Generalized Dichotomies and Hyers-Ulam Stability

We consider a semilinear and nonautonomous differential equation x ' = A(t)x + f(t,x) t >= 0, acting on an arbitrary Banach space X. Provided that the linear part x ' = A(t)x exhibits a very general form of dichotomic behaviour and that the nonlinear term f is Lipschitz in the second variable (with a suitable Lipshitz constant), we prove that (1) admits two different forms of a generalized Hyers-Ulam stability. Moreover, we obtain the converse result which shows that under suitable additional assumptions, the presence of these two forms of a generalized Hyers-Ulam stability for the linear equation x ' = A(t)x implies that it exhibits this general dichotomic behaviour.