  In this paper we present an extension of the Katznelson-Tzafriri Theorem to
the asymptotic behavior of individual solutions of evolution equations $u'(t)
=Au(t)+f(t)$. 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.
