\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 66, 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} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/66\hfil Multiple nonnegative solutions] {Multiple nonnegative solutions for second-order boundary-value problems with sign-changing nonlinearities} \author[S. Xi, M. Jia, H. Ji \hfil EJDE-2009/66\hfilneg] {Shouliang Xi, Mei Jia, Huipeng Ji} % in alphabetical order \address{College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China} \email[S.Xi]{xishouliang@163.com} \email[M.Jia]{jiamei-usst@163.com} \email[H.Ji]{jihuipeng1983@163.com} \thanks{Submitted December 8, 2008. Published May 14, 2009.} \thanks{Supported by grant 05EZ53 from the Foundation of Educational Commission of Shanghai.} \subjclass[2000]{34B10, 34B18} \keywords{Nonnegative solutions; fixed-point theorem in double cones; \hfill\break\indent integral kernel; integral boundary conditions} \begin{abstract} In this paper, we study the existence of multiple nonnegative solutions for second-order boundary-value problems of differential equations with sign-changing nonlinearities. Our main tools are the fixed-point theorem in double cones and the Leggett-Williams fixed point theorem. We present also the integral kernel associated with the boundary-value problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Boundary-value problems with nonnegative solutions describe many phenomena in the applied science, and they are widely used in fields, such as chemistry, biological, etc.; see for example \cite{c2,g1,j1,j2,j3,k1}. Problems with integral boundary conditions have been applied in heat conduction, chemical engineering, underground water flow-elasticity, etc. The existence of nonnegative solutions to these problems have received a lot of attention; see \cite{f1,k1,k2,k3,k4,k5} and reference therein. Recently, by constructing a special cone and using the fixed point index theory, Liu and Yan \cite{k2} proved the existence of multiple solutions to the singular boundary-value problem \begin{gather*} (p(t)x'(t))'+\lambda f(t,x(t),y(t))=0\\ (p(t)y'(t))'+\lambda g(t,x(t),y(t))=0\\ \alpha x(0)-\beta x'(0)=\gamma x(1)+\delta x'(1)=0\\ \alpha y(0)-\beta y'(0)=\gamma y(1)+\delta y'(1)=0, \end{gather*} where the parameter $\lambda$ in $\mathbb{R}^+$, $p\in C([0,1],\mathbb{R}^+)$, $\alpha, \beta, \gamma, \delta \geq 0$, $\beta\gamma+\alpha\delta+\alpha\gamma>0$, $f\in C((0,1) \times \mathbb{R}^+ \times \mathbb{R}, \mathbb{R}^+)$, $g\in C((0,1) \times \mathbb{R}^+ \times \mathbb{R}, \mathbb{R})$, but $g$ must be controlled by $f$. By using fixed point index theory in a cone, Yang \cite{k3} studied the existence of positive solutions to a system of second-order nonlocal boundary value