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Second order linear Volterra integrodifferential equations

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Abstract

In this paper we study mild and classical solutions of the second order linear Volterra integrodifferential equation

$$(VE^f )\left\{ {\begin{array}{*{20}c} {u''(t) = Au(t) + {\text{ }}\int_0^t {B(t - s)u(s)ds + f(t){\text{ }}for{\text{ }}t \in [0,T]} } \\ {u(0) = x{\text{ }}and{\text{ }}u'(0) = y,} \\ \end{array} } \right.$$

whereA is a closed linear operator whose domainD(A) is not necessarily dense in a Banach spaceX, and {B(t)|t≥0} is a family of bounded linear operators from the Banach space,D(A) endowed with the graph norm intoX. We also give two examples to illustrate the abstract results.

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Authors and Affiliations

  1. Department of Mathematics School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, 169-50, Tokyo, Japan

    Hirokazu Oka

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  1. Hirokazu Oka
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Communicated by Jerome A. Goldstein

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Oka, H. Second order linear Volterra integrodifferential equations. Semigroup Forum 53, 25–43 (1996). https://doi.org/10.1007/BF02574118

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