Stability of solutions to some abstract evolution equations with delay

The global existence and stability of the solution to the delay differential equation (*)u(center dot)=A(t)u+G(t,u(t-tau))+f(t), t >= 0, u(t)=v(t), -tau <= t <= 0, are studied. Here A(t):H -> H is a closed, densely defined, linear operator in a Hilbert space H and G(t,u) is a nonlinear operator in H continuous with respect to u and t. We assume that the spectrum of A(t) lies in the half-plane R lambda <= gamma(t), where gamma(t) is not necessarily negative and parallel to G(t,u)parallel to <= alpha(t)parallel to u parallel to(p), p>1, t >= 0. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as t tends to infinity, under the non-classical assumption that gamma(t) can take positive values, are proposed and justified.