Existence of Positive Solutions for Second-Order Neumann Difference System

This paper studies a class of Neumann difference system which is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of the system in terms of the principal eigenvalue of the corresponding linearized system. 1. Introduction There has been a long history of interest in difference equations and difference systems; for example, see [1–8] and the references therein. In particular, a wide variety of nonlinear difference systems have been studied because they model numerous real-life problems in biology, physics, population dynamics, economics, and so on. One of the difference equations that has attracted some attention is where for all , . In 2003, Cabada and Otero-Espinar [4] studied the existence of solution of the problem (1) and obtained optimal existence results by lower and upper solutions methods. Besides, we note that some systems of discrete boundary value problems are investigated by several authors in recent years; see [5, 7, 8] and the references therein. For example, Sun and Li [5] studied the following boundary value problem of discrete system: Under some assumptions on , they obtained some sufficient conditions for the existence of one or two positive solutions to the system by using nonlinear alternative of Leray-Schauder and the fixed point theorem in cones. Henderson et al. [7] considered the following system of three-point discrete boundary value problem: where , , , , , , are nonnegative functions and , . They deduced the existence of the eigenvalues and yielding at least one positive solution to the system (3) under some assumptions on , , , and with weakly coupling behaviors. Their main tool is the fixed point theorem in cones. However, very little work has been done for the existence of positive solutions of second-order Neumann difference systems. Inspired by the above works, we study the existence of positive solutions of the following second order Neumann difference system: where , the coefficients , , , and are positive functions on , and , are positive constants. Through careful analysis, we have found that (4) is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of (4) in terms of the principal eigenvalue of the corresponding linearized system. These conditions are different from those given in [5, 7]. Although system (4) is very simple, it contains an interesting mathematical feature. It is well known that cooperative systems can be analysed by using