Averaging principle and periodic solutions for nonlinear evolution equations at resonance

TL;DR

This paper establishes an index formula for the existence of periodic solutions in nonlinear evolution equations at resonance, generalizing classical conditions and applying to heat equations.

Contribution

It introduces a new index formula relating topological degree to geometrical assumptions, extending Landesman-Lazer conditions for resonance problems.

Findings

Derived an index formula for translation operators at resonance.

Generalized Landesman-Lazer and strong resonance conditions.

Provided criteria for periodic solutions in heat equations at resonance.

Abstract

We study the existence of $T$-periodic solutions $(T > 0)$ for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of well known Landesman-Lazer and strong resonance conditions. Obtained index formula is used to derive the criteria determining the existence of $T$-periodic solutions for the heat equation being at resonance at infinity.