Geodesic
Asymptotic behavior of solutions of delay equations in Hilbert spaces
Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 21 (2007), pp. 77-86 
D. A. Medvedev. Asymptotic behavior of solutions of delay equations in Hilbert spaces. Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 21 (2007), pp. 77-86. http://geodesic.mathdoc.fr/item/CMFD_2007_21_a3/
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     author = {D. A. Medvedev},
     title = {Asymptotic behavior of solutions of delay equations in {Hilbert} spaces},
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     url = {http://geodesic.mathdoc.fr/item/CMFD_2007_21_a3/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We establish unimprovable estimates of solutions of inhomogeneous delay differential-difference equations, the coefficients of which are unbounded operators and operator-functions acting in a Hilbert space. We also present results about expansions of those solutions into a sum of a (finite) linear combination of exponential solutions for the homogeneous equation and a function with a smaller power of the exponential growth.

[1] Bellman R., Kuk K., Differentsialno-raznostnye uravneniya, Mir, M., 1967 | MR | Zbl

[2] Vlasov V. V., “O razreshimosti i svoistvakh reshenii funktsionalno-differentsialnykh uravnenii v gilbertovom prostranstve”, Mat. sb., 186:8 (1995), 67–92 | MR | Zbl

[3] Vlasov V. V., “O razreshimosti i otsenkakh reshenii funktsionalno-differentsialnykh uravnenii v prostranstve Soboleva”, Tr. MIAN, 227, 1999, 109–121 | MR | Zbl

[4] Vlasov V. V., Ivanov S. A., “Otsenki reshenii neodnorodnykh differentsialno-raznostnykh uravnenii neitralnogo tipa”, Izv. vuzov, ser. matem., 2006, no. 3, 24–30 | MR | Zbl

[5] Keldysh M. V., “O polnote sobstvennykh funktsii nekotorykh klassov nesamosopryazhennykh lineinykh operatorov”, Uspekhi mat. nauk, 26:4 (1971), 15–41 | MR | Zbl

[6] Lions Zh. L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[7] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[8] Radzievskii G. V., “Asimptotika raspredeleniya kharakteristicheskikh chisel operator-funktsii, analiticheskikh v ugle”, Mat. sb., 112:3 (1980), 396–420 | MR | Zbl

[9] Di Blasio G., Kunisch K., Sinestrari E., “$L^2$-regularity for parabolic partial integrodifferential equations with delays in the highest order derivatives”, J. Math. Anal. and Appl., 102 (1984), 38–57 | DOI | MR | Zbl

[10] J. A. Burns, T. L. Herdman, H. W. Stech, “Linear functional differential equations as semigroups on product spaces”, SIAM J. Math. Anal., 14:1 (1983), 98–116 | DOI | MR | Zbl

[11] Kappel K., Kunish K., “Invariance results for delay and Volterra equations in fractional order Sobolev spaces”, Trans. Amer. Math. Soc., 304:1 (1987), 1–57 | DOI | MR

[12] Kunish K., Mastinšek M., “Dual semigroups and structural operators for partial differential equations with unbounded operators acting on the delays”, Differ. Integral Equ. Appl., 3:4 (1990), 733–756 | MR

[13] Kunish K., Shappacher W., “Necessary conditions for partial differential eqautions with delay to generate $l_0$-semigroup”, J. Differential Equations, 50 (1983), 49–79 | DOI | MR

[14] Vlasov V. V., Wu J., “Sharp estimates of solutions to neutral equations in Sobolev spaces”, Funct. Differ. Equ., 12:3–4 (2005), 437–461 | MR | Zbl