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).
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Horodnii, M.F. Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space. Ukrainian Mathematical Journal 55, 1071–1085 (2003). https://doi.org/10.1023/B:UKMA.0000010606.57757.4b
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DOI: https://doi.org/10.1023/B:UKMA.0000010606.57757.4b