\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 59, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/59\hfil Existence of solutions] {Existence of solutions to $p$-Laplacian difference equations under barrier strips conditions} \author[C. Gao\hfil EJDE-2007/59\hfilneg] {Chenghua Gao} % in alphabetical order \address{Chenghua Gao \newline College of Mathematics and Information Science, Northwest Normal University \\ Lanzhou, Gansu 730070, China} \email{gaokuguo@163.com} \thanks{Submitted January 24, 2007. Published April 22, 2007.} \subjclass[2000]{39A10} \keywords{Second-order p-Laplacian difference equation; barrier strips; \hfill\break\indent Leray-Schauder principle; existence} \begin{abstract} We study the existence of solutions to the boundary-value problem \begin{gather*} \Delta(\phi_p(\Delta u(k-1)))=f(k,u(k),\Delta u(k)),\quad k\in \mathbb{T}_{[1, N]}, \\ \Delta u(0)=A, \quad u(N+1)=B, \end{gather*} with barrier strips conditions, where $N>1$ is a fixed natural number, $\phi_p(s)=|s|^{p-2}s$, $p>1$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Given $a, b\in\mathbf{Z}$ and $a1$ is a fixed natural number, $f:\mathbb{T}_{[1, N]}\times \mathbb{R}^{2}\to\mathbb{R}$ is continuous, $\phi_p(s)=|s|^{p-2}s, p>1, (\phi_p)^{-1}=\phi_q, \frac{1}{p}+\frac{1}{q}=1$. In recent years, $p$-Laplacian discrete boundary-value problems have been investigated in literature [1,2,4]. But, almost all of the works discussed these problems when $f$ satisfies growth restriction at $\infty$. Now, the question is: Is there still a solution to those problems when $f$ is not restricted at $\infty$? In 1994, Kelevedjiev [3] used Leray-Schauder principle to discuss the solutions to the nonlinear differential boundary-value problem \begin{gather} x{''}(t)=f(t,x(t),x{'}(t)),\quad t\in[0,1],\label{e1.3} \\ x'(0)=A, x(1)=B.\label{e1.4} \end{gather} He established the following results: \begin{theorem} \label{thmA} Let $f:[0,1]\times \mathbb{R}^{2}\to\mathbb{R}$ be continuous. Suppose there are constants $L_{i}$, $i=1,2,3,4$, such that $L_{2}>L_{1}\geq A$, $L_{3}L_{1}\geq A$, $L_{3}L_2,\\ v, & L_3\leq v\leq L_2,\\ L_3, & vL_1\} $$is empty. Suppose it is not empty. Let k_0\in S_0 be fixed. Then \Delta u(k_0)>L_1. From the construction of \Phi, we know$$ L_1<\Phi(\Delta u(k_0))\leq L_2. $$From \eqref{H1} and \Delta(\phi_p(\Delta u(k_0-1)))\leq 0, we have $$|\Delta u(k_0)|^{p-2}\Delta u(k_0)\leq |\Delta u(k_0-1)|^{p-2}\Delta u(k_0-1).\label{e3.3}$$ Now, we prove k_0-1\in S_0. It will be discussed in three cases: \noindent\textbf{Case 1:} \Delta u(k_0)>0. Then from \eqref{e3.3}, we know L_1<\Delta u(k_0)\leq \Delta u(k_0-1); \noindent\textbf{Case 2:} \Delta u(k_0)=0. Then the result is obvious; \noindent\textbf{Case 3:} \Delta u(k_0)<0. Then \Delta u(k_0-1) will be discussed under two cases. \noindent\textbf{Case 3.1:} \Delta u(k_0-1)\geq0. Then from \eqref{e3.3}, \Delta u(k_0-1)>L_1; \noindent\textbf{Case 3.2:} \Delta u(k_0-1)<0. Then p will be discussed under different situations. \noindent\textbf{Case 3.2.1:} p is an odd number. Then (-\Delta u(k_0))^{p-2}=-(\Delta u(k_0))^{p-2}. From \eqref{e3.3}, we know -(\Delta u(k_0))^{p-1}\leq|\Delta u(k_0-1)|^{p-2}\Delta u(k_0-1). Moreover, \Delta u(k_0-1)<0, we have -(\Delta u(k_0))^{p-1}\leq-(\Delta u(k_0-1))^{p-1}. Since p-1 is an even number and \Delta u(k_0), \Delta u(k_0-1)<0, it's not difficult to get$$ L_1<\Delta u(k_0)\leq\Delta u(k_0-1); $$\noindent\textbf{Case 3.2.2:} p is an even number. Then we have (\Delta u(k_0))^{p-1}\leq(\Delta u(k_0-1))^{p-1}, and since p-1 is an odd number, we know that$$ L_1<\Delta u(k_0)\leq\Delta u(k_0-1); $$so, when \Delta u(k_0)<0, \Delta u(k_0-1)<0, there also exists$$ L_1<\Delta u(k_0)\leq\Delta u(k_0-1). $$From Case 1, Case 2, Case 3, we obtain$$ L_1<\Delta u(k_0)\leq\Delta u(k_0-1), $$so k_0-1\in S_0. If we continue the above process, we get$$ \Delta u(0)\geq\Delta u(1)>L_1, $$which contradicts with \Delta u(0)=A, so S_0=\emptyset. Similarly, we can obtain that the set$$ S_1=\{k\in\mathbb{T}_{[0, N]}|\Delta u(k)1$ is a fixed natural number, $B$ is an arbitrary number. Let $f(k,u,p)=p^{4}-6p^{3}+11p^{2}-6p$, $L_{1}=\frac{5}{2}$, $L_{2}=3$, $L_{3}=1$, $L_{4}=\frac{3}{2}$, $A=2$. We can prove that $f(k,u,p)$ satisfies all conditions of Theorem \ref{thm3.1}, so this problem has at least one solution. \end{proof} The next theorem can be proved by similar arguments. \begin{theorem} \label{thm3.2} Let $f:\mathbb{T}_{[1, N]}\times \mathbb{R}^{2}\to\mathbb{R}$ be continuous. Suppose there are constants $L_{i}$, $i=1,2,3,4$ with $L_{2}>L_{1}\geq B$, \$L_{3}