Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Sharp estimates for solutions of systems with aftereffect


Authors: V. V. Vlasov and S. A. Ivanov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 20 (2008), nomer 2.
Journal: St. Petersburg Math. J. 20 (2009), 193-211
MSC (2000): Primary 34K12
DOI: https://doi.org/10.1090/S1061-0022-09-01044-9
Published electronically: January 30, 2009
MathSciNet review: 2423996
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sharp estimates are established for strong solutions of systems of differential-difference equations of both neutral and retarded type.

The approach is based on the study of the resolvent corresponding to the generator of the semigroup of shifts along the trajectories of a dynamical system. In the case of neutral type equations, the Riesz basis property of the subsystem of exponential solutions is used.


References [Enhancements On Off] (What's this?)

  • 1. R. Bellman and K. Cooke, Differential-difference equations, Acad. Press, New York-London, 1963. MR 0147745 (26:5259)
  • 2. A. D. Myshkis, Linear differential equations with retarded argument, ``Nauka'', Moscow, 1972. (Russian) MR 0352648 (50:5135)
  • 3. J. Hale, Theory of functional differential equations, Appl. Math. Sci., vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721 (58:22904)
  • 4. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, Delay equations. Functional, complex, and nonlinear analysis, Appl. Math. Sci., vol. 110, Springer-Verlag, New York, 1995. MR 1345150 (97a:34001)
  • 5. J. Wu, Theory and applications of partial functional-differential equations, Appl. Math. Sci., vol. 119, Springer-Verlag, New York, 1996. MR 1415838 (98a:35135)
  • 6. D. Henry, Linear autonomous neutral functional differential equations, J. Differential Equations 15 (1974), 106-128. MR 0338520 (49:3284)
  • 7. A. P. Khromov, Finite-dimensional perturbations of Volterra operators, Sovrem. Mat. Fundam. Napravl. 10 (2004), 3-163; English transl., J. Math. Sci. (N.Y.) 138 (2006), no. 5, 5893-6066. MR 2120867 (2007m:47028)
  • 8. V. V. Vlasov, Correct solvability of a class of differential equations with deviating argument in Hilbert space, Izv. Vyssh. Uchebn. Zaved. Mat. 1996, no. 1, 22-35; English transl., Russian Math. (Iz. VUZ) 40 (1996), no. 1, 19-32. MR 1424147 (97j:34104)
  • 9. -, Estimates for solutions of differential-difference equations of neutral type, Izv. Vyssh. Uchebn. Zaved. Mat. 2000, no. 4, 14-22; English transl., Russian Math. (Iz. VUZ) 44 (2000), no. 4, 12-20. MR 1782521 (2001i:34137)
  • 10. V. V. Vlasov and J. Wu, Sharp estimates of solutions to neutral equations in Sobolev spaces, Funct. Differ. Equ. 12 (2005), no. 3-4, 437-461. MR 2137522 (2005k:34319)
  • 11. V. V. Vlasov, Some properties of a system of elementary solutions of differential-difference equations, Uspekhi Mat. Nauk 51 (1996), no. 1, 143-144; English transl., Russian Math. Surveys 51 (1996), no. 1, 175-176. MR 1392677 (97b:34088)
  • 12. A. M. Zverkin, Series expansion of solutions of linear difference-differential equations. I. Quasipolynomials, Theory of Differential Equations with Deviating Argument. Vol. 3 (Proc. of Sem.), Univ. Druzhby Narodov Patrisa Lumumby, Moscow, 1965, pp. 3-38. (Russian) MR 0201764 (34:1646)
  • 13. D. Medvedev and V. Vlasov, On certain properties of exponential solutions of difference differential equations in Sobolev spaces, Funct. Differ. Equ. 9 (2002), no. 3-4, 423-435. MR 1971620 (2004c:34183)
  • 14. -, Estimates for the solutions of differential-difference equations of neutral type, Dokl. Akad. Nauk 389 (2003), no. 2, 156-158; English transl. in Dokl. Math. MR 2003230 (2004h:34158)
  • 15. -, On estimates for solutions of differential equations with retarded argument, Izv. Vyssh. Uchebn. Zaved. Mat. 2004, no. 6, 21-29; English transl., Russian Math. (Iz. VUZ) 48 (2004), no. 6, 19-27 (2005). MR 2101715 (2005f:34185)
  • 16. V. V. Vlasov and S. A. Ivanov, Sobolev space estimates for solutions of equations with delay, and the basis of divided differences, Algebra i Analiz 15 (2003), no. 4, 115-141; English transl., St. Petersburg Math. J. 15 (2004), no. 4, 545-561. MR 2068981 (2005c:34139)
