Positive periodic solutions of higher-dimensional functional difference equations with a parameter ✩

By using Krasnoselskii’s fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T -periodic solutions to the higherdimensional functional difference equations of the form x(n + 1) = A(n)x(n)+ λh(n)f x n− τ(n) , n∈ Z, where A(n) = diag[a1(n), a2(n), . . . , am(n)], h(n) = diag[h1(n),h2(n), . . . ,hm(n)], aj ,hj :Z → R+, τ :Z→Z are T -periodic, j = 1, 2, . . .,m, T 1, λ > 0, x :Z →Rm, f :Rm + →Rm +, where Rm + = {(x1, . . . , xm)T ∈ Rm, xj 0, j = 1, 2, . . .,m}, R+ = {x ∈ R, x > 0}.  2003 Elsevier Inc. All rights reserved