Title: Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
Author(s): Toufik Guendouzi, Lamia Bousmaha
Pages: 28-43
Cite as:
Toufik Guendouzi, Lamia Bousmaha, Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations, Int. J. Anal. Appl., 6 (1) (2014), 28-43.

Abstract


In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.

Full Text: PDF

 

References


  1. J.O. Alzabut, J.J. Nieto, G.T. Stamov, Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, Bound Value Probl., 2009 (2009), Article ID 127510. Google Scholar

  2. S. Abbas, M. Benchohra, G.M. N’Gu´er´ekata, Topics in fractional differential equations. Developments in Mathematics, Springer, New York (2012). Google Scholar

  3. R.P. Agarwal, B. Andrade, G. Siracusa, On fractional integro-differential equations with state-dependent delay, Comput. Math. Appl., 62 (2011) 1143-1149. Google Scholar

  4. B. Andrade,J.P.C. Santos, Existence of solutions for a fractional neutral integro-differential equation with unbounded delay, Electron. J. Differ. Equ., 2012 (2012) 1-13. Google Scholar

  5. P. Bezandry, Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statist. Probab. Lett., 78 (2008) 2844-2849. Google Scholar

  6. P. Bezandry, T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Applicable Anal.,86 (7) (2007) 819-827. Google Scholar

  7. J. Cao, Q. Yang, Z. Huang, On almost periodic mild solutions for stochastic functional differential equations, Nonlinear Anal. RWA., 13 (2012) 275-286. Google Scholar

  8. Y.K. Chang, R. Ma, Z.H. Zhao, Almost periodic solutions to a stochastic differential equation in Hilbert spaces, Results in Math., 63, 1-2 (2013) 435-449. Google Scholar

  9. J. Dabas, A. Chauhan, M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ. 2011 (2011), Article ID 793023. Google Scholar

  10. G. Da Prato, J. Zabczyk, Stochastic eqnarrays in infinite dimensions, Cambridge University Press, Cambridge (1992). Google Scholar

  11. A. Debbouche, M.M. El-Borai, Weak almost periodic and optimal mild solutions of fractional evolution equations, Electron. J. Differ. Equ., 46 (2009) 1-8. Google Scholar

  12. M.M. El-Borai, A. Debbouche, Almost periodic solutions of some nonlinear fractional differential equations, Int. J. Contemp. Math. Sci. 4 (2009), 1373-1387. Google Scholar

  13. R. Jahanipur, Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions, J. Differ. Equations 248 (2010), 1230-1255. Google Scholar

  14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006). Google Scholar

  15. J. Liu, C. Zhang, Existence and statbility of almost periodic solutions for impulsive differential equations, Adv. Diff. Equa., 2012 (2012), Article ID 34. Google Scholar

  16. J. Liu, C. Zhang, Existence and statbility of almost periodic solutions to impulsive stochastic differential equations, Cubo, 15, 1 (2013), 77-96. Google Scholar

  17. J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008) 753-760. Google Scholar

  18. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York (1993). Google Scholar

  19. A. Pazy, Semigroups of linear operators and application to partial differential equations, Springer-Verlag, New York, 1983. Google Scholar

  20. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999). Google Scholar

  21. G. Prato, C. Tudor, Periodic and almost periodic solutions for semilinear stochastic evolution equations, Stoch. Anal. Appl., 13 (1995), 13-33. Google Scholar

  22. A.M. Samoilenko, N.A. Perestyuk, Impulsive differential equations, World Scientific, Singapore, 1995. Google Scholar

  23. X.B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. TMA 74 (2011), 2003-2011. Google Scholar

  24. G.T. Stamov, Existence of almost periodic solutions for strong stable impulsive differential equations, IMA J Math Control., 18 (2001), 153-160. Google Scholar

  25. G.T. Stamov, J.O. Alzabut, Almost periodic solutions for abstract impulsive differential equations, Nonlinear Anal. TMA., 72 (2010), 2457-2464. Google Scholar

  26. G.T. Stamov, Almost periodic solutions of impulsive differential equations, Springer, Berlin (2012). Google Scholar

  27. G.T. Stamov, I.M. Stamova, Almost periodic solutions of impulsive fractional differential equations, Dyn. Styt: An Int. J., 29 (2014), 119-132. Google Scholar

  28. T. Taniguchi, K. Liu, A. Truman, Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differ. Equations 181 (2002) 72-91. Google Scholar

  29. J.R. Wang, M. Feckan M, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn Partial Differ Equ. 8 (2011) 345-361. Google Scholar

  30. X. Zhang, X. Huang, Z. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010) 775-781. Google Scholar

  31. Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. Google Scholar