problems \begin{gather*} -u''=f(t,u,v)\\ -v''=g(t,u,v)\\ u(0)=v(0)=0\\ u(1)=H_1 \Big(\int_0^1 u(\tau)d\alpha (\tau)\Big)\\ v(1)=H_2 \Big(\int_0^1 v(\tau)d\beta (\tau)\Big), \end{gather*} where $\alpha$ and $\beta$ are increasing nonconstant functions defined on $[0, 1]$ with $\alpha (0)=0=\beta (0)$ and $f, g\in C((0,1) \times \mathbb{R}^+ \times \mathbb{R}^+, \mathbb{R}^+)$, $H_i\in C(\mathbb{R}^+, \mathbb{R}^+)$. By using fixed point theory in a cone, Feng \cite{k5} studied positive solutions for the boundary-value problem, with integral boundary conditions in Banach spaces, \begin{equation*} x''+f(t,x)=0\\ \end{equation*} with \begin{equation*} x(0)=\int_0^1 g(t)x(t)dt, \quad x(1)=0\\ \end{equation*} or \begin{equation*} x(0)=0, x(1)=\int_0^1 g(t)x(t)dt, \end{equation*} where $f\in C([0,1]\times P, P), g\in L^1[0,1]$, and $P$ is a cone of $E$. All of these, we can find the nonlinear term $f$ is nonnegative. In this paper, by using the fixed point theorem in double cones and the Leggett-Williams fixed point theorem, we study the existence of multiple nonnegative solutions to the \ boundary value problem $$\label{e1.1} \begin{gathered} u_1''(t)+f_1(t,u_1(t),u_2(t))=0\\ u_2''(t)+f_2(t,u_1(t),u_2(t))=0\\ u_1(0)=u_2(0)=0\\ u_1(1)=\int_0^1 g_1(s)u_1(s)ds,u_2(1)=\int_0^1 g_2(s)u_2(s)ds, \end{gathered}$$ where $f_1, f_2\in C((0,1) \times \mathbb{R}^+ \times \mathbb{R}^+, \mathbb{R})$, and $g_1, g_2$ are nonnegative functions in $L^1 [0,1]$. In this paper we assume that the following conditions: \begin{itemize} \item[(H1)] $f_i\in C((0,1) \times \mathbb{R}^+ \times \mathbb{R}^+, \mathbb{R})$, $g_i\in L^1 [0,1]$ is nonnegative, $i=1,2$; \item[(H2)] $1-\int_0^1sg_i(s)ds>0$; \item[(H3)] $f_1(t,0,u_2(t))\geq 0(\not\equiv0)$, $f_2(t,u_1(t),0)\geq 0(\not\equiv0)$, $t\in[0,1]$. \end{itemize} \section{Preliminaries} Let $X$ be a Banach space with norm $\|\cdot\|$ and $K\subset X$ be a cone. For a constant $r>0$, denote $K_r=\{x\in K: \|x\|a, \alpha(x)a>0$ such that \begin{itemize} \item[(C1)] $\|Tx\|b$ for $x\in \partial K'(b)$; \item[(C3)] $Tx=T^*x$, for $x\in K'_{a}(b)\cap\{u:T^*u=u\}$. \end{itemize} Then $T$ has at least two fixed points $y_1$ and $y_2$ in $K$, such that  0\leq \|y_1\|a\} \neq \emptyset $and$\phi (Ax)>a$for$x\in K(\phi ,a,b)$; \item[(C5)]$\| Ax\| a $for$x\in K(\phi ,a,c)$with$\| Ax\| >b$. \end{itemize} Then$A$has at least three fixed points$x_1$,$x_2$,$x_3$in$\overline{K_c}satisfying \| x_1\| d, \quad \phi (x_3)0, there exist P_i (x_i ,y_i )\in X,i=1,2,\dots, m, such that A D\subset\cup_{i=1}^{m}B(P_i,\epsilon), where B(P_i,\epsilon):=\{(u_1,u_2)\in K: \|u_1-x_i\|+\|u_2-y_i\|<\epsilon\}. Then for any Q^{*}(x^{*}_Q , y^{*}_Q)\in (\theta \circ A)(D), there exists Q(x_Q ,y_Q )\in AD, such that (x^{*}_Q , y^{*}_Q )=(\max\{x_Q ,0\},\max\{y_Q ,0\}). We choose a P_i \in \{P_1, P_2,\dots,P_m\}, such that \|x_Q-x_i\|+\|y_Q-y_i\|<\epsilon. Since \|x^{*}_Q-x^{*}_i\|+\|y^{*}_Q-y^{*}_i\|\leq \|x_Q-x_i\|+\|y_Q-y_i\|<\epsilon, we have Q^{*}(x^{*}_Q , y^{*}_Q )\in B(P^{*}_i,\epsilon), and so (\theta \circ A)(D) is relatively compact. For each \epsilon>0, there exists \eta>0, such that \|A(x_1,y_1)-A(x_2,y_2)\|<\epsilon, for \|x_1-x_2\|+\|y_1-y_2\|<\eta. Since \begin{align*} &\|(\theta \circ A)(x_1,y_1)-(\theta \circ A)(x_2,y_2)\|\\ &=\|\Big(\max\{A_1(x_1,y_1),0\}-\max\{A_1(x_2,y_2),0\}, \\ &\quad \max\{A_2(x_1,y_1),0\}-\max\{A_2(x_2,y_2),0\}\Big)\|\\ &\leq\|A(x_1,y_1)-A(x_2,y_2)\|<\epsilon. \end{align*} We have \|(\theta \circ A)(x_1,y_1)-(\theta \circ A)(x_2,y_2)\|<\epsilon, for \|x_1-x_2\|+\|y_1-y_2\|<\eta. Hence, \theta \circ A is continuous in K and \theta \circ A is completely continuous. The proof is complete. \end{proof} Since f_i is continuous, it is clear that A:K\to X and T^{*}:K '\to X are completely continuous. From Lemmas \ref{lem2.6} and \ref{lem2.5}, we have T:K\to K and T^*:K'\to K' are completely continuous. \begin{lemma}\label{lem2.7} If (u_1,u_2) is a fixed point of T, then (u_1,u_2) is a fixed point of A. \end{lemma} \begin{proof} Suppose (u_1,u_2) is a fixed point of T, obviously, we just need to prove that A_i (u_1,u_2)(t)\geq 0, i=1,2, for t\in [0,1]. If there exist t_0\in (0,1) and an i (i=1,2) such that u_i(t_0)=T_i (u_1,u_2)(t_0)=0 but A_i(u_1,u_2)(t_0)<0. Without loss of generalization, let i=1 and (t_1,t_2) be the maximal interval and contains t_0 such that A_1(u_1,u_2)(t)<0 for all t\in(t_1,t_2). Obviously, (t_1,t_2)\neq(0,1). Or else, T_1(u_1,u_2)(t)=u_1(t)=0, for all t\in [0,1]. This is in contradiction with (H3). \textbf{Case i:} If t_2<1, then A_1(u_1,u_2)(t_2)=0. Thus, A_1'(u_1,u_2)(t_2)\geq 0, We obtain A_1''(u_1,u_2)(t)=-f_1(t,0,u_2)\leq 0, \quad\text{for } t\in (t_1,t_2). So A_1'(u_1,u_2)(t)\geq 0, \quad\text{for } t\in [t_1,t_2] We obtain t_1=0, and A_1'(u_1,u_2)(0)\geq 0, A_1(u_1,u_2)(0)<0. This is in contradiction with A_1(u_1,u_2)(0)=0. \textbf{Case ii:} If t_1>0, we have A_1(u_1,u_2)(t_1)=0. Thus A_1'(u_1,u_2)(t_1)\leq 0. We obtain A_1''(u_1,u_2)(t)=-f_1(t,0,u_2)\leq 0, \quad\text{for } t\in (t_1,t_2). So A_1'(u_1,u_2)(t)<0, \quad\text{for } t\in [t_1,t_2]. We obtain t_2=1, A_1'(u_1,u_2)(1)\leq 0. On the other hand, A_1(u_1,u_2)(t)<0, for t\in (t_1,t_2), A_1'(u_1,u_2)(1)\leq 0 imply A_1(u_1,u_2)(1)<0. By (H1), A_1(u_1,u_2)(1)=\int_0^1 g_1(s)u_1(s)ds\geq 0. This is a contradiction. The proof is complete. \end{proof} \section{Main result} Denote M_i=\max_{t\in[0,1]}\int_{0}^{1}H_i(t,s)ds,\quad m_i=\min_{t\in[\delta,1-\delta]}\int_{\delta}^{1-\delta}H_i(t,s)ds, i=1, 