Katznelson-Tzafriri Type Theorems for Individual Solutions of Evolution Equations

Abstract

In this paper we present an extension of the Katznelson-Tzafriri Theorem to the asymptotic behavior of individual solutions of evolution equations $u^{\prime }(t)=Au(t)+f(t)$. The obtained results do not require the uniform continuity of solutions as well as the well-posedness of the equations. The method of study is based on a recently developed approach to the spectral of functions that is direct and free of C_0-semigroups.

This is the revised (and final) version of the version of March 28, 2007. This paper is to appear in the Proceedings of the A.M.S