Positive periodic solutions of functional discrete systems and population models
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We apply a cone-theoretic fixed point theorem to study the existence of positive periodic solutions of the nonlinear system of functional difference equations x(n + 1) = A(n)x(n) + f (n,xn ). 1. Introduction Let R denote the real numbers, Z the integers, Z− the negative integers, and Z+ the nonnegative integers. In this paper we explore the existence of positive periodic solutions of the nonlinear nonautonomous system of difference equations
x(n + 1) = A(n)x(n) + f n,xn ,
(1.1)
where, A(n) = diag[a1 (n),a2 (n),...,ak (n)], a j is ω-periodic, f (n,x) : Z × Rk → Rk is continuous in x and f (n,x) is ω-periodic in n and x, whenever x is ω-periodic, ω ≥ 1 is an integer. Let ᐄ be the set of all real ω-periodic sequences φ : Z → Rk . Endowed with the maximum norm φ = maxθ∈Z kj=1 |φ j (θ)| where φ = (φ1 ,φ2 ,...,φk )t , ᐄ is a Banach space. Here t stands for the transpose. If x ∈ ᐄ, then xn ∈ ᐄ for any n ∈ Z is defined by xn (θ) = x(n + θ) for θ ∈ Z. The existence of multiple positive periodic solutions of nonlinear functional differential equations has been studied extensively in recent years. Some appropriate references are [1, 14]. We are particularly motivated by the work in [8] on functional differential equations and the work of the first author in [4, 11, 12] on boundary value problems involving functional difference equations. When working with certain boundary value problems whether in differential or difference equations, it is customary to display the desired solution in terms of a suitable Green’s function and then apply cone theory [2, 4, 5, 6, 7, 10, 13]. Since our equation (1.1) is not this type of boundary value, we obtain a variation of parameters formula and then try to find a lower and upper estimates for the kernel inside the summation. Once those estimates are found we use Krasnoselskii’s fixed point theorem to show the existence of a positive periodic solution. In [11], the first author studied the existence of periodic solutions of an equation similar to (1.1) using Schauder’s second fixed point theorem. Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 369–380 DOI: 10.1155/ADE.2005.369
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Positive periodic solutions
Throughout this paper, we denote the product of y(n) from n = a to n = b by with the understanding that bn=a y(n) = 1 for all a > b. In [12], the first author considered the scalar difference equation
x(n + 1) = a(n)x(n) + h(n) f x n − τ(n) ,
b
n=a
y(n)
(1.2)
where a(n), h(n), and τ(n) are ω-periodic for ω an integer with ω ≥ 1. Under the assumptions that a(n), f (x), and h(n) are nonnegative with 0 < a(n) < 1 for all n ∈ [0,ω − 1], it was shown that (1.2) possesses a positive periodic solution. In this paper we generalize (1.2) to systems with infinite delay and address the existence of positive periodic solutions of (1.1) in the case a(n) > 1. Let R+ = [0,+∞), for each x = (x1 ,x2 ,...,xn )t ∈ Rn , the norm of x is defined as |x| = n n t n j =1 |x j |. R+ = {(x1 ,x2 ,...,xn ) ∈ R : x j ≥ 0, j = 1,2,...,n}. Also, we
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