Collectanea Mathematica

, Volume 66, Issue 1, pp 63–76 | Cite as

Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces

Article
First Online:
Received:
Accepted:

Abstract

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Keywords

Fractional stochastic evolution equations Nonlocal condition Compact analytic semigroup Fractional power space Wiener process 

Mathematics Subject Classification (2010)

35R11 47J35 60H15 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The authors would like to express his warmest thanks to the anonymous referees for carefully reading the manuscript and giving valuable comments and suggestions to improve the results of the paper. This work is supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).

References

  1. 1.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  2. 2.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  3. 3.
    Eidelman, S.D., Kochubei, A.N.: Cauchy problem for fractional diffusion equations. J. Differ. Equat. 199, 155–211 (2004)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solutions for fractional differential equations with uncertainly. Nonlinear Anal. 72, 2859–2862 (2010)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Darwish, M.A., Ntouyas, S.K.: On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. Nonlinear Anal. 74, 3510–3517 (2011)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons Fractals 14, 433–440 (2002)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real. World Appl. 12, 263–272 (2011)Google Scholar
  8. 8.
    Wang, R.N., Xiao, T.J., Liang, J.: A note on the fractional Cauchy problems with nonlocal conditions. Appl. Math. Lette. 24, 1435–1442 (2011)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1991)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Byszewski, L.: Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems. Nonlinear Anal. 33, 2413–2426 (1998)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Liang, J., Liu, J., Xiao, T.J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57, 183–189 (2004)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Ezzinbi, K., Fu, X., Hilal, K.: Existence and regularity in the \(\alpha \)-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal. 67, 1613–1622 (2007)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Liu, H., Chang, J.: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70, 3076–3083 (2009)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefMATHGoogle Scholar
  16. 16.
    Grecksch, W., Tudor, C.: Stochastic Evolution Equations: a Hilbert space approach. Akademic Verlag, Berlin (1995)MATHGoogle Scholar
  17. 17.
    Ichikawa, A.: Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90, 12–44 (1982)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Sakthivel, R., Luo, J.: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl. 356, 1–6 (2009)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Sakthivel, R., Ren, Y.: Exponential stability of second-order stochastic evolution equations with Poisson jumps. Commun. Nonlinear Sci. Numer. Simul. 17, 4517–4523 (2012)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Ren, Y., Sakthivel, R.: Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps. J. Math. Phys. 53, 14 (2012)MathSciNetGoogle Scholar
  21. 21.
    Yan, Z., Yan, X.: Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces. Z. Angew. Math. Phys. 64, 573–590 (2013)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Yan, Z., Yan, X.: Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay. Collec. Math. 64, 235–250 (2013)CrossRefMATHGoogle Scholar
  23. 23.
    Cui, J., Yan, L., Wu, X.: Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces. J. Korean Stat. Soc. 41, 279–290 (2012)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    EI-Borai, M.M.: On some stochastic fractional integro-differential equations. Advan. Dynam. Syst. Appl. 1, 49–57 (2006)Google Scholar
  25. 25.
    Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A 44, 335201 (2011)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Sakthivel, R., Revathi, P., Mahmudov, N.I.: Asymptotic stability of fractional stochastic neutral differential equations with infinite delays. Abst. Appl. Anal. (2013) Article ID 769257Google Scholar
  27. 27.
    Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  29. 29.
    Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Math, vol. 840. Springer, New York (1981)Google Scholar

Copyright information

© Universitat de Barcelona 2014

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhou People’s Republic of China

Personalised recommendations