An averaging result for periodic solutions of Carath\'{e}odory differential equations

TL;DR

This paper establishes conditions under which periodic solutions exist for perturbative Carathéodory differential equations, using averaging methods and continuation theory, and also discusses convergence to constant solutions.

Contribution

It provides new sufficient conditions for the existence of periodic solutions and their uniform convergence in Carathéodory differential equations, extending previous averaging results.

Findings

Sufficient conditions for periodic solutions are derived.

Conditions for uniform convergence to constant solutions are established.

The main results are proved using an abstract continuation theorem.

Abstract

This paper is concerned with the problem of existence of periodic solutions for perturbative Carath\'{e}odory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations.