STABILITY OF BOUNDED SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH SMALL PARAMETER IN A BANACH SPACE


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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) ∩ iR = ∅ is sufficient for the differential equationwhere ε is a small positive parameter, to have a unique bounded solution xε for an arbitrary bounded function f: RB 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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