Ulam-Hyers stability and exponentially dichotomic evolution equations in Banach spaces.

For finite-dimensional linear differential systems with bounded coefficients we prove that their exponential dichotomy on R is equivalent to their Ulam-Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam-Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant. [ABSTRACT FROM AUTHOR]