\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 94, pp. 1--7.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2011/94\hfil Third-order $q$-difference equations] {Boundary-value problems for nonlinear third-order $q$-difference equations} \author[B. Ahmad\hfil EJDE-2011/94\hfilneg] {Bashir Ahmad} % in alphabetical order \address{Bashir Ahmad \newline Department of Mathematics, Faculty of Science, King Abdulaziz University\\ P. O. Box 80203, Jeddah 21589, Saudi Arabia} \email{bashir\_qau@yahoo.com} \thanks{Submitted December 2, 2010. Published July 28, 2011.} \subjclass[2000]{39A05, 39A13} \keywords{$q$-difference equations; boundary value problems; \hfill\break\indent Leray-Schauder degree theory; fixed point theorems} \begin{abstract} This article shows existence results for a boundary-value problem of nonlinear third-order $q$-difference equations. Our results are based on Leray-Schauder degree theory and some standard fixed point theorems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} The subject of $q$-difference equations, initiated in the beginning of the 19th century \cite{adam,car,jac,mas}, has evolved into a multidisciplinary subject; see for example \cite{ern,fink1,fink2,flo1,flo2,flo3,freu,gas1,han,jaul,kc} and references therein. For some recent work on $q$-difference equations, we refer the reader to \cite{ann,ban,bas,dob,gas2,isma,sha}. However, the theory of boundary-value problems for nonlinear $q$-difference equations is still in the initial stages and many aspects of this theory need to be explored. To the best of our knowledge, the theory of boundary-value problems for third-order nonlinear $q$-difference equations is yet to be developed. In this paper, we discuss the existence of solutions for the nonlinear boundary-value problem (BVP) of third-order $q$-difference equation $$\label{e1} \begin{gathered} D_q^3u(t)=f(t,u(t)), \quad 0\le t \le 1, \\ u(0)=0,\quad D_qu(0)=0, \quad u(1)=0, \end{gathered}$$ where $f$ is a given continuous function. \section{Preliminaries} Let us recall some basic concepts of $q$-calculus \cite{gas1,kc}. For $00$ such that $M_1 G_1 < 1$ and $|f(t,u)| \le M_1|u| +M_2$ for all $t \in [0,1], u \in C([0,1])$, where $G_1$ is given by \eqref{e230}. Then the BVP \eqref{e1} has at least one solution. \end{theorem} \begin{proof} In view of Lemma \ref{lem1}, we just need to prove the existence of at least one solution $u \in C([0,1])$ such that $u =\Gamma u$. Thus, it is sufficient to show that $\Gamma : \overline{B}_R \to C([0,1])$ satisfies $$\label{e31} u \ne \lambda \Gamma u, \quad \forall u \in \partial B_R \quad \forall \lambda \in [0,1],$$ where $B_R \subset C([0,1])$ is a suitable ball with radius $R>0$. Let us define $$H(\lambda, u)=\lambda \Gamma u, \quad u \in C([0,1]),\;\lambda \in [0,1].$$ Then, by Arzela-Ascoli theorem, $h_\lambda(u)=u-H(\lambda, u)=u-\lambda \Gamma u$ is completely continuous. If \eqref{e31} is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that \begin{align*} \deg(h_\lambda, B_R, 0) &=\deg(I-\lambda \Gamma, B_R,0) =\deg(h_1, B_R, 0)\\ &=\deg(h_0, B_R, 0) =\deg(I, B_R, 0)=1\ne 0, \quad 0 \in B_r, \end{align*} where $I$ denotes the unit operator. By the nonzero property of Leray-Schauder degree, $h_1(t)=u- \lambda \Gamma u=0$ for at least one $u \in B_R$. Let us set $$B_R=\{u \in C([0,1]) : \max_{t \in [0,1]}|u(t)|0, we define$$ B_R=\{u \in C([0,1]): \max_{t \in [0,1]}|u(t)|