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:R+m→R+, where R+m={(x1,…,xm)T∈Rm:xj⩾0,j=1,2,…,m}, R+={x∈R:x>0}.