Báo Cáo Báo cáo hóa học POSITIVE PERIODIC SOLUTIONS OF FUNCTIONAL DISCRETE SYSTEMS AND POPULATION MODELS YOU

POSITIVE PERIODIC SOLUTIONS OF FUNCTIONAL DISCRETE SYSTEMS AND POPULATION MODELS YOUSSEF N. RAFFOUL AND CHRISTOPHER C. TISDELL Received 29 March 2004 and in revised form 23 August 2004 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, Zư 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 be the set of all real ω-periodic sequences φ : Z → Rk . Endowed with the maximum norm φ = maxθ∈Z k=1 |φ j (θ)|