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Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions

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Abstract

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory.

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References

  1. Podlubny, I.: Fractional Differential Equations. Acadmic Press, New York (1993)

    MATH  Google Scholar 

  2. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  3. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publisher, Yverdon (1993)

    MATH  Google Scholar 

  4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  5. Li, F., Guérékata, G.M.N’: An existence result for neutral delay integro-differential equations with fractional order and nonlocal conditions. Abstr. Appl. Anal. 2011, 1–20 (2011)

  6. Guendouzi, T., Benzatout, O.: Existence of mild solutions for impulsive fractional stochastic differential inclusions with state-dependent delay. Chin. J. Math. 2014, 1–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yan, Z., Zhang, H.: Existence of solutions to impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay. Electron. J. Differ. Equ. 2013, 1–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benchohra, M., Litimein, S., N’ Guérékata, G.: On fractional integro-differential inclusions with state-dependent delay in Banach spaces. Appl. Anal. 92, 335–350 (2013)

  9. Liu, X., Liu, Z.: Existence results for fractional semilinear differential inclusions in Banach spaces. J. Appl. Math. Comput. 42, 171–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Agarwal, R.P., Santos, J.P.C., Cuevas, C.: Analytic resolvent operator and existence results for fractional order evolutionary integral equations. J. Abstr. Differ. Equ. Appl. 2, 26–47 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Andrade, B.D., Santos, J.P.C.: Existence of solutions for a fractional neutral integro-differential equation with unbounded delay. Electron. J. Differ. Equ. 2012, 1–13 (2012)

    Article  MathSciNet  Google Scholar 

  12. Smirnov, G.V.: Introduction to the Theory of Differential Inclusions. Amer. Math. Soc. Providence, Rhode Island (2002)

    MATH  Google Scholar 

  13. Benchohra, M., Ziane, M.: Impulsive evolution inclusions with state-dependent delay and multivalued jumps. Electron. J. Qual. Theory Differ. Equ. 2013, 1–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yan, Z., Jia, X.: Impulsive problems for fractional partial neutral functional integro-differential inclusions with infinite delay and analytic resolvent operators. Mediterr. J. Math. 2013, 1–36 (2013)

    Google Scholar 

  16. Gunasekar, T., Samuel, F.P., Arjunan, M.M.: Existence of solutions for impulsive partial neutral functional evolution integro-differential inclusions with infinite delay. Int. J. Pure Appl. Math. 85, 939–954 (2013)

    Article  MATH  Google Scholar 

  17. Benchohra, M., Ntouyas, S.: Existence and controllability results for multivalued semilinear differential equations with nonlocal conditions. Soochow J. Math. 29, 157–170 (2003)

    MathSciNet  MATH  Google Scholar 

  18. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 497–505 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chadha, A., Pandey, D.N.: Existence of a mild solution for impulsive neutral fractional differential equations with nonlocal conditions. Differ. Equ. Appl. 7, 151–168 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Lin, A., Hu, L.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Comput. Math. Appl. 59, 64–73 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yan, Z., Zhang, H.: On a nonlocal problem for partial stochastic functional integro-differential equations in Hilbert spaces. Electron. J. Math. Anal. Appl. 1, 212–229 (2013)

    MATH  Google Scholar 

  23. Farahi, S., Guendouzi, T.: Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions. Res. Math. 2014, 1–21 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Guendouzi, T., Farahi, S.: Approximate controllability of semilinear fractional stochastic dynamic systems with nonlocal conditions in Hilbert spaces. Mediterr. J. Math. 2015, 20 (2015)

    MATH  Google Scholar 

  25. Chang, Y.-K., Nieto, J.J.: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numer. Funct. Anal. Optim. 30, 227–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Benchohra, M., Ntouyas, S.K.: Nonlocal Cauchy problems on semi-infinite intervals for neutral functional differential and integro-differential inclusions in Banach spaces. Math. Slovaca 51, 529–545 (2011)

    MATH  Google Scholar 

  27. Ezzinbi, K., Fu, X., Hilal, K.: Existence and regularity in the \(\alpha \)-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal. TMA 67, 1613–1622 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chalishajar, D.N., Acharya, F.S.: Controllability of neutral impulsive differential inclusions with non-local conditions. Appl. Math. 2, 1486–1496 (2011)

    Article  Google Scholar 

  29. Debbouchea, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yan, Z.: Controllability of fractional-order partial neutral functional integro-differential inclusions with infinite delay. J. Frankl. Inst. 348, 2156–2173 (2011)

    Article  MATH  Google Scholar 

  31. Liu, Z., Li, X.: On the controllability of impulsive fractional evolution inclusions in Banach spaces. J. Optim. Theory Appl. 156, 167–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vijayakumar, V., Ravichandran, C., Murugesu, R., Trujillo, J.J.: Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators. Appl. Math. Comput. 247, 152–161 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Fečkan, M., Wang, J.-R., Zhou, Y.: Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory Appl. 165, 79–95 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Balasubramaniam, P., Vembarasan, V., Senthilkumar, T.: Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. Numer. Funct. Anal. Optim. 35, 177–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zang, Y., Li, J.: Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions. Bound. Value Probl. 2013, 13 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fan, Z.: Approximate controllability of fractional differential equations via resolvent operators. Adv. Differ. Equ. 2014, 11 (2014)

