%0 Journal Article
%T Global Behavior of
%A Chenquan Gan
%A Xiaofan Yang
%A Wanping Liu
%J Discrete Dynamics in Nature and Society
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/963757
%X This paper aims to investigate the global stability of negative solutions of the difference equation , , where the initial conditions is a positive integer, and the parameters . By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end. 1. Introduction The study of nonlinear difference equations has always attracted a considerable attention (see, e.g., [1¨C30] and the references cited therein). In particular, some references investigated the dynamical behavior of positive solutions of difference equations (see, e.g., [3¨C5, 11]), and some references examined the dynamical behavior of negative solutions of some difference equations (see, e.g., [6, 7]). Gibbons et al. [4] studied the behavior of nonnegative solutions to the recursive sequence with , and also presented an open problem, which had been solved by in [8]. and Ladas, in addition, considered (1) in their book [9]. [10] considered the behavior of nonnegative solutions of the following second-order difference equation where is a nonnegative increasing mapping. Douraki et al. [11] studied the qualitative behavior of positive solutions of the difference equation where the initial values , is a positive integer, and . Moreover, (3) is a special case of the following open problem (see also in [11]), which was proposed by and Ladas in [9]. Open Problem (equation in [9]). Assume that . Investigate the global behavior of positive solutions of (3). [12] considered the boundedness, oscillatory behavior, and global stability of nonnegative solutions of the difference equation where , and is a continuous function nondecreasing in each variable such that . It is worthwhile to note that the above mentioned references ([4, 10¨C12]), especially [4, 11], only discussed the dynamical behavior of positive solutions of difference equation. Furthermore, inspired by the above work and [6, 7], the main goal of this paper is to study the global behavior of negative solutions of the difference equation where is a positive integer, , and the initial conditions . In fact, it is easy to see that (5) is an extension of an open problem introduced by and Ladas in [9] and also is a special case of (4) by a simple change. However, here we establish some results regarding the global stability, invariant interval, and periodic character of negative solutions of (5). 2. Linearized Stability and Period 2 Solutions The aim of this
%U http://www.hindawi.com/journals/ddns/2013/963757/