On asymptotic properties of solutions of functional differential equations of neutral type
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 1, Tome 15 (2006), pp. 112-125.

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In this paper, we obtain sharp estimates for strong solutions of functional differential equations of neutral type. Our result is closely connected with our previous results devoted to the initial-value problem for above-mentioned equations in the scale of Sobolev spaces. To obtain our estimates of the solutions, we essentially use Riesz basis properties of the system of exponential solutions. The fact that they form a Riesz basis is one of the main results of this article.
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V. V. Vlasov; D. A. Medvedev. On asymptotic properties of solutions of functional differential equations of neutral type. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 1, Tome 15 (2006), pp. 112-125. http://geodesic.mathdoc.fr/item/CMFD_2006_15_a7/

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

[2] Vlasov V. V., “Korrektnaya razreshimost odnogo klassa differentsialnykh uravnenii v gilbertovom prostranstve”, Izv. vuzov, ser. mat., 1996, no. 1, 22–35 | MR | Zbl

[3] Vlasov V. V., “Ob odnom klasse differentsialno-raznostnykh uravnenii neitralnogo tipa”, Izv. vuzov, ser. mat., 1999, no. 2, 20–29 | MR | Zbl

[4] Vlasov V. V., “Ob otsenkakh reshenii differentsialno-raznostnykh uravnenii neitralnogo tipa”, Izv. vuzov, ser. mat., 2000, no. 4, 14–22 | MR | Zbl

[5] Vlasov V. V., Ivanov S. A., “Otsenki reshenii uravnenii s posledeistviem v prostranstvakh Soboleva i bazis iz razdelennykh raznostei”, Mat. zametki, 72:2 (2002), 303–306 | MR | Zbl

[6] Vlasov V. V., Ivanov S. A., “Otsenki reshenii uravnenii s posledeistviem v shkale prostranstv Soboleva i bazis iz razdelennykh raznostei”, Algebra i analiz, 15:4 (2003), 115–141 | MR | Zbl

[7] Vlasov V. V., Medvedev D. A., “Otsenki reshenii differentsialno-raznostnykh uravnenii neitralnogo tipa”, Dokl. RAN, 389:2 (2003), 156–158 | MR | Zbl

[8] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[9] Gromova S. G., Zverkin A. M., “O trigonometricheskikh ryadakh, summoi kotorykh yavlyaetsya nepreryvnaya neogranichennaya na chislovoi osi funktsiya – reshenie uravneniya s otklonyayuschimsya argumentom”, Diff. ur-ya, 4:10 (1986), 1774–1784 | MR

[10] Markus A. S., Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtinitsa, Kishinev, 1986 | MR | Zbl

[11] Miloslavskii A. I., “Ob ustoichivosti nekotorykh klassov evolyutsionnykh uravnenii”, Sib. mat. zh., 26 (1985), 118–132 | MR

[12] Myshkis A. D., Lineinye differentsialnye uravneniya s zapazdyvayuschim argumentom, Nauka, M., 1972 | MR | Zbl

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

[14] Sedletskii A. M., “Biortogonalnye razlozheniya funktsii v ryady eksponent na intervalakh veschestvennoi osi”, Usp. mat. nauk, 37:5 (1982), 51–95 | MR

[15] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984 | MR

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

[17] Hale J., Verduyn Lunel S., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993 | MR

[18] Henry D., “Linear autonomous neutral functional differential equations”, J. Differ. Equations, 15 (1974), 106–128 | DOI | MR | Zbl

[19] Kolmanovskii V., Nosov V., Stability of Functional Differential Equations, Academic Press, San Diego, 1986 | MR | Zbl

[20] Rabath R., Sklyar G., Resounenko A., “Generalized Riesz basis property in the analysis of neutral type systems”, C. R. Acad. Sci. Paris, 1337 (2003), 19–24 | MR

[21] Verduyn Lunel S. M., Yakubovich D. V., “A functional model approach to linear neutral functional difference equations”, Int. Equ. Oper. Theory, 27 (1997), 347–378 | DOI | MR | Zbl

[22] Vlasov V. V., “On spectral problems arising in the theory of functional differential equations”, Funct. Differ. Equ., 8:3–4 (2001), 435–446 | MR | Zbl

[23] Vlasov V. V., Medvedev D. A., “On certain properties of exponential solutions of difference differential equations in Sobolev spaces”, Funct. Differ. Equ., 9:3–4 (2002), 423–435 | MR | Zbl

[24] 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

[25] Wu J., “Theory and applications of partial functional differential equations”, Appl. Math. Sci., 1996, 119 | MR