Exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay

We show that if U(t,s)U(t,s) is an exponentially dichotomic evolution operator, then the unique solution of the Volterra equation V(t,s)=U(t,s)+∫stU(t,τ)B(τ)V(τ,s)dτ is also an exponentially dichotomic evolution operator, for small B(t)B(t). As a consequence, we prove that any exponentially dichotomic evolution family is structurally stable. We improve the previous results, showing that the condition of bounded growth and decay, present everywhere in the existing literature, can be completely removed.