Ulam-Hyers stability and exponentially stable evolution equations in Banach spaces

We show that uniformly exponentially stable abstract linear evolution equations are Ulam-Hyers stable on [a,\infty). Moreover, we prove that this property is maintained when perturbing this type of equations with a nonlinear term having a small Lipschitz constant. These results complement the literature on Ulam-Hyers stability, a special relation having with some works of I. A. Rus.