Abstract
We study the well-posed solvability of initial value problems for Volterra integro-differential equations in a Hilbert space with kernels of integral operators represented by Stieltjes integrals. The approach used in the paper is related to the application of the theory of operator semigroups.
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REFERENCES
Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovykh prostranstvakh (Linear Differential Equations in Banach Spaces), Moscow: Nauka, 1967.
Engel, K.J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, New York: Springer, 2000.
Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Fundamentals of the Mathematical Theory of Thermoviscoelasticity), Moscow: Nauka, 1970.
Christensen, R.M., Theory of Viscoelasticity.An Introduction, NewYork– London:Academic Press, 1971.
Vlasov, V.V. and Rautian, N.A., Spektral’nyi analiz funktsional’no-differentsial’nykh uravnenii (Spectral Analysis of Functional Differential Equations), Moscow: Maks Press, 2016.
Vlasov, V.V. and Rautian, N.A., On Volterra integro-differential equations with kernels representable by Stieltjes integrals, Differ. Equations, 2021, vol. 57, no. 4, pp. 517–532.
Rautian, N.A., Semigroups generated by Volterra integro-differential equations, Diff. Equations, 2020, vol. 56, no. 9, pp. 1193–1211.
Rautian, N.A., On the properties of semigroups generated by Volterra integro-differential equations with kernels representable by Stieltjes integrals, Differ. Equations, 2021, vol. 57, no. 9, pp. 1231–1248.
Gel’fand, I.M. and Vilenkin, N.Ya., Nekotorye primeneniya garmonicheskogo analiza. Osnashchennye gil’bertovy prostranstva (Some Applications of Harmonic Analysis. Framed Hilbert Spaces), Moscow: Gos. Izd. FIz.-Mat. Lit., 1961.
Rabotnov, Yu.N., Elementy nasledstvennoi mekhaniki tverdykh tel (Elements of Hereditary Solid Mechanics), Moscow: Nauka, 1977.
Funding
This work was supported in part by the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284 and with the partial financial support of the Interdisciplinary Scientific and Educational School of Moscow University “Mathematical Methods of Analysis of Complex Systems.”
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Translated by V. Potapchouck
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Vlasov, V.V., Rautian, N.A. Well-Posed Solvability of Volterra Integro-Differential Equations in Hilbert Spaces. Diff Equat 58, 1410–1426 (2022). https://doi.org/10.1134/S0012266122010010X
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DOI: https://doi.org/10.1134/S0012266122010010X