A class of integrodifferential equations and applications

Pages: 386 - 396, Issue Special, August 2005

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Min He - Department of Mathematical Sciences, Kent State University-Trumbull Campus, Warren, Ohio 44483, United States (email)

Abstract: This work considers an abstract integrodifferential equation in Banach space: \be & &u'(t)=A(\ep)\left[u(t)+\int_0^t F(t-s)u(s)\,ds\right]+Ku(t)+f(t), \,\,t\ge0, \nonumber\\ & &u(0)=u_0, \nonumber \ee where $A(\ep)$ is a closed, linear, and non-densely defined operator, $\ep$ is a multi-parameter, and $K$ and $F(t)$ are bounded operators. The purpose of this work is to study effect of the parameter on the solution of the equation. The approach used is based on the integrated semigroup theory. Two methods are employed to determine the continuity in parameter $\ep$ of integrated semigroup, which is generated by $A(\ep)$. The main theorems on the integrated semigroup can be effectively used to obtain the similar results for the solution of the equation. The applications of these results to some equations of viscoelasticity are discussed.

Keywords:  Integrated semigroup, Parameter, Continuity .
Mathematics Subject Classification:  Primary: 47D62, 45K05; Secondary: 35L20.

Received: September 2004;      Revised: April 2005;      Published: September 2005.