\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 31, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/31\hfil Multiple positive solutions] {Multiple positive solutions for integro-differential equations with integral boundary conditions and sign changing nonlinearities} \author[M. Jia, P. Wang \hfil EJDE-2012/31\hfilneg] {Mei Jia, Pingyou Wang} % in alphabetical order \address{Mei Jia \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{jiamei-usst@163.com} \address{Pingyou Wang \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{wpingy2008@163.com} \thanks{Submitted November 16, 2011. Published February 23, 2012.} \thanks{Supported by grants 10ZZ93 from the Innovation Program of Shanghai Municipal Education \hfill\break\indent Commission, and 11171220 from the National Natural Sciences Foundation of China} \subjclass[2000]{34B15, 34B18} \keywords{Boundary value problems; $p$-Laplacian operator; \hfill\break\indent integral boundary conditions; fixed point theorem; multiple positive solutions} \begin{abstract} In this article, we show the existence of multiple positive solutions for integro-differential equations with one-dimensional $p$-Laplacian operator, sign changing nonlinearities, and integral boundary conditions. By using the Schauder fixed point theorem and the Krasnosel'skii fixed point theorem, we obtain sufficient conditions for the existence of at least two positive solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we study the existence of positive solutions for the following integro-differential equation with integral boundary conditions, and sign changing nonlinearities: \label{e1.1} \begin{gathered} (\varphi_p(u'(t)))'+w(t)f(t,u(t),Au(t),Bu(t))=0, \quad 01$,$\varphi_q=(\varphi_p)^{-1}$, and$\frac{1}{q}+\frac{1}{p}=1$. By using the Schauder fixed point theorem and the Krasnosel'skii fixed point theorem, we obtain sufficient conditions for the existence of at least two positive solutions under suitable conditions assumed on the nonlinear terms$f$and$w$. The theory of boundary-value problems for integro-differential equations arises in different areas of applied mathematics, fluid dynamics, plasma physics, biological sciences and chemical kinetics (for details, see \cite{alm,as,waz} and the references therein). Since boundary-value problems with integral boundary conditions include two, three, multi-point and nonlocal boundary-value problems as special cases, the existence and multiplicity of positive solutions for such problems have been put emphasis on continuously (see \cite{g1,ma,wi1,wi2,x1,kt,lj,jw,l1} and references therein). Because of the wide mathematical and physical background, the existence of positive solutions for nonlinear boundary-value problems with$p$-Laplacian has also received wide attention. For details, we can refer to see \cite{b1,lg,f2,c1,m1,w1,lh,lj,jw,jtg,Ge}. The main tools for such problems are various kinds of fixed-point theorem in cones (see \cite{b1,lg,f2,w1,lh,lj,jw,jtg}), the monotone iterative technique (see \cite{m1}) and the fixed point index theory (see \cite{c1,jtg}). If the nonlinear term is nonnegative, we can apply the concavity of solutions in the proofs. Under the assumption that the nonlinear term is nonnegative, authors obtained the existence of at least one positive solutions or multiple positive solutions, see \cite{b1,lg,f2,m1,w1,lh,jw}. By using the upper and lower solution approach and the growth