    Article  MathSciNet  Google Scholar 

  37. Sakthivel, R., Nieto, J.J., Mahmudov, N.I.: Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay. Taiwan. J. Math. 14, 1777–1797 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Park, J.Y., Jeong, J.U.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays. Adv. Differ. Equ. 2014, 1–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chang, Y.K., Zhao, Z.H., Guérékata, G.M.: \(N^{\prime }\): Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces. Comput. Math. Appl. 61, 384–391 (2011)

    Article  MathSciNet  Google Scholar 

  40. Fu, M.M., Liu, Z.X.: Square-mean almost automorphic solutions for some stochastic differential equations. Proc. Am. Math. Soc. 138, 3689–3701 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Oksendal, B.: Stochastic Differential Equations, 5th edn. Springer, Berlin (2002)

    Google Scholar 

  42. Mao, X.R.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)

    MATH  Google Scholar 

  43. Sakthivel, R., Luo, J.: Asymptotic stability of nonlinear impulsive stochastic differential eqnarrays. Stat. Probab. Lett. 79, 1219–1223 (2009)

    Article  MATH  Google Scholar 

  44. Balasubramaniam, P., Vinayagam, D.: Existence of solutions of nonlinear stochastic integro-differential inclusions in a Hilbert space. Comput. Math. Appl. 50, 809–821 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. Balasubramaniam, P., Ntouyas, S.K., Vinayagam, D.: Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space. J. Math. Anal. Appl. 305, 438–451 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, Y., Liu, B.: Existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay. Stoch. Anal. Appl. 25, 397–415 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yan, Z., Lu, F.: On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay. Appl. Anal. 2014, 1–26 (2014)

    Article  Google Scholar 

  48. Yan, Z., Jia, X.: Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Collect. Math. 2014, 1–32 (2014)

    Google Scholar 

  49. Kerboua, M., Debbouche, A., Baleanu, D.: Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. Abstr. Appl. Anal. 2013, 10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kerboua, M., Debbouche, A., Baleanu, D.: Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces. Electron. J. Qual. Theory Differ. Equ. 2014, 16 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mahmudov, N.I.: Existence and approximate controllability of Sobolev type fractional stochastic evolution equations. Bull. Pol. Acad. Sci. 62, 205–215 (2014)

    Google Scholar 

  52. Sakthivel, R., Ren, Y., Mahmudov, N.I.: Approximate controllability of second-order stochastic differential equations with impulsive effects. Mod. Phy. Lett. B 24, 1559–1572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sakthivel, R., Ganesh, R., Suganya, S.: Approximate controllability of fractional neutral stochastic system with infinite delay. Rep. Math. Phys. 70, 291–311 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252, 6163–6174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Li, K., Peng, J., Gao, J.: Controllability of nonlocal fractional differential systems of order \(\alpha \in (1,2]\) in Banach spaces. Rep. Math. Phys. 71, 11 (2013)

    MathSciNet  Google Scholar 

  56. Li, K., Peng, J.: Controllability of fractional neutral stochastic functional differential systems. Z. Angew. Math. Phys. 2013, 19 (2013)

    Google Scholar 

  57. Mahmudov, N.I., Zorlu, S.: Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions. Bound. Value Probl. 2013, 16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Palanisamy, M., Chinnathambi, R.: Approximate boundary controllability of Sobolev-type stochastic differential systems. J. Egypt. Math. Soc. 22, 201–208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63, 660–668 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  60. Ren, Y., Dai, H., Sakthivel, R.: Approximate controllability of stochastic differential systems driven by a Lévy process. Int. J. Control 86, 1158–1164 (2013)

    Article  MATH  Google Scholar 

  61. Balachandran, K., Sakthivel, R.: Controllability of integro-differential systems in Banach spaces. Appl. Math. Comput. 118, 63–71 (2001)

    MathSciNet  MATH  Google Scholar 

  62. Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    MATH  Google Scholar 

  63. Pazy, A.: Semi-groups of Linear Operator and Applications of Partial Differential Equations. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  64. Prato, G.D., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  65. Deimling, K.: Multivalued Differential Equations. de Gruyter, Berlin (1992)

    Book  MATH  Google Scholar 

  66. Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Vol. 7 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyte, Berlin (2001)

  67. Mahmudov, N.I., Denker, A.: On controllability of linear stochastic systems. Int. J. Control 73, 144–151 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  68. Dauer, J.P., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl. 290, 373–394 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  69. Dhage, B.C.: Fixed-point theorems for discontinuous multi-valued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi, and Indian Institute of Technology, Roorkee.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Alka Chadha & Dwijendra N. Pandey

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  1. Alka Chadha
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  2. Dwijendra N. Pandey
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Correspondence to Alka Chadha.

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Chadha, A., Pandey, D.N. Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions. J Theor Probab 31, 705–740 (2018). https://doi.org/10.1007/s10959-016-0732-2

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