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Existence of square-mean almost automorphic solutions to stochastic functional integrodifferential equations in Hilbert spaces. (English) Zbl 1469.60188

Summary: The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K30 Functional-differential equations in abstract spaces
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
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[1] Bochner, S., A new approach to almost periodicity, Proceedings of the National Academy of Sciences of the United States of America, 48, 2039-2043 (1962) · Zbl 0112.31401 · doi:10.1073/pnas.48.12.2039
[2] Bezandry, P. H.; Diagana, T., Existence of quadratic-mean almost periodic solutions to some stochastic hyperbolic differential equations, Electronic Journal of Differential Equations, 111, 1-14 (2009) · Zbl 1185.35345
[3] Caicedo, A.; Cuevas, C.; Mophou, G. M.; N’Guérékata, G. M., Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces, Journal of the Franklin Institute: Engineering and Applied Mathematics, 349, 1, 1-24 (2012) · Zbl 1260.34142 · doi:10.1016/j.jfranklin.2011.02.001
[4] Diagana, T.; Nguerekata, G.; van Minh, N., Almost automorphic solutions of evolution equations, Proceedings of the American Mathematical Society, 132, 11, 3289-3298 (2004) · Zbl 1053.34050 · doi:10.1090/S0002-9939-04-07571-9
[5] Diagana, T.; Hernández, E. M.; dos Santos, J. P. C., Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71, 1-2, 248-257 (2009) · Zbl 1172.45002 · doi:10.1016/j.na.2008.10.046
[6] Ding, H.-S.; Xiao, T.-J.; Liang, J., Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 338, 1, 141-151 (2008) · Zbl 1142.45005 · doi:10.1016/j.jmaa.2007.05.014
[7] dos Santos, J. P. C.; Cuevas, C., Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Applied Mathematics Letters, 23, 9, 960-965 (2010) · Zbl 1198.45014 · doi:10.1016/j.aml.2010.04.016
[8] Liu, Y. J.; Liu, A. M., Almost periodic solutions for a class of stochastic differential equations, Journal of Computational and Nonlinear Dynamics, 8, 4 (2013) · doi:10.1115/1.4023914
[9] NGuerekata, G. M., Almost Automorphic and Almost Periodic Functions in Abstract Spaces (2001), Plenum Publishing Corporation · Zbl 1001.43001
[10] N’Guérékata, G. M., Topics in Almost Automorphy (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1073.43004
[11] Veech, W. A., Almost automorphic functions, Proceedings of the National Academy of Sciences of the United States of America, 49, 4, 462-464 (1963) · Zbl 0173.33402 · doi:10.1073/pnas.49.4.462
[12] Huang, Z.; Yang, Q.-G., Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay, Chaos, Solitons & Fractals, 42, 2, 773-780 (2009) · Zbl 1198.60024 · doi:10.1016/j.chaos.2009.02.008
[13] Cao, J.; Yang, Q.; Huang, Z., Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics, 83, 3, 259-275 (2011) · Zbl 1221.60078 · doi:10.1080/17442508.2010.533375
[14] Bezandry, P. H., Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statistics & Probability Letters, 78, 17, 2844-2849 (2008) · Zbl 1156.60046 · doi:10.1016/j.spl.2008.04.008
[15] Chang, Y.-K.; Zhao, Z.-H.; N’Guérékata, G. M., A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations, Nonlinear Analysis: Theory, Methods & Applications, 74, 6, 2210-2219 (2011) · Zbl 1217.60043 · doi:10.1016/j.na.2010.11.025
[16] Desch, W.; Grimmer, R.; Schappacher, W., Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, Journal of Differential Equations, 74, 2, 391-411 (1988) · Zbl 0663.45008 · doi:10.1016/0022-0396(88)90011-3
[17] Fu, M.; Liu, Z., Square-mean almost automorphic solutions for some stochastic differential equations, Proceedings of the American Mathematical Society, 138, 10, 3689-3701 (2010) · Zbl 1202.60109 · doi:10.1090/S0002-9939-10-10377-3
[18] Grecksch, W.; Tudor, C., Stochastic Evolution Equations:A Hilbert Space Approach (1995), Berlin, Germany: Akademie, Berlin, Germany · Zbl 0831.60069
[19] Keck, D. N.; McKibben, M. A., Functional integro-differential stochastic evolution equations in Hilbert space, Journal of Applied Mathematics and Stochastic Analysis, 16, 2, 141-161 (2003) · Zbl 1031.60061 · doi:10.1155/S1048953303000108
[20] Keck, D. N.; McKibben, M. A., Abstract stochastic integrodifferential delay equations, Journal of Applied Mathematics and Stochastic Analysis, 2005, 3, 275-305 (2005) · Zbl 1105.60045 · doi:10.1155/JAMSA.2005.275
[21] Ding, H.-S.; Liang, J.; Xiao, T.-J., Pseudo almost periodic solutions to integro-differential equations of heat conduction in materials with memory, Nonlinear Analysis: Real World Applications, 13, 6, 2659-2670 (2012) · Zbl 1253.35195 · doi:10.1016/j.nonrwa.2012.03.009
[22] Grimmer, R. C., Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273, 1, 333-349 (1982) · Zbl 0493.45015 · doi:10.2307/1999209
[23] Chen, G., Control and stabilization for the wave equation in a bounded domain, SIAM Journal, 17, 1, 66-81 (1979) · Zbl 0402.93016 · doi:10.1137/0317007
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