Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2015, Article ID 925209, 26 pages
http://dx.doi.org/10.1155/2015/925209
Research Article

On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

Received 20 January 2015; Revised 16 April 2015; Accepted 19 April 2015

Academic Editor: Mark A. McKibben

Copyright © 2015 Zuomao Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  2. V. Lakshmikanthan, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishers, Singapore, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  3. W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  4. E. Hernández, M. Pierri, and G. Goncalves, “Existence results for an impulsive abstract partial differential equation with state-dependent delay,” Computers & Mathematics with Applications, vol. 52, no. 3-4, pp. 411–420, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A. Anguraj and A. Vinodkumar, “Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional differential equations with infinite delays,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 67, pp. 1–13, 2009. View at Google Scholar
  6. A. Lin, Y. Ren, and N. Xia, “On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 413–424, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Z. Yan and X. Yan, “Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay,” Collectanea Mathematica, vol. 64, no. 2, pp. 235–250, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. R. Triggiani, “A note on the lack of exact controllability for mild solutions in Banach spaces,” SIAM Journal on Control and Optimization, vol. 15, no. 3, pp. 407–411, 1977. View at Publisher · View at Google Scholar · View at MathSciNet
  9. N. I. Mahmudov and A. Denker, “On controllability of linear stochastic systems,” International Journal of Control, vol. 73, no. 2, pp. 144–151, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. J. P. Dauer and N. I. Mahmudov, “Controllability of stochastic semilinear functional differential equations in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 373–394, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. N. I. Mahmudov, “Controllability of linear stochastic systems,” IEEE Transactions on Automatic Control, vol. 46, no. 5, pp. 724–731, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. P. Balasubramaniam, J. Y. Park, and P. Muthukumar, “Approximate controllability of neutral stochastic functional differential systems with infinite delay,” Stochastic Analysis and Applications, vol. 28, no. 2, pp. 389–400, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. R. Sakthivel, J. J. Nieto, and N. I. Mahmudov, “Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay,” Taiwanese Journal of Mathematics, vol. 14, no. 5, pp. 1777–1797, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. R. Subalakshmi and K. Balachandran, “Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces,” Chaos, Solitons & Fractals, vol. 42, no. 4, pp. 2035–2046, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. Shen and J. Sun, “Approximate controllability of stochastic impulsive functional systems with infinite delay,” Automatica, vol. 48, no. 10, pp. 2705–2709, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Y. Zang and J. Li, “Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions,” Boundary Value Problems, vol. 2013, article 193, 14 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Y. Ren and D. Sun, “Second-order neutral impulsive stochastic evolution equations with delay,” Journal of Mathematical Physics, vol. 50, Article ID 102709, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. J. Cui and L. Yan, “Existence results for impulsive neutral second-order stochastic evolution equations with nonlocal conditions,” Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2378–2387, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. N. I. Mahmudov and M. A. McKibben, “Approximate controllability of second-order neutral stochastic evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems B, vol. 13, pp. 619–634, 2006. View at Google Scholar
  20. P. Muthukumar and P. Balasubramaniam, “Approximate controllability of second-order damped McKean-Vlasov stochastic evolution equations,” Computers & Mathematics with Applications, vol. 60, no. 10, pp. 2788–2796, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. P. Balasubramaniam and P. Muthukumar, “Approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 143, no. 2, pp. 225–244, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. R. Sakthivel, Y. Ren, and N. I. Mahmudov, “Approximate controllability of second-order stochastic differential equations with impulsive effects,” Modern Physics Letters B: Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 24, no. 14, pp. 1559–1572, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. R. Pettersson, “Existence theorem and Wong-Zakai approximations for multivalued stochastic differential equations,” Probability and Mathematical Statistics, vol. 17, pp. 29–45, 1997. View at Google Scholar
  24. N. U. Ahmed, “Nonlinear stochastic differential inclusions on Banach space,” Stochastic Analysis and Applications, vol. 12, no. 1, pp. 1–10, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. P. Balasubramaniam and S. K. Ntouyas, “Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 161–176, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. Z. Yan and X. Yan, “Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces,” Zeitschrift für Angewandte Mathematik und Physik, vol. 64, no. 3, pp. 573–590, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. Z. Yan and H. Zhang, “Existence of impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay in Hilbert spaces,” Electronic Journal of Differential Equations, vol. 2013, pp. 1–21, 2013. View at Google Scholar
  28. B. C. Dhage, “Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications,” Computers & Mathematics with Applications, vol. 51, no. 3-4, pp. 589–604, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. K. Deimling, Multi-Valued Differential Equations, De Gruyter, Berlin, Germany, 1992.
  30. S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
  31. J. Kisyński, “On cosine operator functions and one parameter group of operators,” Studia Mathematica, vol. 49, pp. 93–105, 1972. View at Google Scholar
  32. C. C. Travis and G. F. Webb, “Cosine families and abstract nonlinear second order differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 32, no. 1-2, pp. 75–96, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. C. C. Travis and G. F. Webb, “Compactness, regularity, and uniform continuity properties of strongly continuous cosine families,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 555–567, 1977. View at Google Scholar · View at MathSciNet
  34. J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj. Serio Internacia, vol. 21, no. 1, pp. 11–41, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. Y. Hino, S. Murakami, and T. Naito, Functional-Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991. View at MathSciNet
  36. H. R. Henrĺquez and C. H. Vásquez, “Differentiability of solutions of second-order functional differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 284–312, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  38. A. Lasota and Z. Opial, “An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations,” Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 781–786, 1965. View at Google Scholar