Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

Abstract

For a sectorial operator A with spectrum σ(A) that acts in a complex Banach space B, we prove that the condition σ(A) ∩ i R = Ø is sufficient for the differential equation $\varepsilon x_\varepsilon^\prime\prime(t)+x_\varepsilon^\prime(t)=Ax_\varepsilon(t)+f(t), t \in R,$ where ε is a small positive parameter, to have a unique bounded solution x _ε for an arbitrary bounded function f: R → B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as ε → 0+ to the unique bounded solution of the differential equation x′(t) = Ax(t) + f(t).