Positive solutions for nonlinear discrete periodic boundary value problems

We are concerned with the nonlinear discrete periodic boundary value problem equation(P)−Δ[p(t−1)Δu(t−1)]+q(t)u(t)=rg(t)f(u(t)),t∈[1,T]Z,u(0)=u(T),p(0)Δu(0)=p(T)Δu(T), where rr is a positive parameter. Optimal interval will be given to the parameter rr to ensure that (P) has at least one positive solution.