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Studying Volterra Integro-Differential Equations by Methods of the Theory of Operator Semigroups

  • INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
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Abstract

We study abstract Volterra integro-differential equations with kernels of integral operators represented by Stieltjes integrals. The results are based on an approach related to the study of one-parameter semigroups for linear evolution equations. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space. The existence of a contraction \(C_0\)-semigroup is proved. The properties of the generator of the semigroup are established based on the properties of the operator function that is the symbol of the original integro-differential equation.

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Funding

The proof of Theorems 6 and 7 was supported by the Russian Foundation for Basic Research, project no. 20-01-00288 A. The proof of Theorems 9 and 10 was supported by the Interdisciplinary Scientific and Educational School of Lomonosov Moscow State University “Mathematical Methods for the Analysis of Complex Systems.”

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. A. Rautian

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    N. A. Rautian

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  1. N. A. Rautian
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Correspondence to N. A. Rautian.

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Translated by V. Potapchouck

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Rautian, N.A. Studying Volterra Integro-Differential Equations by Methods of the Theory of Operator Semigroups. Diff Equat 57, 1665–1684 (2021). https://doi.org/10.1134/S0012266121120120

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  • DOI: https://doi.org/10.1134/S0012266121120120

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