2 \begin{theorem}\label{thm3.1} Suppose that condition {\rm (H1)--(H3)} hold. Assume that there exist positive numbers \delta, a, b, \lambda_i, \mu_i, i=1, 2, such that \delta\in (0,\frac{1}{2}), 01, and satisfy \begin{itemize} \item[(H4)] f_i(t,u_1,u_2)\geq0, for t\in[0,1], u_1+u_2\in[0,b]; \item[(H5)] f_i(t,u_1,u_2)<\frac{\lambda_ia}{M_i}, for t\in [0,1], u_1+u_2\in[0,a]; \item[(H6)] f_i(t,u_1,u_2)\geq\frac{\mu_i\delta b}{m_i}, for t\in[\delta,1-\delta], u_1+u_2\in[\delta b, b]. \end{itemize} Then, \eqref{e1.1} has at least two nonnegative solutions (u_1,u_2) and (u_1^{*},u_2^{*}) such that 0\leq \|(u_1,u_2)\|\delta b. \end{align*} Therefore (C2) of Theorem \ref{thm2.1} is satisfied. Finally, we show that (C3) of Theorem \ref{thm2.1} is satisfied. Let (u_1,u_2)\in K'_a(\delta b)\cap\{(u_1,u_2): T^*(u_1,u_2)=(u_1,u_2)\}, we have \alpha(u_1,u_2)<\delta b, \|(u_1,u_2)\|>a. From Lemma \ref{lem2.5}, we know that \begin{gather*} \|(u_1,u_2)\|\leq \frac{1}{\delta}\alpha(u_1,u_2)0, i=1, 2, such that 01, and {\rm (H5), (H6)} hold, and satisfy \begin{itemize} \item[(H7)] f_i(t,u_1,u_2)\geq 0, for t\in[0,1], u_1+u_2\in[\delta b,b]. \item[(H8)] f_i(t,u_1,u_2)\leq \frac{\lambda_ib}{M_i}, for t\in [0,1], u_1+u_2\in[0,b]. \end{itemize} Then, \eqref{e1.1} has at least three nonnegative solutions (u_1,u_2), (u_1^*,u_2^*), (u_1^{**},u_2^{**}), such that 0\leq \|(u_1,u_2)\|\delta b\}\neq \emptyset. Assume (u_1,u_2)\in K(\phi, \delta b, b), for any t\in [\delta,1-\delta], we have \delta b\leq u_1+u_2\leq b. From (H6) and (H7) we obtain \begin{align*} \phi(T(u_1,u_2)) &=\min_{t\in[\delta,1-\delta]} \Big(\int_0^1H_1(t,s)f_1(s,u_1(s),u_2(s))ds\Big)^{+}\\ &\quad +\min_{t\in[\delta,1-\delta]} \Big(\int_0^1H_2(t,s)f_2(s,u_1(s),u_2(s))ds\Big)^{+}\\ &\geq \min_{t\in[\delta,1-\delta]}\int_{0}^{1} H_1(t,s)f_1(s,u_1(s),u_2(s))ds\\ &\quad +\min_{t\in[\delta,1-\delta]}\int_{0}^{1} H_2(t,s)f_2(s,u_1(s),u_2(s))ds\\ &\geq \min_{t\in[\delta,1-\delta]}\int_{\delta}^{1-\delta}H_1(t,s)f_1(s,u_1(s),u_2(s))ds\\ &\quad +\min_{t\in[\delta,1-\delta]}\int_{\delta}^{1-\delta}H_2(t,s)f_2(s,u_1(s),u_2(s))ds\\ &\geq \frac{\mu_1\delta b}{m_1}\min_{t\in[\delta,1-\delta]}\int_{\delta}^{1-\delta} H_1(t,s)ds+\frac{\mu_2\delta b}{m_2}\min_{t\in[\delta,1-\delta]}\int_{\delta}^{1-\delta} H_2(t,s)ds\\ &=\mu_1\delta b+\mu_2\delta b>\delta b. \end{align*} Finally, for (u_1,u_2)\in K(\phi, \delta b, b) and \|T(u_1,u_2)\|>b, it is easy to prove that \[ \phi(T(u_1,u_2))\geq \delta \|T(u_1,u_2)\|>\delta b. Then (C6) of Theorem \ref{thm2.2} is satisfied. Therefore from Theorem \ref{thm2.2} and Lemma \ref{lem2.7} we know that \eqref{e1.1} has at least three nonnegative solutions(u_1,u_2)$,$(u_1^*,u_2^*)$,$(u_1^{**},u_2^{**})\$, such that \[ 0\leq \|(u_1,u_2)\|