Periodic solutions for nonlinear evolution equations at resonance

TL;DR

This paper establishes conditions under which nonlinear evolution equations at resonance have periodic solutions, using fixed point index theory and averaging techniques in Banach spaces.

Contribution

It introduces a formula linking the fixed point index of the translation operator to the average of the nonlinear term on the kernel of A, under Landesman-Lazer conditions.

Findings

Proves existence of periodic solutions at resonance.

Derives a fixed point index formula involving time averaging.

Shows the fixed point index is nonzero under certain conditions.

Abstract

We are concerned with periodic problems for nonlinear evolution equations at resonance of the form $\dot u(t) = - A u(t) + F (t,u(t))$, where a densely defined linear operator $A\colon D(A)\to X$ on a Banach space $X$ is such that $-A$ generates a compact $C_0$ semigroup and $F\colon [0,+\infty)\times X \to X$ is a nonlinear perturbation. Imposing appropriate Landesman--Lazer type conditions on the nonlinear term $F$, we prove a formula expressing the fixed point index of the associated translation along trajectories operator, in the terms of a time averaging of $F$ restricted to $\mathrm{Ker} \, A$. By the formula, we show that the translation operator has a nonzero fixed point index and, in consequence, we conclude that the equation admits a periodic solution.