Abstract
The paper deals with a T ‐periodically perturbed autonomous system in ℝ^n^ of the form
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 $ \dot x $ = ϕ (x). The starting point of our approach is the following property due to Malkin: if for any T ‐periodic limit cycle x 0 of $ \dot x $ = ϕ (x) belonging to ∂W the so‐called bifurcation function f (θ), θ ∈ [0, T ], associated to x~0~, see (1.11), satisfies the condition f(0) ≠ 0 then the integral operator
equation image
does not have fixed points on ∂W for all ε > 0 sufficiently small. By means of the Malkin's bifurcation function we then establish a formula to evaluate the Leray–Schauder topological degree of I – Q~ε~ on W. 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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)