\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 219, pp. 1--9.\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-2012/219 \hfil Monotone iterative method ] {Monotone iterative method for obtaining positive solutions of integral boundary-value problems with $\phi$-Laplacian operator} \author[Y. Ding \hfil EJDE-2012/219\hfilneg] {Yonghong Ding} % in alphabetical order \address{Yonghong Ding \newline Department of Mathematics, Tianshui Normal University, Tianshui 741000, China} \email{dyh198510@126.com} \thanks{Submitted October 11, 2012. Published November 29, 2012.} \subjclass[2000]{34B15, 34B18} \keywords{$\phi$-Laplacian; monotone iterative; cone; positive solutions} \begin{abstract} This article concerns the existence, multiplicity of positive solutions for the integral boundary-value problem with $\phi$-Laplacian, \begin{gather*} \big(\phi(u'(t))\big)'+f(t,u(t),u'(t))=0,\quad t\in[0,1],\\ u(0)=\int_0^1 u(r)g(r)\,\mathrm{d}r,\quad u(1)=\int_0^1u(r)h(r)\,\mathrm{d}r, \end{gather*} where $\phi$ is an odd, increasing homeomorphism from $\mathbb{R}$ to $\mathbb{R}$. Using a monotone iterative technique, we obtain the existence of positive solutions for this problem, and present iterative schemes for approximating the solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Consider the integral boundary-value problem $$\begin{gathered} \big(\phi(u'(t))\big)'+f(t,u(t),u'(t))=0,\quad t\in[0,1],\\ u(0)=\int_0^1u(r)g(r)\,\mathrm{d}r,\quad u(1)=\int_0^1u(r)h(r)\,\mathrm{d}r, \end{gathered} \label{e1}$$ where $\phi,f,g$ and $h$ satisfy \begin{itemize} \item[(H1)] $\phi$ is an odd, increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ and there exist two increasing homeomorphisms $\psi_1$ and $\psi_2$ of $(0, \infty)$ onto $(0, \infty)$ such that $$\psi_1(u)\phi(v) \leq \phi(uv) \leq \psi_2(u)\phi(v), u, v > 0.$$ Moreover, $\phi, \phi^{-1}\in C^1(R)$, where $\phi^{-1}$ denotes the inverse of $\phi$. \item[(H2)] $f:[0, 1]\times [0, +\infty)\times (-\infty, +\infty)\to (0, +\infty)$ is continuous. $g, h\in L^1[0, 1]$ are nonnegative, and $0<\int_0^1g(t)\,\mathrm{d}t<1, 0<\int_0^1h(t)\,\mathrm{d}t<1$. \end{itemize} The assumption (H1) on the function $\phi$ was first introduced by Wang [1, 2]. It covers two special cases: $\phi(u)=u$ and $\phi(u)=|u|^{p-2}u, p>1$. Many authors have studied the positive solutions of two-point and multi-point boundary value problems when $\phi(u)=u$ or $\phi(u)=|u|^{p-2}u, p>1$. For details and references see \cite{a1,f1,l1,s1,w3,w4}. In a recent paper \cite{d1}, the author proved the existence and multiplicity of positive solutions of \eqref{e1} by applying the Krasnoselskii fixed point theorem and Avery-Peterson fixed point theorem. However there is an interesting question which is showing how to find these solutions, since they exist. Motivated by the question and all the works above, in this article, by applying classical monotone iterative techniques, we not only obtain the