PERIODIC SOLUTIONS FOR A COUPLED PAIR OF DELAY DIFFERENCE EQUATIONS GUANG ZHANG, SHUGUI KANG, AND SUI SUN CHENG Received 31 January 2005 and in revised form 24 March 2005 Based on the fixed-point index theory for a Banach space, positive periodic solutions are found for a system of delay difference equations. By using such results, the existence of nontrivial periodic solutions for delay difference equations with positive and negative terms is also considered. 1. Introduction The existence of positive periodic solutions for delay difference equations of the form xn+1 = an xn + hn f n,xn−τ(n) , n ∈ Z = {.,. | PERIODIC SOLUTIONS FOR A COUPLED PAIR OF DELAY DIFFERENCE EQUATIONS GUANG ZHANG SHUGUI KANG AND SUI SUN CHENG Received 31 January 2005 and in revised form 24 March 2005 Based on the fixed-point index theory for a Banach space positive periodic solutions are found for a system of delay difference equations. By using such results the existence of nontrivial periodic solutions for delay difference equations with positive and negative terms is also considered. 1. Introduction The existence of positive periodic solutions for delay difference equations of the form Xn 1 anXn hnf n Xn-T nf n e Z . -2 -1 0 1 2 . has been studied by many authors see for example 1 3 5 7 8 9 and the references contained therein. The above equation may be regarded as a mathematical model for a number of dynamical processes. In particular Xn may represent the size of a population in the time period n. Since it is possible that the population may be influenced by another factor of the form -hnf2 n Xn-T n we are therefore interested in a more general equation of the form Xn 1 anxn hnf1 n Xn-T n - hn_ 2 n Xn-T n 1-2 which includes the so-called difference equations with positive and negative terms see . 6 . In this paper we will approach this equation see Section 4 by treating it as a special case of a system of difference equations of the form n w-1 Un y G n s hsf1 s Us-T s - Vs-T s s n n co-1 Vn y G n s hsf2 s Us-T s - Vs-T s s n Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 3 2005 215-226 DOI 216 Periodic solutions of coupled equations where n e Z. We will assume that w is a positive integer G and G are double sequences satisfying G n s G n w s w and G n s G n w s w for n s e Z h hn nez and h hn neZ are positive w-periodic sequences t n neZ is an integer-valued w-periodic sequence f1 f2 Z X R R are continuous functions and f1 n w u f1 n u as well as f2 n w u f2 n u for any u e R and n e Z. By a solution of we mean a pair u
