Averaging Principle on Semi-axis for Semi-linear Differential Equations

TL;DR

This paper proves an averaging principle for semi-linear differential equations on the semi-axis with unbounded operators and asymptotically recurrent coefficients, showing solutions converge to averaged solutions uniformly as the small parameter tends to zero.

Contribution

It establishes an averaging principle for semi-linear equations with unbounded operators and asymptotically recurrent coefficients on the semi-axis, extending existing theory.

Findings

Existence of solutions with the same asymptotic recurrence as coefficients

Uniform convergence of solutions to the averaged equation's stationary solution

Applicable to equations with various asymptotic behaviors

Abstract

We establish an averaging principle on the real semi-axis for semi-linear equation \begin{equation}\label{eqAb1} x'=\varepsilon (\mathcal A x+f(t)+F(t,x))\nonumber \end{equation} with unbounded closed linear operator $\mathcal A$ and asymptotically Poisson stable (in particular, asymptotically stationary, asymptotically periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent) coefficients. Under some conditions we prove that there exists at least one solution, which possesses the same asymptotically recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this solution converges to the stationary solution of averaged equation uniformly on the real semi-axis when the small parameter approaches to zero.