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On an abstract integro-differential problem

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Abstract

In an arbitrary Banach space we consider the Cauchy problem for an integro-differential equation with unbounded operator coefficients. We establish the solvability of the problem in the weight spaces of integrable functions under certain conditions on the data. To prove this, the solution of the problem is written in explicit form with the help of analytic semigroups generated by fractional powers of the operator coefficients. Here an important role is played by the conditions on the coefficients ensuring the coercive estimates of the corresponding integral operators.

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Authors and Affiliations

  1. Voronezh State University, Voronezh, USSR

    V. P. Orlov

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  1. V. P. Orlov
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Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 887–896, December, 1999.

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Orlov, V.P. On an abstract integro-differential problem. Math Notes 66, 733–740 (1999). https://doi.org/10.1007/BF02674331

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  • DOI: https://doi.org/10.1007/BF02674331

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