in a Banach space X. The functions u(t) andf(t) are defined on the real half line [0, co) and have values in X. The function A(t) has the same domain of definition and has values in a set of unbounded linear operators acting in X. Tanabe [I] investigated the behavior as t + co of the solution of Eq. (1.1). He stated that under certain natural conditions on the behavior of A(t) and f(t), one can prove that if both converge in some sense as t -+ 03, then the solution u(t)of Eq. (1.1) aJs 0 converges to some element-of X as t -+ 00. In the present note we assume a certain asymptotic behavior of A(t) and f(t) as t -+ co and obtain a corresponding asymptotic behavior of the solution u(t). This result is used in Section 4 to obtain an asymptotic expansion of the solution of a parabolic equation in a cylindrical domain. Results similar to the results of Section 4 were obtained by Friedman [4].