Positive periodic solutions of higher-dimensional nonlinear functional difference equations ✩

By using a well-known fixed point index theorem, we study the existence, multiplicity and nonexistence of positive T -periodic solution(s) to the higher-dimensional nonlinear 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 +→R+, where Rm + = {(x1, . . . , xm)T ∈ Rm: xj 0, j = 1, 2, . . . , m}, R+ = {x ∈ R: x >0}.  2005 Elsevier Inc. All rights reserved.