restriction approach, in \cite{ar} the author presented some general existence theorems second-order boundary-value problems with sign changing nonlinearities: \label{1.2} \begin{gathered} y''+q(t)f(t,y)=0, \quad 0Su(t_0)$. Let $(t_1,t_2)$ be the maximal interval which contains $t_0$ and such that $Su(t)<0$, $t\in (t_1,t_2)$. It follows $[t_1,t_2]\neq [0,1]$ from (H2). Case 1: If $t_2<1$, we have $u(t)=0$ for $t\in [t_1,t_2]$, $Su(t)<0$ for $t\in (t_1,t_2)$ and $Su(t_2)=0$. Thus $(Su)'(t_2)\geq 0$. From (H2), we know $[\varphi_p((Su)'(t))]'=-w(t)f(t,0,Au(t),Bu(t))\leq 0$ for $t\in[t_1,t_2]$ and we can get $(Su)'(t)$ is monotone decreasing on $[t_1,t_2]$. So $t_1=0$, $Su(t_1)<0$ and $$(Su)'(t)\geq (Su)'(t_2)\geq 0,\; t\in [0,t_2].$$ On the other hand, if $\xi\leq t_2$, we have $$0>Su(0)=\int_0^\xi g(\xi,s)(Su)'(s)\,\mathrm{d}s\geq 0,$$ which is a contradiction. If $\xi> t_2$, by using mean value theorem of integral, we have \begin{align*} Su(0) & =\int_0^{t_2} g(\xi,s)(Su)'(s)\,\mathrm{d}s+\int_{t_2}^\xi g(\xi,s)(Su)'(s)\,\mathrm{d}s\\ & =g(\xi,\xi_1)\int_0^{t_2} (Su)'(s)\,\mathrm{d}s+g(\xi,t_2)\int_{t_2}^{\xi_2} (Su)'(s)\,\mathrm{d}s\\ & = g(\xi,\xi_1)(Su(t_2)-Su(0))+g(\xi,t_2)(Su(\xi_2)-Su(t_2))\\ & = g(\xi,\xi_1)(-Su(0))+g(\xi,t_2)Su(\xi_2), \end{align*} and $$\label{2.6} 0>(1+g(\xi,\xi_1))Su(0)=g(\xi,t_2)Su(\xi_2),$$ where $\xi_1\in [0,t_2]$ and $\xi_2\in [t_2,\xi]$. It follows that \eqref{2.6} is a contradiction if $Su(\xi_2)\geq 0$. If $Su(\xi_2)< 0$, let $(t_3,t_4)$ be the maximal interval which contains $\xi_2$ and such that $Su(t)<0$, $t\in (t_3,t_4)$. It is obvious that $[t_3,t_4]\subset [t_2,1]$. If $t_4<1$, we have $u(t)=0$ for $t\in [t_3,t_4]$, $Su(t)<0$ for $t\in (t_3,t_4)$ and $Su(t_3)=0$. Thus $(Su)'(t_3)\leq 0$. From (H2), we know $[\varphi_p((Su)'(t))]'=-w(t)f(t,0,Au(t),Bu(t))\leq 0$ and $\varphi_p((Su)'(t))$ is monotone decreasing on $[t_3, t_4]$, we can obtain $(Su)'(t)$ is monotone decreasing on $[t_3, t_4]$. It is easy to show that $$(Su)'(t)\leq (Su)'(t_3)\leq 0,\; t\in [t_3, t_4].$$ Hence, $t_4=1$ and $Su(1)<0$. Since $\xi\leq\eta$, we have $(Su)'(t)\leq 0,\; t\in [\eta,1]$ and $$0>Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s\geq 0,$$ which is a contradiction. Therefore, $t_2<1$ is not true. We have $t_2=1$. Case 2: If $t_1>0$, we have $Su(t)=0$ for $t\in [t_1,1]$, $Su(t)<0$ for $t\in (t_1,1)$ and $Su(t_1)=0$. Thus $(Su)'(t_1)\leq 0$. We have $[\varphi_p((Su)'(t))]'=-f(t,0,Au(t),Bu(t))\leq 0$ by (H2). This implies $(Su)'(t)\leq0$ and $Su(t)<0$ for $t\in (t_1,1]$ and $Su(1)=\min_{t\in [t_1,1]}Su(t)$. We can prove that $$\label{2.08} Su(t)\geq 0\;\mathrm{for}\;t\in [0,t_1].$$ If there exists a $t_5\in [0,t_1]$ such that $Su(t_5)<0$ and there is a maximal interval $[t_6,t_7]$ which contains $t_5$ such that $Su(t)<0$ for $t\in (t_6,t_7)$. Obviously $[t_6,t_7)\cap [t_1,1]=\emptyset$, so $1\not\in(t_6,t_7)$; i.e., $t_7<1$, this is a contradiction with the above discussion. Thus we can show $Su(t)\geq 0$ for $t\in [0,t_1]$. For $Su(1)<0$, we have $$Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s.