existence of positive solutions of \eqref{e1}, but also give iterative schemes for approximating the solutions. It is worth stating that we will not require the existence of lower and upper solutions, and the first term of iterative scheme is a constant function or a simple function. Therefore, the iterative scheme is significant and feasible. \section{Preliminaries} The basic space used in this article is the real Banach space $C^1[0, 1]$ with norm $\|u\|_1=\max\{\|u\|_{c}, \|u'\|_{c}\}$, where $\|u\|_{c}=\max_{0\leq t\leq 1}|u(t)|$. Let $P=\big\{u\in C^1[0, 1]: u(t)\geq 0, u \text{ is concave on [0, 1]}\big\}.$ It is obvious that $P$ is a cone in $C^1[0, 1]$. For any $x\in C^1[0, 1]$, $x(t)\geq 0$, $t\in [0, 1]$, we suppose that $u$ is a solution of the boundary-value problem $$\begin{gathered} \big(\phi(u'(t))\big)'+f(t,x(t),x'(t))=0, \quad t\in[0,1],\\ u(0)=\int_0^1u(r)g(r)\,\mathrm{d}r, \quad u(1)=\int_0^1u(r)h(r)\,\mathrm{d}r. \end{gathered} \label{e2}$$ By integrating \eqref{e2} on $[0, t]$, we have $$\phi(u'(t))=A_x-\int_0^t f(s,x(s),x'(s))\,\mathrm{d}s,$$ then $$u'(t)=\phi^{-1}\Big(A_x-\int_0^t f(s,x(s),x'(s))\,\mathrm{d}s\Big). \label{e3}$$ Thus $$u(t)=u(0)+\int_0^t\phi^{-1}\Big(A_x-\int_0^s f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s \label{e4}$$ or $$u(t)=u(1)-\int_t^1\phi^{-1}\Big(A_x-\int_0^s f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s. \label{e5}$$ According to the boundary condition, we have $$u(0)=\frac{1}{1-\int_0^1g(r)\,\mathrm{d}r}\int_0^1g(r)\int_0^r\phi^{-1} \Big(A_x-\int_0^s f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big)\, \mathrm{d}s\,\mathrm{d}r$$ and $$u(1)=-\frac{1}{1-\int_0^1h(r)\,\mathrm{d}r}\int_0^1h(r)\int_r^1\phi^{-1} \Big(A_x-\int_0^sf(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big) \,\mathrm{d}s\,\mathrm{d}r,$$ where $A_x$ satisfies the equation \begin{aligned} H_x(c) &=\frac{1-\int_0^1h(r)\,\mathrm{d}r}{1-\int_0^1g(r)\,\mathrm{d}r} \int_0^1g(r)\int_0^r\phi^{-1}\Big(c-\int_0^s f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\big)\,\mathrm{d}s\,\mathrm{d}r\\ &\quad +\Big(1-\int_0^1h(r)\,\mathrm{d}r\Big)\int_0^1\phi^{-1}\Big(c-\int_0^s f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\ &\quad +\int_0^1h(r)\int_r^1\phi^{-1}\Big(c-\int_0^sf(\tau,x(\tau),x'(\tau)) \,\mathrm{d}\tau\Big) \,\mathrm{d}s\,\mathrm{d}r=0. \end{aligned} \label{e6} \begin{lemma} \label{lem2.1} Assume {\rm (H1)} and {\rm (H2)} hold, for any $x\in C^1[0, 1]$ with $x(t)\geq 0, t\in [0, 1]$, there exists a unique $A_x\in (-\infty, +\infty)$ satisfying \eqref{e6}. Moreover, there is a unique $\delta_x\in (0,1)$ such that $$A_x=\int_0^{\delta_x} f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau.$$ \end{lemma} \begin{proof} From the expression of $H_x(c)$ it is easy to see that $H_x: \mathbb{R}\to\mathbb{R}$ is continuous and strictly increasing, and $$H_x(0)<0, \quad H_x\Big(\int_0^1 f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau\Big)>0.