  • 17. S. A. Avdonin and S. A. Ivanov, Riesz bases formed by exponentials and divided differences, Algebra i Analiz 13 (2001), no. 3, 1-17; English transl., St. Petersburg Math. J. 13 (2002), no. 3, 339-351. MR 1850184 (2002g:42009)
  • 18. A. A. Lesnykh, Estimates for solutions of differential-difference equations of a neutral type, Mat. Zametki 81 (2007), no. 4, 569-585; English transl., Math. Notes 81 (2007), no. 3-4, 503-517. MR 2352023 (2008m:34183)
  • 19. V. V. Vlasov and S. A. Ivanov, Estimates for solutions of nonhomogeneous differential-difference equations of neutral type, Izv. Vyssh. Uchebn. Zaved. Mat. 2006, no. 3, 24-30; English transl., Russian Math. (Iz. VUZ) 50 (2006), no. 3, 22-28. MR 2249016 (2007d:34153)
  • 20. -, On estimates for solutions of functional-differential equations of neutral type in Sobolev space, Dokl. Akad. Nauk 396 (2004), no. 4, 443-445; English transl. in Dokl. Math. MR 2115516 (2005h:34213)
  • 21. -, Sharp estimates for solutions of differential-difference equations of neutral type, Dokl. Akad. Nauk 406 (2006), no. 5, 583-585; English transl. in Dokl. Math. MR 2347314 (2008j:34125)
  • 22. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vols. I, II, Grundlehren Math. Wiss., Bd. 181, 182, Springer-Verlag, New York-Heidelberg, 1972; Vol. III, Grundlehren Math. Wiss., Bd. 183, Springer-Verlag, New York-Heidelberg, 1973. MR 0350177 (50:2670); MR 0350178 (50:2671); MR 0350179 (50:2672)
  • 23. K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Grad. Texts in Math., vol. 194, Springer-Verlag, New York, 2000. MR 1721989 (2000i:47075)
  • 24. S. G. Kreĭn, Linear differential equations in a Banach space, ``Nauka'', Moscow, 1967. (Russian) MR 0247239 (40:508)
  • 25. T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., Bd. 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617 (53:11389)
  • 26. Yu. L. Daletskiĭ and M. G. Kreĭn, Stability of solutions of differential equations in Banach space, ``Nauka'', Moscow, 1970; English transl., Transl. Math. Monogr., vol. 43, Amer. Math. Soc., Providence, RI, 1974. MR 0352638 (50:5125); MR 0352639 (50:5126)
  • 27. A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, ``Shtiintsa'', Kishinev, 1986; English transl., Transl. Math. Monogr., vol. 71, Amer. Math. Soc., Providence, RI, 1988. MR 0861412 (88d:47022); MR 0971506 (89h:47023)
  • 28. A. M. Sedletskiĭ, Biorthogonal expansions of functions in exponential series on intervals of the real axis, Uspekhi Mat. Nauk 37 (1982), no. 5, 51-95; English transl., Russian Math. Surveys 37 (1982), no. 5, 57-108. MR 0676613 (84g:42025)
  • 29. A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, Trudy Sem. Petrovsk. No. 9 (1983), 190-229; English transl. in J. Soviet Math. 33 (1986), no. 5, 6. MR 0731903 (85m:34028)
  • 30. B. Ya. Levin, Interpolation by entire functions of exponential type, Trudy Fiz.-Tekhn. Inst. Nizkikh Temperatur Akad. Nauk Ukrain. SSR, vyp. 1, Khar'kov. Mat. Obshch., Khar'kov, 1962, pp. 136-146. (Russian) MR 0466562 (57:6439)
  • 31. F. Kappel and K. P. Zhang, Equivalence of functional-differential equations of neutral type and abstract Cauchy problems, Monatsh. Math. 101 (1986), no. 2, 115-133. MR 0843296 (87f:34079)
  • 32. J. A. Burns, T. L. Herdman, and H. W. Stech, Linear functional-differential equations as semigroups on product spaces, SIAM J. Math. Anal. 14 (1983), no. 1, 98-116. MR 0686237 (84e:34096)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 34K12

Retrieve articles in all journals with MSC (2000): 34K12


Additional Information

V. V. Vlasov
Affiliation: Moscow State University, Leninskiye Gory, Moscow, 119992, Russia
Email: vlasov@math.mipt.ru, vikvvlasov@rambler.ru

S. A. Ivanov
Affiliation: St. Petersburg State University, Russian Center of Laser Physics, Ulyanovskaya 1, St. Petersburg, 198904, Russia
Email: Sergei.Ivanov@pobox.spbu.ru

DOI: https://doi.org/10.1090/S1061-0022-09-01044-9
Keywords: Differential-difference equation, retarded type, neutral type, neutral-neutral type
Received by editor(s): November 14, 2006
Published electronically: January 30, 2009
Additional Notes: Supported by RFBR (grant no. 05-01-00989) and by grant NSh-5247.2006.1
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society