$$ Then, if $\eta\geq t_1$, we have $$0>Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s\geq 0,$$ which is a contradiction. If $\eta(1+h(\eta,\eta_2))Su(1)=h(\eta,t_1)Su(\eta_1). By \eqref{2.08}, we have$Su(\eta_1)\geq 0$. Hence, \eqref{2.7} is a contradiction. Therefore$t_1=0$. The above also contradicts$[t_1,t_2]\neq [0,1]$. Thus$Su(t)\geq 0$for$t\in[0,1]$. That is$u(t)$is a fixed point of operator$S$. \end{proof} Next we state the Krasnosel'skii Fixed Point Theorem \cite{gl}. \begin{lemma} \label{lemma2.10} Let$E$be a Banach space and$K \subset E$be a cone in$E$. Assume$\Omega_1$and$\Omega_2$are open subsets of$E$with$0 \in \Omega_1$and$\overline{\Omega}_1 \subset \Omega_2$and$A: K \cap (\overline{\Omega}_2 \backslash \Omega_1) \rightarrow K$be a completely continuous operator. In addition, suppose either $$\|Au\|\leq\|u\|,\indent u \in K \cap \partial \Omega_1,\;and \; \|Au\|\geq \|u\|,\indent u \in K \cap \partial \Omega_2;$$ or $$\|Au\|\geq\|u\|, \indent u \in K \cap \partial \Omega_1,\; and\;\|Au\|\leq \|u\|,\indent u \in K \cap \partial \Omega_2.$$ hold. Then$A$has a fixed point in$K \cap (\overline{\Omega}_2 \backslash \Omega_1). \end{lemma} \section{Main result} Denote \begin{gather*} M=\min\Big\{\int_{\delta}^{1/2}\varphi_q\Big(\int_s^{1/2} w(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s,\int_{1/2}^{1-\delta} \varphi_q\Big(\int_{1/2}^sw(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big\}, \\ \begin{aligned} N=\max\Big\{&(1+\int_0^{1}g(\xi,s)\,\mathrm{d}s)\varphi_q \Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big), \\ &(1+\int_0^{1}h(\eta,s)\,\mathrm{d}s)\varphi_q\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big) \Big\}. \end{aligned} \end{gather*} For the next theorem, we assume thatf$satisfies the following growth conditions: \begin{itemize} \item[(H4)]$f(t,u,x,y)\geq 0$for$(t,u,x,y)\in [0,1]\times [c_1,c_3]\times\mathbb{R}^2$; \item[(H5)]$f(t,u,x,y)<\varphi_p(\frac{c_2}{N})$for$(t,u,x,y)\in [0,1]\times [0,c_2]\times\mathbb{R}^2$; \item[(H6)]$f(t,u,x,y)\geq \varphi_p(\frac{c_3}{M})$for$(t,u,x,y)\in [\delta,1-\delta]\times [\delta c_3,c_3]\times\mathbb{R}^2$. \end{itemize} \begin{theorem} \label{thm3.1} Suppose {\rm (H1)--(H6)} hold. There exist constants$c_1, c_2, c_3\$ such that 0\sigma _u, we have \begin{align*} Tu(\overline{t}) &= \Big[\int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau) f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\ &\quad + \int_{\bar{t}}^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f(\tau,u(\tau),Au(\tau), Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big]^+\\ &< \int_0^{1}h(\eta,s)\varphi_q\Big(\int_0^{1}w(\tau) \varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s +\int_0^{1}\varphi_q\Big(\int_0^{1}w(\tau)\varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\ &= \frac{c_2}{N}\varphi_q\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big) \Big[1+\int_0^{1}h(\eta,s)\,\mathrm{d}s\Big]\\ &\leq c_2. \end{align*} We have \label{3.1} \|Tu\|\sigma_u, by (H3) and mean value theorem of integral, there exist \xi_2\in [0,\sigma_u] and \xi_3\in [\sigma_u,\xi] such that \begin{align*} u(0)= & Au(\xi)=\int_0^{\sigma_u}g(\xi,s)u'(s)\,\mathrm{d}s +\int_{\sigma_u}^{\xi}g(\xi,s)u'(s)\,\mathrm{d}s\\ &= g(\xi,\xi_2)\int_0^{\sigma_u}u'(s)\,\mathrm{d}s +g(\xi,\xi_3)\int_{\sigma_u}^{\xi}u'(s)\,\mathrm{d}s\\ &= g(\xi,\xi_2)(u(\sigma_u)-u(0))+g(\xi,\xi_3)(u(\xi)-u(\sigma_u))\\ \geq & g(\xi,\xi)u(\xi)-g(\xi,0)u(0) \end{align*} and $$\label{eq3.2} u(0)\geq\frac{g(\xi,\xi)u(\xi)}{1+g(\xi,0)} \geq\frac{\delta g(\xi,\xi)}{1+g(\xi,0)}\|u\| \geq\frac{\delta g(\xi,\xi)c_2}{1+g(\xi,0)}.