Electron. J. Differential Equations, Vol. 2024 (2024), No. 39, pp. 113.
Caratheodory periodic perturbations of degenerate systems
Alessandro Calamai, Marco Spadini
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 Caratheodorytype regularity.
Employing degreetheoretic 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 \(\phi\)Laplaciantype
perturbed equations.
Submitted January 12, 2024. Published July 9, 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.
DOI: 10.58997/ejde.2024.39
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Alessandro Calamai
Dipartimento di Ingegneria Civile, Edile e Architettura
Università Politecnica delle Marche
Via Brecce Bianche, I60131 Ancona, Italy
email: a.calamai@univpm.it


Marco Spadini
Dipartimento di Matematica e Informatica "Ulisse Dini''
Università degli Studi di Firenze
Via S. Marta 3, I50139 Florence, Italy
email: marco.spadini@unifi.it

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