Download PDF - Existence of solutions for second order stochastic differential inclusions in Hilbert spacesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Existence of solutions for second order stochastic differential inclusions in Hilbert spaces

Authors 1, 2

Affiliations

  1. Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram - 624 302, Tamil Nadu, India
  2. Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Abstract

In this paper, sufficient conditions are given for the existence of solutions for a class of second order stochastic differential inclusions in Hilbert space with the help of Leray-Schauder Nonlinear Alternative.

Keywords

ENG
existence, multivalued map, stochastic differential inclusions, fixed point, Hilbert space

Bibliography

  1. N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stochastic Anal. Appl. 12 (1994), 1-10.
  2. J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
  3. P. Balasubramaniam, Existence of solutions of functional stochastic differential inclusions, Tamkang J. Math. 33 (2002), 35-43.
  4. P. Balasubramaniam, S.K. Ntouyas and D. Vinayagam, Existence of solutions of nonlinear stochastic differential inclusions in a Hilbert space, Comm. Appl. Nonlinear Anal. 12 (2005), 1-15.
  5. P. Balasubramaniam and J.Y. Park, Nonlocal Cauchy problem for second order stochastic evolution equations in Hilbert spaces, Dynamic Syst. Appl. (in press).
  6. J. Ball, Initial boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90.
  7. J. Bochenek, An abstract nonlinear second order differential equation, Ann. Polon. Math. 2 (1991), 155-166.
  8. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
  9. W.E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal. 13 (1982), 739-745.
  10. K. Deimling, Multivalued Differential Equations, de Gruyter, New York, 1992.
  11. A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  12. S. Hu and N.S. Papageorgiou, On the existence of periodic solutions for non-convex valued differential inclusions in ℝⁿ, Proc. Amer. Math. Soc. 123 (1995), 3043-3050.
  13. S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 1997.
  14. D.N. Keck and M.A. McKibben, Functional integro-differential stochastic evolution equations in Hilbert space, J. Appl. Math. Stochastic Anal. 16 (2003), 127-147.
  15. P. Kree, Diffusion equation for multivalued stochastic differential equations, J. Funct. Anal. 49 (1982), 73-90.
  16. A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.
  17. N.I. Mahmudov and M.A. McKibben, Abstract second-order damped McKean-Vlasov stochastic evolution equations, Stochastic Anal. Appl. 24 (2006), 303-328.
  18. M.A. McKibben, Second-order neutral stochastic evolution equations with heredity, J. Appl. Math. Stochastic Anal. 2 (2004), 177-192.
  19. M. Michta and J. Motyl, Second order stochastic inclusion, Stochastic Anal. Appl. 22 (2004), 701-720.
  20. M. Martelli, A Rothe's type theorem for non-compact acyclic-valued map, Boll. Unione Mat. Ital. 4 (11) (1975), 70-76.
  21. S.K. Ntouyas, Global existence results for certain second order delay integrodifferential equations with nonlocal conditions, Dynam. Systems Appl. 7 (1998), 415-425.
  22. S.K. Ntouyas and P.Ch. Tsamatos, Global existence for second order functional semilinear equations, Period. Math. Hungar. 31 (1995), 223-228.
  23. N. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Univ. Carol. 29 (1988), 355-362.
  24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
  25. R. Pettersson, Yosida approximations for multivalued stochastic differential equations, Stochastics and Stochastics Reports 52 (1995), 107-120.
  26. R. Pettersson, Existence theorem and Wong-Zakai approximations for multivalued stochastic differential equations, Probability and Mathematical Statistics 17 (1997), 29-45.
  27. T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations 181 (2002), 72-91.
  28. C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Mathematica Academiae Scientiarum Hungaricae 32 (1978), 75-96.
  29. C.C. Travis and G.F. Webb, An abstract second order semilinear Volterra integrodifferential equation, SIAM J. Math. Anal. 10 (1979), 412-424.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2007/27/2
Export to PBN, Opens in new tab
Pages:
365-384
Main language of publication
English
Received
2006-09-18
Published
2007

DOI: 10.7151/dmdico.1090

Exact and natural sciences