Periodic solutions to parameter-dependent equations with a $\phi$-Laplacian type operator

TL;DR

This paper investigates the existence and multiplicity of periodic solutions for a broad class of $ ext{phi}$-Laplacian boundary value problems with a real parameter, extending classical results under general conditions.

Contribution

It generalizes classical and recent results by analyzing a broad family of nonlinearities in a unified framework for parameter-dependent $ ext{phi}$-Laplacian equations.

Findings

Existence of zero, one, or two solutions depending on the parameter $s$.

Extension of Ambrosetti-Prodi type results to $ ext{phi}$-Laplacian equations.

Analysis under non-uniform conditions on $g(t,u)$ as $u o o \

Abstract

We study the periodic boundary value problem associated with the $\phi$-Laplacian equation of the form $(\phi(u'))'+f(u)u'+g(t,u)=s$, where $s$ is a real parameter, $f$ and $g$ are continuous functions, and $g$ is $T$-periodic in the variable $t$. The interest is in Ambrosetti-Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter $s$. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on $g(t,u)$ as $u\to \pm\infty$. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.