Delete the last sentence in the introduction: Moreover, \dots by O'Regan and Zima \cite{o1}". Then insert the following paragraph: Using the Legget-Williams norm-type theorem for coincidences, which is a tool introduced by O'Regan and Zima \cite{o1}, Infante and Zima \cite{i1} studied the multi-point boundary-value problem \begin{gather*} x''(t)=f(t,x(t))=0,\\ x'0)=0, \quad x(1)=\sum_{i=1}^{m-2} \alpha_i x(\eta_i)\,. \end{gather*} Inspired by the work in \cite{i1,o1}, we follow their steps, use the Legget-Williams norm-type theorem, and quote some of their results. \medskip Page 6, Line$-3$: Replace$b\leq \|x^*\|\leq B$by$\|x^*\|\leq B\$. \medskip The authors want to thank the anonymous reader for the suggestions. \end{document}