Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 39, pp. 1-13.

Title: Caratheodory periodic perturbations of degenerate systems

Authors: Alessandro Calamai (Univ. Politecnica delle Marche, Ancona, Italy)
         Marco Spadini (Univ. degli Studi di Firenze, Florence, Italy)

Abstract: 
We study the structure of the set of harmonic solutions to $T$-periodically 
perturbed coupled differential equations on differentiable manifolds, where the 
perturbation is allowed to be of Caratheodory-type regularity. 
Employing degree-theoretic methods, we prove the existence of a noncompact connected 
set of nontrivial $T$-periodic solutions that, in a sense, emanates from the set of
zeros of the unperturbed vector field. The latter is assumed to be degenerate:
Meaning that, contrary to the usual assumptions on the leading vector field, 
it is not required to be either trivial nor to have a compact set of zeros. 
In fact, known results in the nondegenerate case can be recovered from our ones. 
We also provide some illustrating examples of Lienard- and $\phi$-Laplacian-type 
perturbed equations.


Submitted January 12, 2024. Published July 09, 2024.
Math Subject Classifications: 34C25, 34C40, 34C23, 47H11.
Key Words: Coupled differential equations on manifolds; topological degree; branches of periodic solutions; Carath\'eodory vector field.