Integrated semigroups and integrodifferential equations

Abstract

A class of partial integrodifferential equation from viscoelasticity is formulated as the abstract integrodifferental equations in Banach space$\mathbb{X}$:

$$\begin{gathered} x'(t) = A\left[ {x(t) + \int_0^t {F(t - s)x(s)ds} } \right] + Kx(t) + f(t),t \geqslant 0, \hfill \\ x(0) = x'_0 \hfill \\ \end{gathered} $$

((a))

and

$$\begin{gathered} x'(t) = A\left[ {x(t) + \int_{ - \infty }^t {F(t - s)x(s)ds} } \right] + Kx(t) + f(t),t \geqslant 0, \hfill \\ x(s) = \varphi (s),s \leqslant 0, \hfill \\ \end{gathered} $$

((b))

withA a not necessarily densely defined operator andK andF(t) bounded operators fort≥0. We will assume that the operatorA satisfies a Hille-Yosida condition so that it is the generator of a non-degenerate, locally Lipschitz continuous integrated semigroup. We will then use the associated integrated semigroup theory to prove the existence, uniqueness and continuity of solutions with respect to initial data. A variation of constants formula for (a) is also obtained.

The results are than applied to equations of viscoelasticity.