On the asymptotic behavior of a semilinear functional differential equation in Banach space (Q1079728)

Let X be a Banach space, \(\{\) A(t):t\(\geq 0\}^ a \)family of linear operators on X that generates a linear evolution system, r a positive real number, and \(\phi: [-r,0]\to X\) a continuous function. This paper is concerned with the asymptotic behavior of the solutions to the semilinear functional differential equation \[ u'(t)+A(t)u(t)=f(t,u_ t),\quad t\geq 0,\quad u_ 0=\phi, \] where f is a sufficiently small nonlinear perturbation and the ''history'' \(u_ t: [-r,0]\to X\) is defined by \(u_ t(s)=u(t+s)\) for -r\(\leq s\leq 0\). It is shown that if the linear equation \(u'(t)+A(t)u(t)=0\) exhibits asymptotic stability, so does the nonlinear one. this result improves upon that of \textit{S. M. Rankin} [ibid. 88, 531- 542 (1982; Zbl 0519.34045)] because the assumed bound on f does not depend on the delay r.