Square-mean asymptotically almost automorphic mild solutions to non-autonomous stochastic differential equations

Zhi-Hong Li; Zhi-Han Zhao

doi:10.26637/mjm203/012

Abstract

This paper is mainly concerned with square-mean asymptotically almost automorphic mild solutions to a class of non-autonomous stochastic differential equations in a real separable Hilbert space. Some existence results of square-mean asymptotically almost automorphic mild solutions have been established by properties and composition theroems of square-mean asymptotically almost automorphic functions and fixed point theorems.

Keywords:

Non-autonomous differential equation, Square-mean asymptotically almost automorphic

Mathematics Subject Classification:

34K14, 60H10, 35B15, 34F05

Authors Imprint References Funding Related Articles Downloads

Zhi-Hong Li Department of Mathematics, Lanzhou Jiaotong University, Lanzhou - 730070, P. R. China.
Zhi-Han Zhao Department of Information Engineering, Sanming University, Sanming - 365004, P. R. China

Pages: 263-272
Date Published: 01-07-2014
Vol. 2 No. 03 (2014): Malaya Journal of Matematik (MJM)

P. Acquistapace, Evolution operators and strong solution of abstract linear parabolic equations, Differential Integral Equations, 1(1988), 433-457. DOI: https://doi.org/10.57262/die/1372451947

P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78(1987), 47-107.

D. Bugajewski and G. M. N’Guérékata, On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Anal., 59(2004), 1333-1345. DOI: https://doi.org/10.1016/S0362-546X(04)00329-3

P. Bezandry and T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal., 86(2007), 819-827. DOI: https://doi.org/10.1080/00036810701397788

P. Bezandry and T. Diagana, Square-mean almost periodic solutions to nonautonomous stochastic differential equations, Electron. J. Differertial Equations, 2007, 1-10.

Y. K. Chang, Z. H. Zhao, G. M. N’Guérékata, Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61(2011), 384-391. DOI: https://doi.org/10.1016/j.camwa.2010.11.014

Y. K. Chang, Z. H. Zhao, G. M. N’Gu’er’ekata, Square-mean almost automorphic mild solutions to some stochastic differential equations in Hilbert space, Advances in Difference Equations, 2011, 2011:9. DOI: https://doi.org/10.1186/1687-1847-2011-9

Y. K. Chang, Z. H. Zhao, G. M. N’Guérékata and R. Ma, Stepanov-like almost automorphic for stochastic processes and applications to stochastic differential equations, Nonlinear Anal. RWA, 12(2011), 1130-1139. DOI: https://doi.org/10.1016/j.nonrwa.2010.09.007

Y. K. Chang, Z. H. Zhao and G. M. N’Guérékata, A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations, Nonlinear Anal., 74(2011), 2210-2219. DOI: https://doi.org/10.1016/j.na.2010.11.025

Z. Chen, W. Lin, Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261(2011), 69-89. DOI: https://doi.org/10.1016/j.jfa.2011.03.005

H. S. Ding, T. J. Xiao and J. Liang, Asymptotically almost automorphic solutions for some integral-differential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338(2008), 141-151. DOI: https://doi.org/10.1016/j.jmaa.2007.05.014

T. Diagana, E. Hernández and J. P. C. dos Santos, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integral-differential equations, Nonlinear Anal., 71(2009), 248-257. DOI: https://doi.org/10.1016/j.na.2008.10.046

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

G. M. N’Guérékata, Sur les solutions presqu’automorphes d”equations diff’erentielles abstraites, Ann. Sci. Math. Qu’ebec, 5(1981), 69-79.

A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90(1982), 12-14. DOI: https://doi.org/10.1016/0022-247X(82)90041-5

Z. H. Zhao, Y. K. Chang and J. J. Nieto, square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynam. Syst. Appl., 22(2013), 269-284.

Z. H. Zhao, Y. K. Chang and J. J. Nieto, Asymptotic behavior of solutions to abstract stochastic fractional partial integrodifferential equations, Abstr. Appl. Anal., Vol. 2013, Article ID 138068, 8 pages, 2013. DOI: https://doi.org/10.1155/2013/138068

Z. H. Zhao, Y. K. Chang and J. J. Nieto, Almost automorphic solutions to some stochastic functional differential equations with delay, Afr. Diaspora J. Math., 15(2013), 7-25.

Research Fund for Young Teachers of Sanming University (B201107/Q) and Science Foundation of the Education Department of Fujian Province (Grant No.JB12227)