In this paper, we consider recurrence of bounded solutions to semilinear stochastic integro-differential equations via uniformly exponentially stable resolvent operators family. Concretely, we first establish the existence and uniqueness of the -bounded solution to a semilinear stochastic integro-differential equations driven by Lévy processes, and then we show this kind of -bounded solution is almost automorphic in distribution or weighted pseudo almost automorphic in distribution under suitable conditions respectively.
S.Abbas, Pseudo almost automorphic solutions of some nonlinear integro-differential equations, Computers and Mathematics with Applications62 (2011), 2259–2272. doi:10.1016/j.camwa.2011.07.013.
2.
S.Abbas and Y.Xia, Existence and attractivity of k-almost automorphic solutions of a model of cellular neural network with delay, Acta Mathematica Scientia1 (2013), 290–302. doi:10.1016/S0252-9602(12)60211-2.
3.
D.Applebaum, Lévy Process and Stochastic Calculus, 2nd edn, Cambridge University Press, 2009.
4.
A.Benchaabane and R.Sakthivel, Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients, Journal of Computational and Applied Mathematics312 (2017), 65–73. doi:10.1016/j.cam.2015.12.020.
5.
P.Bezandry and T.Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011.
6.
Y.T.Bian, Y.K.Chang and J.J.Nieto, Weighted asymptotic behavior of solutions to a semilinear integro-differential equation in Banach spaces, Electronic Journal of Differential Equations2014 (2014), 91.
7.
J.F.Cao, Q.G.Yang and Z.T.Huang, Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics83 (2011), 259–275. doi:10.1080/17442508.2010.533375.
8.
Y.K.Chang and R.Ponce, Uniform exponential stability and its applications to bounded solutions of integro–differential equations in Banach spaces, to appear in Journal of Integral Equations and Applications.
9.
Y.K.Chang and C.Tang, Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process, Stochastics88 (2016), 980–1011. doi:10.1080/17442508.2016.1178748.
10.
J.Chen, T.Xiao and J.Liang, Uniform exponential stability of solutions to abstract Volterra equations, Journal of Evolution Equations4 (2009), 661–674. doi:10.1007/s00028-009-0028-4.
11.
Z.Chen and W.Lin, Square-mean almost automorphic process and its application to stochastic evolution equations, Journal of Functional Analysis261 (2011), 69–89. doi:10.1016/j.jfa.2011.03.005.
12.
T.Diagana, Weighted pseudo almost periodic functions and applications, Comptes Rendus Mathematique343 (2006), 643–646. doi:10.1016/j.crma.2006.10.008.
13.
M.M.Fu, Almost automorphic solutions for nonautonomous stochastic differential equations, Journal of Mathematical Analysis and Applications393 (2012), 231–238. doi:10.1016/j.jmaa.2012.04.017.
14.
M.M.Fu and Z.X.Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proceedings of the American Mathematical Society138 (2010), 3689–3701. doi:10.1090/S0002-9939-10-10377-3.
15.
M.Kamenskii, O.Mellah and P.Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, Journal of Mathematical Analysis and Applications427 (2015), 336–364. doi:10.1016/j.jmaa.2015.02.036.
16.
K.X.Li, Weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise, Journal of Mathematical Analysis and Applications427 (2015), 686–721. doi:10.1016/j.jmaa.2015.02.071.
17.
Z.X.Liu and K.Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, Journal of Functional Analysis266 (2014), 1115–1149. doi:10.1016/j.jfa.2013.11.011.
18.
C.Lizama and R.Ponce, Bounded solutions to a class of semilinear integro-differential equations in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications74 (2011), 3397–3406. doi:10.1016/j.na.2011.02.018.
19.
M.A.Meyers, K.K.Chawla and Chawla, Mechanical Behavior of Materials, Cambridge University Press, 2009.
20.
G.M.N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Space, Kluwer, New York, 2001.
21.
G.M.N’Guérékata, Topics in Almost Automorphy, Springer, New York, 2005.
22.
S.Peszat and J.Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007.
23.
J.Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, 1993.
24.
Y.Ren, Reflected backward doubly stochastic differential equations driven by a Lévy process, Comptes Rendus Mathematique348 (2010), 439–444. doi:10.1016/j.crma.2009.11.004.
25.
Y.Ren and R.Sakthivel, Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, Journal of Mathematical Physics53 (2012), 073517. doi:10.1063/1.4739406.
26.
R.Sakthivel, Y.Ren, A.Debbouche and N.I.Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Applicable Analysis95 (2016), 2361–2382. doi:10.1080/00036811.2015.1090562.
27.
Z.Xia, Pseudo asymptotically periodic solutions for Volterra integro-differential equations, Mathematical Methods in the Applied Sciences38 (2015), 799–810. doi:10.1002/mma.3108.
