Volterra equations in Banach spaces with completely monotone kernels |
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Authors: | Stefano Bonaccorsi Gertrud Desch |
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Affiliation: | 1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123, Povo, Trento, Italy 2. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstrasse 36, 8010, Graz, Austria
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Abstract: | We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized. |
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