Semigroup Forum

, Volume 73, Issue 3, pp 427-443

First online:

A Class of Integrodifferential Equations with Memory

  • Min HeAffiliated withDepartment of Mathematical Sciences, Kent State University-Trumbull Campus Email author 

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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 \(A(\varepsilon)\) is a closed, linear, and non-densely defined operator which depends on a multi-parameter \(\varepsilon\), and F(t) and K are bounded operators for \(t\ge0\). \(\varphi\) 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.