A continuation principle for a class of periodically perturbed autonomoussystems

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Abstract

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)

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