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Asymptotic stability of fractional stochastic neutral differential equations with infinite delays. (English) Zbl 1269.60059
Summary: We study the existence and asymptotic stability in the $p$th moment of a mild solution to a class of nonlinear fractional neutral stochastic differential equations with infinite delays in Hilbert spaces. A set of novel sufficient conditions are derived with the help of semigroup theory and the fixed point technique for achieving the required result. The uniqueness of the solution of the considered problem is also studied under suitable conditions. Finally, an example is given to illustrate the obtained theory.

##### MSC:
 60H10 Stochastic ordinary differential equations 93E15 Stochastic stability
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##### References:
 [1] J. Bao, Z. Hou, and C. Yuan, “Stability in distribution of mild solutions to stochastic partial differential equations,” Proceedings of the American Mathematical Society, vol. 138, no. 6, pp. 2169-2180, 2010. · Zbl 1195.60087 · doi:10.1090/S0002-9939-10-10230-5 [2] M. M. Fu and Z. X. Liu, “Square-mean almost automorphic solutions for some stochastic differential equations,” Proceedings of the American Mathematical Society, vol. 138, no. 10, pp. 3689-3701, 2010. · Zbl 1202.60109 · doi:10.1090/S0002-9939-10-10377-3 [3] Y.-K. Chang, Z.-H. Zhao, G. M. N’Guérékata, and R. Ma, “Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1130-1139, 2011. · Zbl 1209.60034 · doi:10.1016/j.nonrwa.2010.09.007 [4] 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 Analysis. Theory, Methods and Applications A, vol. 74, no. 6, pp. 2210-2219, 2011. · Zbl 1217.60043 · doi:10.1016/j.na.2010.11.025 [5] Y.-K. Chang, Z.-H. Zhao, and G. M. N’Guérékata, “Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 384-391, 2011. · Zbl 1211.60025 · doi:10.1016/j.camwa.2010.11.014 [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2000. · Zbl 0998.26002 [7] M. M. El-Borai, K. EI-Said EI-Nadi, and H. A. Fouad, “On some fractional stochastic delay differential equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1165-1170, 2010. · Zbl 1189.60117 · doi:10.1016/j.camwa.2009.05.004 [8] H. Chen, “Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays,” Statistics and Probability Letters, vol. 80, no. 1, pp. 50-56, 2010. · Zbl 1177.93075 · doi:10.1016/j.spl.2009.09.011 [9] M. M. El-Borai, O. L. Moustafa, and H. M. Ahmed, “Asymptotic stability of some stochastic evolution equations,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 273-286, 2003. · Zbl 1033.60074 · doi:10.1016/S0096-3003(02)00406-X [10] T. Caraballo and K. Liu, “Exponential stability of mild solutions of stochastic partial differential equations with delays,” Stochastic Analysis and Applications, vol. 17, no. 5, pp. 743-763, 1999. · Zbl 0943.60050 · doi:10.1080/07362999908809633 [11] L. Wan and J. Duan, “Exponential stability of non-autonomous stochastic partial differential equations with finite memory,” Statistics and Probability Letters, vol. 78, no. 5, pp. 490-498, 2008. · Zbl 1141.37030 · doi:10.1016/j.spl.2007.08.003 [12] R. Sakthivel, Y. Ren, and H. Kim, “Asymptotic stability of second-order neutral stochastic differential equations,” Journal of Mathematical Physics, vol. 51, no. 5, article 005005, pp. 1-9, 2010. · Zbl 1310.35248 · doi:10.1063/1.3397461 [13] R. Sakthivel and J. Luo, “Asymptotic stability of nonlinear impulsive stochastic differential equations,” Statistics and Probability Letters, vol. 79, no. 9, pp. 1219-1223, 2009. · Zbl 1166.60316 · doi:10.1016/j.spl.2009.01.011 [14] R. Sakthivel and J. Luo, “Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 1-6, 2009. · Zbl 1166.60037 · doi:10.1016/j.jmaa.2009.02.002 [15] D. Zhao and D. Han, “Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations,” Journal of Inequalities and Applications, vol. 2011, article 9, 2011. · Zbl 1264.60049 · doi:10.1186/1029-242X-2011-9 [16] M. M. El-Borai, K. E.-S. El-Nadi, O. L. Mostafa, and H. M. Ahmed, “Volterra equations with fractional stochastic integrals,” Mathematical Problems in Engineering, vol. 2004, no. 5, pp. 453-468, 2004. · Zbl 1081.45007 · doi:10.1155/S1024123X04312020 · eudml:52103 [17] H. M. Ahmed, “Controllability of fractional stochastic delay equations,” Lobachevskii Journal of Mathematics, vol. 30, no. 3, pp. 195-202, 2009. · Zbl 05948809 · doi:10.1134/S1995080209030019 [18] Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1063-1077, 2010. · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026 [19] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 [20] J. Luo, “Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 753-760, 2008. · Zbl 1157.60065 · doi:10.1016/j.jmaa.2007.11.019 [21] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008