Asymptotic Behaviour of Nonlinear Evolution Equations in Banach Spaces

TL;DR

This paper develops new results for nonlinear evolution equations in Banach spaces using Yosida approximation, focusing on existence, asymptotic behavior, and perturbations of solutions with almost periodicity properties.

Contribution

It introduces a Yosida approximation-based approach to analyze nonlinear evolution systems, including multivalued perturbations and solutions in translation invariant subspaces.

Findings

Established criteria for existence of solutions in almost periodic function spaces.

Derived asymptotic behavior results for solutions close to invariant subspaces.

Applied methods to functional differential equations.

Abstract

We show how the approach of Yosida approximation of the derivative serves to obtain new results for evolution systems. Using this method we obtain multivalued time dependent perturbation results. Additionally, translation invariant subspaces $Y$ of the bounded and uniformly continuous functions are considered, to obtain criteria for the existence of solutions $u\in Y$ to the equation $$ u^{\prime}(t)\in A(t)u(t)+ \om u(t) + f(t), t\in \re, $$ or of solutions $u$ asymptotically close to $Y$ for the inhomogeneous differential equation \begin{eqnarray*} u^{\prime}(t)&\in& A(t)u(t) + \om u(t) + f(t), \ \ t > 0, u(0)&=&u_0, \end{eqnarray*} in general Banach spaces, where $A(t)$ denotes a possibly nonlinear time dependent dissipative operator. Particular examples for the space $Y$ are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior. Further, an application to functional differential equations is given.