# Solvability on the real line of a class of linear volterra integrodifferential equations of parabolic type

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## Summary

We consider an abstract parabolic integrodifferential equation with infinite delay in general Banach space X: where A: D(A) ⊂ X → X generates an analytic semigroup, and K(t) ε L(D(A), X) for every t ⩾ 0. Under suitable assumptions on the kernel K, we extend to equation (*) the well known results about bounded solutions, periodic solutions, and solutions with exponential growth, of the abstract parabolic equation:

$$u'(t) = Au(t) + \int\limits_{ - \infty }^t {K(t - s)u(s)ds + f(t),t \in R}$$

(*)

$$v'(t) = Av(t) + f(t),t \in R.$$

(**)

## Keywords

Banach Space Periodic Solution Exponential Growth Parabolic Equation Real Line
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