Some Characters of a Generalized Rational Difference Equation

Abstract

Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2^r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results.