A continuation principle for a class of periodically perturbed autonomous systems

The paper deals with a T-periodically perturbed autonomous system in R^n of the form x˙ = ψ(x) + ε φ(t, x, ε) (PS) with ε > 0 small. The main goal of the paper is to provide conditions ensuring the existence of T-periodic solutions to (PS) belonging to a given open set W ⊂ C([0, T],R^n). This problem is considered in the case when the boundary ∂W of W contains at most a finite number of nondegenerate T-periodic solutions of the autonomous system x˙ = ψ(x). By means of the Malkin’s bifurcation function we establish a formula to evaluate the Leray–Schauder topological degree of the integral equation associated to (PS). This formula permits to state existence results that generalize or improve several results of the existing literature. In particular, we extend a continuation principle due to Capietto, Mawhin and Zanolin where it is assumed that ∂W does not contain any T-periodic solutions of the unperturbed system. Moreover, we obtain generalizations or improvements of some existence results due to Malkin and Loud.