Stable perturbations of linear differential equations generating a uniformly bounded group

Stability problems for solutions of the differential equation u'(t) = Au + epsilon B(t,u) in a Banach space are considered. It is assumed that for epsilon = 0 this equation generates a uniformly bounded group of class C-0. Sufficient conditions on B and A are found under which the solutions of this equation are bounded for small epsilon. A linearization principle is proved for this equation under certain conditions on the operator B.