$$ Hence there exists a unique $A_x\in (0, \int_0^1 f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau) \subset(-\infty, +\infty)$ satisfying \eqref{e6}. Let $$F(t)=\int_0^t f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau,$$ then $F(t)$ is continuous and strictly increasing on $[0, 1]$, and $F(0)=0$, $F(1)=\int_0^1 f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau$. So $$0=F(0)0 is a constant such that$$ \int_0^1f(\tau,x(\tau),x'(\tau))\,\mathrm{d}\tau< m, \quad \forall ~x\in D. $$From the definition of T, for any x\in D, we obtain$$ |Tx(t)|<\begin{cases} \frac{\phi^{-1}(m)}{1-\int_0^1g(r)\,\mathrm{d}r}, & 0\leq t\leq\delta_x,\\ \frac{\phi^{-1}(m)}{1-\int_0^1h(r)\,\mathrm{d}r}, & \delta_x\leq t\leq 1, \end{cases}  |(Tx)'(t)|<\phi^{-1}(m), \quad 0\leq t\leq 1. $$Hence, TD is uniformly bounded and equicontinuous. So we have TD is compact on C[0, 1]. From \eqref{e8} we know that for all \varepsilon>0 there exists \kappa>0, such that when |t_1-t_2|<\kappa, we have$$ |\phi(Tx)'(t_1)-\phi(Tx)'(t_2)|<\varepsilon. $$So \phi(TD)' is compact on C[0, 1], it follows that (TD)' is compact on C[0, 1]. Therefore, TD is compact on C^1[0, 1]. Thus, T:P\to P is completely continuous. The proof is complete. \end{proof} \section{Existence of positive solutions} For convenience, we denote$$ A=\max\Big\{\frac{1}{1-\int_0^1g(r)\,\mathrm{d}r}, \frac{1}{1-\int_0^1h(r)\,\mathrm{d}r}\Big\}.  Our result is as follows. \begin{theorem} \label{thm3.1} Assume {\rm (H1)} and {\rm (H2)} hold. If there exists $a>0$ such that \begin{itemize} \item[(i)] $f(t,x_1,y_1)\leq f(t,x_2,y_2)$ for any $0\leq t\leq 1$, $0\leq x_1\leq x_2\leq a$, $0\leq|y_1|\leq| y_2|\leq a$; \item[(ii)] $\max_{0\leq t\leq 1}f(t,a,a)\leq\phi(\frac{a}{A})$. \end{itemize} Then \eqref{e1} has at least two positive, concave solutions $w^{\ast}$ and $v^{\ast}$ satisfying \begin{gather*} 00$such that \begin{itemize} \item[(iii)]$\lim_{\ell\to\infty}\max_{0\leq t\leq 1}f(t,\ell,a) \leq\phi(\frac{1}{A})$. \end{itemize} Then \eqref{e1} has at least two positive, concave solutions$w^{\ast}$and$v^{\ast}$such that the conclusion of Theorem \ref{thm3.1} hold. \end{corollary} \begin{corollary} \label{coro2.2} Assume {\rm (H1), (H2)} and Theorem \ref{thm3.1}(i) hold. If there exists$00$. Choosing$a=4$. By calculations we obtain$A=2$. It is easy to verify that$f(t,u,v)$satisfies \begin{itemize} \item[(1)]$f(t,x_1,y_1)\leq f(t,x_2,y_2)$for any$0\leq t\leq 1, 0\leq x_1\leq x_2\leq 4, 0\leq|y_1|\leq| y_2|\leq 4$; \item[(2)]$\max_{0\leq t\leq 1}f(t,a,a)=f(\frac{1}{2},4,4) \leq\phi(\frac{a}{A})=4$. \end{itemize} Hence, by Theorem \ref{thm3.1}, \eqref{e12} has two positive solutions$w^{\ast}$and$v^{\ast}$. For$n=0,1,2,\dots $, the two iterative schemes are: \begin{gather*} w_0(t)=\begin{cases} 2+2t, & 0\leq t\leq \frac{1}{2},\\ 4-2t, & \frac{1}{2}\leq t\leq 1, \end{cases} \\ w_{n+1}(t)=\begin{cases} \int_0^1\int_0^r\phi^{-1}\Big(\int_s^{\delta_n} (-\tau^{2}+\tau+\frac{1}{8}w_n(\tau)+\frac{1}{16}w_n'^{2}(\tau)) \,\mathrm{d}\tau\Big)\,\mathrm{d}s\,\mathrm{d}r \\ +\int_0^t\phi^{-1}\Big(\int_s^{\delta_n} (-\tau^{2}+\tau+\frac{1}{8}w_n(\tau) +\frac{1}{16}w_n'^{2}(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s, & 0\leq t\leq \delta_n,\\ \int_0^1\int_r^1\phi^{-1}\Big(\int_{\delta_n}^s (-\tau^{2}+\tau+\frac{1}{8}w_n(\tau)+\frac{1}{16}w_n'^{2}(\tau)) \,\mathrm{d}\tau\Big)\,\mathrm{d}s\,\mathrm{d}r\\ +\int_t^1\phi^{-1}\Big(\int_{\delta_n}^s(-\tau^{2}+\tau+\frac{1}{8}w_n(\tau) +\frac{1}{16}w_n'^{2}(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s, & \delta_n\leq t\leq 1. \end{cases} \end{gather*} \begin{gather*} v_0(t)=0, \\ v_{n+1}(t)=\begin{cases} \int_0^1\int_0^r\phi^{-1}\Big(\int_s^{\delta_n} (-\tau^{2}+\tau+\frac{1}{8}v_n(\tau)+\frac{1}{16}v_n'^{2}(\tau)) \,\mathrm{d}\tau\Big)\,\mathrm{d}s\,\mathrm{d}r \\ +\int_0^t\phi^{-1}\Big(\int_s^{\delta_n} (-\tau^{2}+\tau +\frac{1}{8}v_n(\tau)+\frac{1}{16}v_n'^{2}(\tau))\,\mathrm{d}\tau\Big) \,\mathrm{d}s, & 0\leq t\leq \delta_n, \\ \int_0^1\int_r^1\phi^{-1}\Big(\int_{\delta_n}^s (-\tau^{2}+\tau+\frac{1}{8}v_n(\tau) +\frac{1}{16}v_n'^{2}(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\,\mathrm{d}r\\ +\int_t^1\phi^{-1}\Big(\int_{\delta_n}^s(-\tau^{2}+\tau +\frac{1}{8}v_n(\tau)+\frac{1}{16}v_n'^{2}(\tau))\, \mathrm{d}\tau\Big)\,\mathrm{d}s, &\delta_n\leq t\leq 1 . \end{cases} \end{gather*} \begin{thebibliography}{9} \bibitem{a1} R. Avery, J. Henderson; \emph{Existence of three positive pseudo-symmetric solutions for a one-dimensional$p$-Laplacian}, J. Math. Anal. Appl. 277~(2003)~395-404. \bibitem{d1} Y. Ding; \emph{Positive solutions for integral boundary value problem with$\phi$-Laplacian operator}, Boundary Value Problem. Volume(2011), Article ID 827510. \bibitem{f1} H. Feng, W. Ge, M. Jiang; \emph{Multiple positive solutions for m-point boundary value problems with a one-dimensional$p$-Laplacian}, Nonlinear Anal. 68 (2008) 2269-2279. \bibitem{l1} B. Liu; \emph{Positive solutions of three-point boundary value problems for one-dimensional$p$-Laplacian with infinitely many singularities}, Appl. Math. Lett. 17 (2004) 655-661. \bibitem{s1} B. Sun, W. Ge; \emph{Existence and iteration of positive solutions for some$p$-Laplacian boundary value problems}, Nonlinear Anal. 67 (2007) 1820-1830. \bibitem{w1} H. Wang; \emph{On the number of positive solutions of nonlinear systems}, J. Math. Anal. Appl. 281(2003) 287-306. \bibitem{w2} H. Wang; \emph{On the structure of positive radial solutions for quasilinear equations in annular domain}, Adv. Differential Equations. 8 (2003) 111-128. \bibitem{w3} J. Wang; \emph{The existence of positive solutions for the one-dimensional$p$-Laplacian}, Proc. Amer. Math. Soc. 125 (1997) 2275-2283. \bibitem{w4} Z. Wang, J. Zhang; \emph{Positive solutions for one-dimensional$p\$-Laplacian boundary value problems with dependence on the first order derivative}, J. Math. Anal. Appl. 314 (2006) 618-630. \end{thebibliography} \end{document}