Existence of mild solutions for impulsive fractional stochastic equations with infinite delay

Toufik Guendouzi; Khadem Mehdi

doi:10.26637/mjm104/004

Abstract

This paper is mainly concerned with the existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. A new set of sufficient conditions are formulated and proved for the existence of mild solutions by means of Sadovskii’s fixed point theorem. An example is given to illustrate the theory.

Keywords:

Existence result, fractional stochastic differential equation, fixed point technique, infinite delay, resolvent operators

Mathematics Subject Classification:

34K30, 34K50, 26A33

Authors Imprint References Funding Related Articles Downloads

Toufik Guendouzi Laboratory of Stochastic Models, Statistic and Applications, Tahar Moulay University PO.Box 138 En-Nasr, 20000 Saida, Algeria.
Khadem Mehdi Laboratory of Stochastic Models, Statistic and Applications, Tahar Moulay University PO.Box 138 En-Nasr, 20000 Saida, Algeria.

Pages: 30-43
Date Published: 01-10-2013
Vol. 1 No. 04 (2013): Malaya Journal of Matematik (MJM)

P. Balasubramaniam, J.Y. Park, A. Vincent Antony Kumar, Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions, Nonlinear Anal. TMA., 71 (2009), 1049-1058. DOI: https://doi.org/10.1016/j.na.2008.11.032

J. Bao, Z. Hou, and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Pro. American Math. Soc., 138(6)(2010), 2169-2180. DOI: https://doi.org/10.1090/S0002-9939-10-10230-5

M. Benchohra, J. Henderson, S.K. Ntouyas, Existence results for impulsive semilinear neutral functional differential equations in Banach spaces, Memoirs on Diff. Equ. Math. Phys., 25(2002), 105-120.

Y.K. Chang, Z.H. Zhao, G. M. N’Gu´ er´ ekata, Squaremean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Computers & Mathematics with Applications, 61(2)(2011), 384-391. DOI: https://doi.org/10.1016/j.camwa.2010.11.014

J. Cui, L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A: Math. Theor. 44 (2011) 335201 (16pp) DOI: https://doi.org/10.1088/1751-8113/44/33/335201

J. Dabas, A. Chauhan, M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ., (2011) 20. Article ID 793023. DOI: https://doi.org/10.1155/2011/793023

J. Dabas, A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Computer Modelling, 57(2013), 754-763. DOI: https://doi.org/10.1016/j.mcm.2012.09.001

M. M. Fu and Z. X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Pro. American Math. Soc., 138(10) (2010), 3689-3701. DOI: https://doi.org/10.1090/S0002-9939-10-10377-3

T. Guendouzi, I. Hamada, Existence and controllability result for fractional neutral stochastic integro-differential equations with infinite delay, AMO - Advanced Modeling and Optimization, 15(2)(2013), 281-300.

E. Hernández. M. Pierri, G. Goncalves, Existence results for an impulsive abstract partial differential equation with state-dependent delay, Computers & Mathematics with Applications, 52(3-4)(2006), 411-420. DOI: https://doi.org/10.1016/j.camwa.2006.03.022

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2000. DOI: https://doi.org/10.1142/3779

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

A. Lin, L. Hu, Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions, Computers & Mathematics with Applications, 59(2010), 64-73. DOI: https://doi.org/10.1016/j.camwa.2009.09.004

G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Vol 44 Encyclopedia of Mathematics and its Applications, Cambridge university Press, Cambridge, Mass, USA, 1992. DOI: https://doi.org/10.1017/CBO9780511666223

Y. Ren, R. Sakthivel, Existence, uniqueness and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys., 53(2012), 073517. DOI: https://doi.org/10.1063/1.4739406

Y. Ren, Q. Zhou, L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl., 149(2011), 315-331. DOI: https://doi.org/10.1007/s10957-010-9792-0

B.N. Sadovskii, On a fixed point principle, Funct. Anal. Appl., 1 (1967), 71-74. DOI: https://doi.org/10.1007/BF01076087

R. Sakthivel, P. Revathi, Yong Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal. TMA., 81(2013), 70-86. DOI: https://doi.org/10.1016/j.na.2012.10.009

R. Sakthivel, P. Revathi, N. I.Mahmudov, Asymptotic stability of fractional stochastic neutral differential equations with infinite delays, Abstract and Applied Analysis V. 2013, Article ID 769257, 9. DOI: https://doi.org/10.1155/2013/769257

R. Sakthivel, P. Revathi, S. Marshal Anthoni, Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations, Nonlinear Anal. TMA., 75(7)(2012), 3339-3347. DOI: https://doi.org/10.1016/j.na.2011.12.028

R. Sakthivel, Yong Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Comm. Nonlinear Sci. Numerical Simulation, 17(2012), 4517-4523. DOI: https://doi.org/10.1016/j.cnsns.2012.04.020

R. Sakthivel, Yong Ren, Hyunsoo Kim, Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phys., 51(2010), 052701. DOI: https://doi.org/10.1063/1.3397461

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. DOI: https://doi.org/10.1016/j.na.2010.11.007