Abstract and Applied Analysis

Asymptotic Stability of Fractional Stochastic Neutral Differential Equations with Infinite Delays

R. Sakthivel, P. Revathi, and N. I. Mahmudov

Full-text: Open access

Abstract

We study the existence and asymptotic stability in pth 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 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 769257, 9 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306764

Digital Object Identifier
doi:10.1155/2013/769257

Mathematical Reviews number (MathSciNet)
MR3035189

Zentralblatt MATH identifier
1269.60059

Citation

Sakthivel, R.; Revathi, P.; Mahmudov, N. I. Asymptotic Stability of Fractional Stochastic Neutral Differential Equations with Infinite Delays. Abstr. Appl. Anal. 2013 (2013), Article ID 769257, 9 pages. doi:10.1155/2013/769257. https://projecteuclid.org/euclid.aaa/1366306764


Export citation

References

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2000.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • H. M. Ahmed, “Controllability of fractional stochastic delay equations,” Lobachevskii Journal of Mathematics, vol. 30, no. 3, pp. 195–202, 2009.
  • 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.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  • 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.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.