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Stability properties of linear Volterra integrodifferential equations in a Banach space. (English) Zbl 1113.45013

Summary: For linear Volterra integrodifferential equations we characterize the uniform asymptotic stability property of the zero solution by a property for the resolvent operator. In particular, for equations of convolution type, we characterize the uniform asymptotic stability property in terms of the integrability of the resolvent operator, as well as the invertibility of the characteristic operator. Furthermore, we apply our results to nonhomogeneous equations with asymptotically almost periodic forcing terms, and establish some results on the existence of asymptotically almost periodic solutions.

MSC:

45M10 Stability theory for integral equations
45M05 Asymptotics of solutions to integral equations
45J05 Integro-ordinary differential equations
45M15 Periodic solutions of integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
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