Global continuum of solutions for systems of ODEs with periodic boundary conditions and generalized variable-exponent operators

Raúl Manásevich; Jorge Novoa

doi:10.3934/cpaa.2026032

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2026, Volume 30: 151-166. Doi: 10.3934/cpaa.2026032

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Global continuum of solutions for systems of ODEs with periodic boundary conditions and generalized variable-exponent operators

Raúl Manásevich^{1, 2,} and
Jorge Novoa^1, ,

1.
Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile
2.
Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile, Santiago, Chile

^*Corresponding author: Jorge Novoa
^*Corresponding author: Jorge Novoa

Received: November 4, 2025

Revised: January 15, 2026

Early access: February 10, 2026

Published online: February 10, 2026

The authors were partially supported by Centro de Modelamiento Matemático (CMM) BASAL fund FB210005 for centers of excellence from ANID-Chile.

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Abstract

We study nonlinear systems of boundary-value problems in one dimension of the form

$ (\mathcal S(t, u(t)'))' = f(t, u(t), u'(t), \lambda), $

subject to periodic boundary value conditions. The function $ \mathcal S:[0, T]\times\mathbb{R}^N\to\mathbb{R}^N $ is continuous and satisfies $ \mathcal S(0, x) = \mathcal S(T, x) $ for all $ x\in\mathbb{R}^N $, and the function $ f $ is Carathéodory. Furthermore, we assume $ \mathcal{S} $ satisfies some monotonicity and coercivity conditions, thereby defining a class of generalized variable-exponent operators that strictly contains the Musielak–Orlicz case. Using an abstract continuation framework and the Leray–Schauder degree, we establish the existence of a global continuum of nontrivial solutions depending on the parameter $ \lambda $. Our approach builds on a degree-theoretic alternative of Fitzpatrick, Massabò, and Pejsachowicz, reducing the evaluation of the Leray–Schauder degree to a Brouwer degree. This work shows how monotonicity and coercivity conditions in one dimension allow for continuum of solutions in a setting strictly more general than the Musielak–Orlicz framework. Furthermore, our results are new even if $ \mathcal S $ is independent of $ t $.

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Mathematics Subject Classification: Primary: 34A34, 34B08; Secondary: 35J92, 47H11, 47J15.

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