Abstract
We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hale, J., Theory of Functional Differential Equations, New York: Springer, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984.
Azbelev, N.V., Maksimov, V.P., and Rakhmatullina, L.F., Elementy sovremennoi teorii funktsional’nodifferentsial’nykh uravnenii (Elements of Modern Theory of Functional-Differential Equations), Moscow, 2002.
Antonevich, A.B., Lineinye funktsional’nye uravneniya. Operatornyi podkhod (Linear Functional Equations. The Operator Approach), Minsk: Universitetskoe, 1988.
Vlasov, V.V., On the Solvability and Estimates for the Solutions of Functional-Differential Equations in Sobolev Spaces, Tr. Mat. Inst. Steklova, 1999, vol. 227, no. 4, pp. 109–121.
Vlasov, V.V. and Medvedev, D.A., Functional-Differential Equations in Sobolev Spaces and Related Problems in Spectral Theory, Sovrem. Mat. Fundam. Napravl., 2008, vol. 30, pp. 3–173.
Vlasov, V.V., Medvedev, D.A., and Rautian, N.A., Functional-Differential Equations in Sobolev Spaces and Their Spectral Analysis, Sovrem. Probl. Mat. Mekh. Mat., 2011, vol. 8, no. 1.
Lions, J.-L. and Magenes, E., Problèmes aux limites non homogénes et applications, Paris: Dunod, 1968. Translated under the title Neodnorodnye granichnye zadachi i ikh prilozheniya, Moscow: Mir, 1971.
Datko, R., Uniform Asymptotic Stability of Evolutionary Processes in a Banach Space, SIAM J. Math. Anal., 1973, vol. 3, pp. 428–445.
Datko, R., Representation of Solutions and Stability of Linear Differential-Difference Equation in a Banach Space, J. Differential Equations, 1978, vol. 29, no. 1, pp. 105–166.
Di Blasio, G., Kunisch, K., and Sinestrari, E., L 2-Regularity for Parabolic Partial Integro-Differential Equations with Delays in the Highest Order Derivatives, J. Math. Anal. Appl., 1984, vol. 102, pp. 38–57.
Di Blasio, G., Kunisch, K., and Sinestrari, E., Stability for Abstract Linear Functional Differential Equations, Israel J. Math., 1985, vol. 50, no. 3, pp. 231–263.
Kunisch, K. and Mastinšek, M., Dual Semigroups and Structural Operators for Partial Differential Equations with Unbounded Operators Acting on the Delays, Differential Integral Equations, 1990, vol. 3, no. 4, pp. 733–756.
Kunisch, K. and Schappacher, W., Necessary Conditions for Partial Differential Equations with Delay to Generate C 0-Semigroups, J. Differential Equations, 1983, vol. 50, pp. 49–79.
Staffans, O.J., Some Well-Posed Functional Equations Which Generate Semigroups, J. Differential Equations, 1985, vol. 58, no. 2, pp. 157–191.
Wu, J., Semigroup and Integral Form of Class of Partial Differential Equations with Infinite Delay, Differential Integral Equations, 1991, vol. 4, no. 6, pp. 1325–1351.
Engel, K.J. and Nagel, R., One Parameter Semigroups for Linear Evolution Equations, Springer, 1999.
Krasnosel’skii, M.A., Zabreiko, P.P., Pustyl’nik, E.I., and Sobolevskii, P.E., Integral’nye operatory v prostranstvakh summiruemykh funktsii (Integral Operators in Spaces of Integrable Functions), Moscow: Nauka, 1996.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R.Kh. Akylzhanov, V.V. Vlasov, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 9, pp. 1175–1186.
Rights and permissions
About this article
Cite this article
Akylzhanov, R.K., Vlasov, V.V. Well-posed solvability of functional-differential equations with unbounded operator coefficients. Diff Equat 50, 1161–1172 (2014). https://doi.org/10.1134/S0012266114090043
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266114090043