Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues

TL;DR

This paper develops new Lyapunov inequalities for periodic boundary value problems at higher eigenvalues, providing optimal constants and applications to stability analysis and nonlinear problems at resonance.

Contribution

It introduces novel Lyapunov inequalities at higher eigenvalues, with detailed zero distribution analysis and applications to Hill's equation and nonlinear resonance problems.

Findings

Derived optimal Lyapunov constants for higher eigenvalues

Established new stability conditions for Hill's equation

Proved existence and uniqueness results for nonlinear problems at resonance

Abstract

This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.