Higher order functional differential systems with parameter dependence

In this paper, we consider a class of high order functional differential systems depending on a parameter λ. By using the fixed point index theory, we prove that there exists a positive number λ˜ separating (0,+∞)⧹{λ˜} into two disjoint subintervals Λ1 and Λ2 such that the system has at least two positive periodic solutions for λ∈Λ1, one positive periodic solution for λ=λ˜, and no positive periodic solution for λ∈Λ2. Our results extend and improve some results in the literature. As an application, an example is given to demonstrate our main results.