#### Title

A Class of Integrodifferential Equations With Memory

#### Publication Title

Semigroup Forum

#### Publication Date

12-2006

#### Document Type

Article

#### DOI

10.1007/S00233-006-0627-0

#### Keywords

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

#### Disciplines

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-parameter*Formula 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.

#### Recommended Citation

He, Min (2006). A Class of Integrodifferential Equations With Memory. *Semigroup Forum* 73(3), 427-443. doi: 10.1007/S00233-006-0627-0 Retrieved from https://digitalcommons.kent.edu/mathpubs/6