Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

TL;DR

This paper establishes the existence of two positive periodic solutions for a class of nonlinear second-order differential equations with indefinite weight, using topological degree theory, particularly Mawhin's coincidence degree, in the super-sublinear case.

Contribution

It introduces a novel application of Mawhin's coincidence degree to find multiple positive solutions for nonlinear ODEs with indefinite weights and super-sublinear nonlinearities.

Findings

Proves existence of two positive solutions under certain integral and parameter conditions.

Utilizes Mawhin's coincidence degree and index calculations for the analysis.

Addresses super-sublinear growth conditions at zero and indefinite weight functions.

Abstract

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $\lambda > 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.