Positive Periodic Solutions for Higher Order Functional Difference Systems Depending on a Parameter

In this paper, we consider a type of nth order functional difference systems depending on a parameter λ. By using the fixed point index theory, we prove that there exists a positive number \({\widetilde{\lambda}}\) separating \({(0,+ \infty) \backslash \widetilde{\lambda}}\) into two disjoint subintervals Λ1 and Λ2 such that the system has zero, at least one or at least two positive periodic solutions according to \({\lambda \in \Lambda_1, \lambda = \widetilde{\lambda}}\), or \({\lambda \in \Lambda_2}\). This work improves and extends some recent results in the literature for the first order systems. In addition, we obtain conditions for a general system to have nontrivial periodic solutions. The results are illustrated with an example.