Periodic solutions for a coupled pair of delay difference equations
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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 = {..., −2, −1,0,1,2,...},
(1.1)
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 −hn f2 (n,xn−τ(n) ), we are therefore interested in a more general equation of the form
xn+1 = an xn + hn f1 n,xn−τ(n) − hn f2 n,xn−τ(n) ,
(1.2)
which includes the so-called difference equations with positive and negative terms (see, e.g., [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 un =
n+ω −1 s=n
vn =
n+ω −1 s=n
G(n,s)hs f1 s,us−τ(s) − vs−τ(s) , G(n,s) hs f2 s,us−τ(s) − vs−τ(s) ,
Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 215–226 DOI: 10.1155/ADE.2005.215
(1.3)
216
Periodic solutions of coupled equations
where n ∈ Z. We will assume that ω is a positive integer, G and G are double sequences + ω,s + ω) for n,s ∈ Z, h = {hn }n∈Z satisfying G(n,s) = G(n + ω,s + ω) and G(n,s) = G(n and h = {hn }n∈Z are positive ω-periodic sequences, {τ(n)}n∈Z is an integer-valued ωperiodic sequence, f1 , f2 : Z × R → R are continuous functions, and f1 (n + ω,u) = f1 (n,u) as well as f2 (n + ω,u) = f2 (n,u) for any u ∈ R and n ∈ Z. By a solution of (1.3), we mean a pair (u,v) of sequences u = {un }n∈Z and v = {vn }n∈Z which renders (1.3) into an identity for each n ∈ Z after substitution. A solution (u,v) is said to be ω-periodic if un+ω = un and vn+ω = vn for n ∈ Z. Let X be the set of all real ω-periodic sequences of the form u = {un }n∈Z and endowed with the usual linear structure and ordering (i.e., u ≤ v if un ≤ vn for n ∈ Z). When equipped with the norm u = max un ,
u ∈ X,
0≤n≤ω−1
(1.4)
X is an ordered Banach space with cone Ω0 = {u = {un }n∈Z ∈ X | un ≥ 0, n ∈ Z}. X × X will denote the product (Banach) space equipped with the norm (u,v) = max u, v ,
u,v ∈ X,
(1.5)
and ordering defined by (u,v) ≤ (x, y) if u ≤ x and v ≤ y for any u,v,x, y ∈ X. We remark that a recent paper [4] is concerned with the differential system
y = −a(t)y(t) + f t, y t − τ(t) ,
(1.6)
x = −a(t)x(t) + f t,x t − τ(t) .
There are some ideas in the proof of Theorem 2.1 which are similar to those in [4]. But the techniques in the other results are new. 2. Main result In this s
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