×

zbMATH — the first resource for mathematics

Existence of almost automorphic mild solutions to non-autonomous neutral stochastic differential equations. (English) Zbl 1410.34247
Summary: In this paper, we consider the existence and uniqueness of square-mean and weighted pseudo almost automorphic mild solutions to a class of nonautonomous stochastic neutral differential equations with infinite delay in real separable Hilbert spaces. A new set of sufficient conditions are established for obtaining the required result. To prove the main result, we use Banach contraction principle together with evolution family of operators. Some known results are generalized and improved. An application to the stochastic nonlinear heat equation with infinite delay is provided to illustrate the obtained theory.

MSC:
34K50 Stochastic functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K40 Neutral functional-differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] NGuérékata, G. M., Almost automorphic and almost periodic functions in abstract spaces, (2001), Kluwer Academic, Plenum Publishers New York, London, Moscow
[2] NGuérékata, G. M., Topics in almost automorphy, (2005), Springer-Verlag New York Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[3] Blot, J.; Mophou, G. M.; Guérékata, G. M.N.; Pennequin, D., Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal., 71, 903-909, (2009) · Zbl 1177.34077
[4] Boukli-Hacene, N.; Ezzinbi, K., Weighted pseudo-almost automorphic solutions for some partial functional differential equations, Nonlinear Anal.: Real World Appl., 12, 562-570, (2011) · Zbl 1223.35032
[5] Caraballo, T.; Garrido-Atienza, M. J.; Taniguchi, T., The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74, 3671-3684, (2011) · Zbl 1218.60053
[6] Cao, J.; Yang, Q.; Huang, Z., Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stoch.: An Int. J. Probab. Stoch. Processes, 83, 259-275, (2011) · Zbl 1221.60078
[7] Zhang, R.; Chang, Y.-K.; Guérékata, G. M.N., New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal.: Real World Appl., 13, 2866-2879, (2012) · Zbl 1316.34064
[8] Chang, Y.-K.; Zhao, Z.-H.; Nieto, J. J., Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. Mat. Complut., 24, 421-438, (2011) · Zbl 1232.34087
[9] Chang, Y.-K.; Zhang, R.; Guérékata, G. M.N., Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Comput. Math. Appl., 64, 3160-3170, (2012) · Zbl 1268.34010
[10] Chang, Y.-K.; Zhao, Z.-H.; Guérékata, G. M.N.; Ma, R., Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Anal.: Real World Appl., 12, 1130-1139, (2011) · Zbl 1209.60034
[11] Chang, Y.-K.; Zhao, Z.-H.; Guérékata, G. M.N., Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61, 384-391, (2011) · Zbl 1211.60025
[12] Chang, Y.-K.; Zhao, Z.-H.; NGuérékata, G. M., Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space, Adv. Differ. Equ., 2011, 9, (2011)
[13] Diagana, T., Evolution equations in generalized Stepanov-like pseudo almost automorphic spaces, Electron. J. Differ. Equ., 2012, 1-19, (2012) · Zbl 1238.58006
[14] Diagana, T., Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations, Semigroup Forum, 82, 455-477, (2011) · Zbl 1229.34095
[15] Diagana, T., Existence of pseudo-almost automorphic mild solutions to some nonautonomous partial evolution equations, Adv. Differ. Equ., 2011, 895079, (2011) · Zbl 1207.35007
[16] Diagana, T., Existence of almost automorphic solutions to some neutral functional differential equations with infinite delay, Electron. J. Differ. Equ., 2008, 1-14, (2008) · Zbl 1175.34090
[17] Diagana, T.; Hernández, E.; Rabello, M., Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay, Math. Comput. Model., 45, 1241-1252, (2007) · Zbl 1133.34042
[18] Sakthivel, R.; Revathi, P.; Marshal Anthoni, S., Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations, Nonlinear Anal., 75, 3339-3347, (2012) · Zbl 1243.34006
[19] Sakthivel, R.; Ren, Y.; Kim, H., Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phys., 51, 052701, (2010) · Zbl 1310.35248
[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, 073517, (2012) · Zbl 1287.60080
[21] Fan, Z.; Liang, J.; Xiao, T.-J., Composition of Stepanov-like pseudo almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal.: Real World Appl., 13, 131-140, (2012) · Zbl 1238.34112
[22] Chen, Z.; Lin, W., Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261, 69-89, (2011) · Zbl 1233.60030
[23] Cui, J.; Yan, L., On almost automorphic mild solutions for nonautonomous stochastic evolution equations, Abstract Appl. Anal., 2012, 870831, (2012) · Zbl 1257.34044
[24] Xia, Z.; Fan, M., Weighted Stepanov-like pseudo almost automorphy and applications, Nonlinear Anal., Theory, Methods Appl., 75, 2378-2397, (2012) · Zbl 1306.35140
[25] Ding, H.-S.; Longa, W.; Guérékata, G. M.N., A composition theorem for weighted pseudo-almost automorphic functions and applications, Nonlinear Anal., 73, 2644-2650, (2010) · Zbl 1210.43008
[26] Fu, M. M.; Liu, Z. X., Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Am. Math. Soc., 138, 3689-3701, (2010) · Zbl 1202.60109
[27] Hino, Y.; Murakami, S.; Naito, T., Functional differential equations with infinite delay, Lecture Notes in Mathematics, 1473, (1991), Springer-Verlag Berlin · Zbl 0732.34051
[28] Liu, J.-H.; Song, X.-Q., Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. Funct. Anal., 258, 196-207, (2010) · Zbl 1194.47047
[29] Mophou, G. M., Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Appl. Math. Comput., 217, 7579-7587, (2011) · Zbl 1221.34015
[30] Mishra, I.; Bahuguna, D., Existence of almost automorphic solutions of neutral differential equations, J. Nonlinear Evol. Equ. Appl., 2012, 17-28, (2012) · Zbl 1238.34132
[31] Lin, A.; Ren, Y.; Xia, N., On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators, Math. Comput. Model., 51, 413-424, (2010) · Zbl 1190.60045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.