Generalized Dichotomies and Hyers–Ulam Stability

We consider a semilinear and nonautonomous differential equation 1 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.