$$ If \min_{0\leq t \leq 1}u(t)=u(1), when \eta\leq\sigma_u, by (H3) and mean value theorem of integral, there exist \eta_1\in [\eta,\sigma_u] and \eta_2\in [\sigma_u,1] such that \begin{align*} u(1)&= -Bu(\eta)=-\int_{\eta}^{\sigma_u}h(\eta,s)u'(s)\,\mathrm{d}s -\int_{\sigma_u}^{1}h(\eta,s)u'(s)\,\mathrm{d}s\\ &= -h(\eta,\eta_1)\int_{\eta}^{\sigma_u}u'(s)\,\mathrm{d}s -h(\eta,\eta_2)\int_{\sigma_u}^{1}u'(s)\,\mathrm{d}s\\ &= -h(\eta,\eta_1)(u(\sigma_u)-u(\eta))-h(\eta,\eta_2)(u(1)-u(\sigma_u))\\ &= h(\eta,\eta_2)u(\eta)-h(\eta,\eta_1)u(1) +(h(\eta,\eta_2)-h(\eta,\eta_1))u(\sigma_u)\\ &\geq h(\eta,\eta)u(\eta)-h(\eta,1)u(1) \end{align*} and $$\label{eq3.3} u(1)\geq\frac{h(\eta,\eta)u(\eta)}{1+h(\eta,1)} \geq\frac{\delta h(\eta,\eta)}{1+h(\eta,1)}\|u\| \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,1)}.$$ When \eta>\sigma_u, by (H3) and mean value theorem for integrals, there exists \eta_3\in [\eta,1] such that \begin{align*} u(1)&= -Bu(\eta)=-\int_{\eta}^{1}h(\eta,s)u'(s)\,\mathrm{d}s\\ &= h(\eta,\eta_3)(u(\eta)-u(1)) \geq h(\eta,\eta)(u(\eta)-u(1)) \end{align*} and $$\label{eq3.4} u(1)\geq\frac{h(\eta,\eta)u(\eta)}{1+h(\eta,\eta)} \geq\frac{\delta h(\eta,\eta)}{1+h(\eta,\eta)}\|u\| \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,\eta))} \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,1)}.$$ Hence, it follows \min_{0\leq t \leq 1}u(t)\geq\min\Big\{\frac{g(\xi,\xi)}{1+g(\xi,0)}, \frac{h(\eta,\eta)}{1+h(\eta,1)}\Big\}\delta c_2\geq c_1 $$from \eqref{eq3.1}--\eqref{eq3.4}. Therefore, if u\in\overline{\Omega}_2\setminus\Omega_1 and T^*u=u, we have$$ c_1\leq u(t)\leq \|u\| \leq c_3. $$It follows f(t,u(t),Au(t),Bu(t))\geq 0, t\in [0,1] from (H4). Thus, T^*u=Su. That is, the fixed point of T^* on \overline{\Omega}_2\setminus\Omega_1 is also a fixed point of S. We can get the boundary-value problem \eqref{e1.1} has at least one positive solutions u_2 such that c_2\leq\|u_2 \|\leq c_3. The proof is complete. \end{proof} For the next theorem, we have a new assumption: \begin{itemize} \item[(H7)] For each t\in [0,1], g(t,s) and h(t, s) are monotone with respect to s. \end{itemize} We denote$$ M_g=\max\{\max_{t\in[0,1]}g(t,0),\max_{t\in[0,1]}g(t,t)\},\quad M_h=\max\{\max_{t\in[0,1]}h(t,1),\max_{t\in[0,1]}h(t,t)\}. $$If (H7) holds, for each t\in [0,1], when g(t,s) is decreasing with respect to s, there exists a \bar{\xi}_1\in[0,t] such that$$ |(Au)(t)|=|\int_0^{t}g(t,s)u'(s)\,\mathrm{d}s| =g(t,0)|\int_0^{\bar{\xi}_1}u'(s)\,\mathrm{d}s| \leq 2g(t,0)\|u\|\leq 2M_g\|u\|, $$and when g(t,s) is monotone creasing with respect to s, there exists a \bar{\xi}_2\in[0,t] such that$$ |(Au)(t)|=|\int_0^{t}g(t,s)u'(s)\,\mathrm{d}s| =g(t,t)|\int_{\bar{\xi}_2}^{t}u'(s)\,\mathrm{d}s| \leq 2g(t,t)\|u\|\leq 2M_g\|u\|. $$As above, if (H7) holds, we can show that$$ |(Bu)(t)|\leq 2M_h\|u\|,\quad \text{for each } t\in[0,1]. $$As in the proof of Theorem \ref{thm3.1}, we obtain the following theorem under the assumption taht f satisfies the following growth conditions: \begin{itemize} \item[\rm{(H8)}] f(t,u,x,y)\geq 0 for (t,u,x,y)\in [0,1]\times [c_1,c_3]\times[-2M_gc_3,2M_gc_3]\\ \times [-2M_hc_3,2M_hc_3]; \item[\rm{(H9)}] f(t,u,x,y)<\varphi_p(\frac{c_2}{N}) for (t,u,x,y)\in [0,1]\times [0,c_2]\times[-2M_gc_2,2M_gc_2] \times[-2M_hc_2,2M_hc_2]; \item[\rm{(H10)}] f(t,u,x,y)\geq \varphi_p(\frac{c_3}{M}) for (t,u,x,y)\in [\delta,1-\delta]\times [\delta c_3,c_3]\times[-2M_gc_3,2M_gc_3] \times[-2M_hc_3,2M_hc_3]. \end{itemize} \begin{theorem} \label{thm3.2} Suppose {\rm (H1)--(H3)} and {\rm (H7)--(H10)} hold. Then there exist constants c_1, c_2, c_3 such that such that$$ 0