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On the Properties of Semigroups Generated by Volterra Integro-Differential Equations with Kernels Representable by Stieltjes Integrals

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Abstract

Abstract Volterra integro-differential equations with integral operator kernels representable by Stieltjes integrals of the exponential function are studied. The approach is based on the study of one-parameter semigroups for linear evolution equations. A method 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 is presented. The existence of a contraction \(C_0 \)-semigroup is proved. As a corollary, we establish the well-posed solvability of the resulting Cauchy problem for the first-order differential equation in an extended function space and the initial value problem for the original abstract integro-differential equation and indicate a relation between their solutions.

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Funding

This work was supported by the Interdisciplinary Scientific and Educational School “Mathematical Methods for the Analysis of Complex Systems” at Lomonosov Moscow State University (Theorems 1 and 2) and by the Russian Foundation for Basic Research, project no. 20-01-00288-A (Theorems 3 and 4).

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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. On the Properties of Semigroups Generated by Volterra Integro-Differential Equations with Kernels Representable by Stieltjes Integrals. Diff Equat 57, 1231–1248 (2021). https://doi.org/10.1134/S0012266121090111

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  • DOI: https://doi.org/10.1134/S0012266121090111

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