An averaging method for periodic solutions of quasilinear equations in Banach spaces
Abstract
The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations (ϕ(t,u'))' = ε f(t,u,u',ε), u(0) = u(T), u'(0) = u'(T), where u and u' take values in a Banach space X and ϕ is a suitable homeomorphism between a neighborhood of $0$ in X and X. The approach follows from a given extension of the averaging method to first order systems of the form u' = g(t,v), v' = h(t,u,v), u(0) = u(T), v(0) = v(T), in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.
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BibTeX
@misc{mawhin2026averaging,
title = {{An averaging method for periodic solutions of quasilinear equations in Banach spaces}},
author = {Jean Mawhin and Jorge Novoa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02156},
eprint = {2610.02156},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02156},
}
RIS
TY - UNPB
TI - An averaging method for periodic solutions of quasilinear equations in Banach spaces
AU - Jean Mawhin
AU - Jorge Novoa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02156
UR - https://arxiv.org/abs/2610.02156
AB - The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations (ϕ(t,u'))' = ε f(t,u,u',ε), u(0) = u(T), u'(0) = u'(T), where u and u' take values in a Banach space X and ϕ is a suitable homeomorphism between a neighborhood of $0$ in X and X. The approach follows from a given extension of the averaging method to first order systems of the form u' = g(t,v), v' = h(t,u,v), u(0) = u(T), v(0) = v(T), in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.
ER -