## Mathematical Sciences Publications

#### Title

A Class of Integrodifferential Equations With Memory

Semigroup Forum

12-2006

Article

#### DOI

10.1007/S00233-006-0627-0

#### Keywords

banach spaces, integro-differential equations, viscoelasticity, integral equations, banach algebras

Mathematics

#### Abstract

This work considers an abstract integrodifferential equation in Banach space: $u'(t) = A(\varepsilon) \left[u(t)+\int_{-\infty}^t F(t-s)u(s)\,ds\right]+Ku(t)+f(t), \quad t\ge0,$ $u(s) = \varphi(s),\quad s\le0,$ where Formula Not Shown is a closed, linear, and non-densely defined operator which depends on a multi-parameterFormula Not Shown , and F(t) and K are bounded operators for Formula Not Shown . Formula Not Shown is refereed to as the "memory" of the equation. This study investigates the effect of the parameter on the solution of this equation. In particular, this work attempts to determine conditions for continuity with respect to parameters of solutions of this equation. Methods are employed to treat two different cases and to obtain results on continuity in parameters of integrated semigroups. With the aid of the theory of integrated semigroups, these results simply lead to the analogous results for the solution of this equation. In the last section, the applications of the obtained results to some equations of viscoelasticity are discussed.

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