Periodic Solutions of Second Order Nonlinear Functional Difference Equations

The development of the study of periodic solution of functional difference equations is relatively rapid. There has been many approaches to study periodic solutions of difference equations, such as critical point theory, fixed point theorems in Banach spaces or in cones of Banach spaces, coincidence degree theory, KaplanYorke method, and so on, one may see [3-7,11,13-15] and the references therein. In papers [5,7,11,13,14], the authors studied the existence of periodic solutions of first order functional difference equations using different fixed point theorems in cones of Banach spaces. Zhu and Li in [15] used fixed point theorems in cones of Banach spaces to obtain positive periodic solutions of higher order functional difference equations. In [4], the authors studied the existence of periodic solutions of a second order nonlinear difference equation by using the critical point theory. Papers [1,2,8-10,12] concerned with the solvability (existence of positive solutions) of periodic boundary value problems for second order difference equations on a finite discrete segment. In this paper, we, by using coincidence degree theory, study the second order nonlinear functional difference equation