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Existence of Solutions and Approximate Controllability of Fractional Nonlocal Neutral Impulsive Stochastic Differential Equations of Order 1 < q < 2 with Infinite Delay and Poisson Jumps

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Abstract

In this paper, we study the existence of mild solutions and the approximate controllability of nonlinear fractional nonlocal neutral impulsive stochastic differential equations of order 1 < q < 2 with infinite delay and Poisson jumps in which the initial value belong to the abstract phase space C h . The existence of mild solutions is derived with the help of Sadovskii’s fixed point theorem. The approximate controllability of the nonlinear fractional nonlocal neutral impulsive stochastic differential systems of order 1 < q < 2 with infinite delay and Poisson jumps is discussed under the assumption that the corresponding linear system is approximately controllable. Moreover, the approximate controllability of the above control system is established by using Lebesgue dominated convergence theorem. An example is provided to illustrate the theory.

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References

  1. Abada N, Benchohra M, Hammouche H. Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J Diff Eqn 2009;246(10):3834–3863.

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal RP, dos Santos JPC, Cuevas C. Analytic resolvent operator and existence results for fractional integro-differential equations. J Abstr Differ Equ Appl 2012;2(2):26–47.

    MathSciNet  MATH  Google Scholar 

  3. de Andrade B, dos Santos JPC. Existence of solutions for a fractional neutral integro-differential equations with unbounded delay. Electron J Differ Equ 2012;90:1–13.

    MathSciNet  MATH  Google Scholar 

  4. Arthi G, Balachandran K. Controllability of second-order impulsive differential equations with state-dependent delay. Bull Korean Math Soc 2011;48(6):1271–1290.

    Article  MathSciNet  MATH  Google Scholar 

  5. Balachandran K, Sakthivel R. Controllability of functional semilinear integro-differential systems in Banach spaces. J Comput Math Appl 2001;255(2):447–457.

    MATH  Google Scholar 

  6. Benchohra M, Handeson J, Ntouyas SK. Existence results for impulsive multivalued semilinear neutral functional inclusions in Banach spaces. J Math Anal Appl 2001;263(2):763–780.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang YK, Anguraj A, Mallika Arjunan M. Existence results for impulsive neutral fractional differential equations with infinite delay. Nonlinear Anal Hybrid Systems 2008;2(1):209–218.

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang YK, Nieto JJ, Li WS. Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. J Optim Theory App 2009; 142(2):267–273.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cui J, Yan L. Successive approximation of neutral stochastic evolution equations with infinite delay and Poisson jumps. Comput Math Appl 2012;218(2):6776–6784.

    MathSciNet  MATH  Google Scholar 

  10. Guendouzi T, Hamada I. Existence and controllability result for fractional neutral stochastic integro-differential equations with infinite delay. Advanced Modeling and Optimization 2013;15(2):281–299.

    Google Scholar 

  11. Guendouzi T, Hamada I. Relative controllability of fractional stochastic dynamical systems with multiple delays in control. Malaya Journal of Matematik 2013;1 (1):86–97.

    MATH  Google Scholar 

  12. Guendouzi T, Idrissi S. Approximate controllability of fractional stochastic functional evolutions driven by a fractional Brownian motion. Romai J 2012;8(2):103–117.

    MathSciNet  MATH  Google Scholar 

  13. Hernandez E. A Second order impulsive cauchy problem. Inst J Math Sci 2002; 31(8):451–461.

    Article  MathSciNet  MATH  Google Scholar 

  14. Hernandez E, Henriquez HR. Impulsive partial neutral differential equations. Appl Math Lett 2006;19(3):215–222.

    Article  MathSciNet  MATH  Google Scholar 

  15. Klamka J. Constrained controllability of semilinear systems. Nonlinear Anal 2001; 47(5):2939–2949.

    Article  MathSciNet  MATH  Google Scholar 

  16. Klamka J. Constrained exact controllability of semilinear systems. Syst Control Lett 2002;47(2):139–147.

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin A, Ren Y, Xia N. On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators. Math Comput Model 2010;51(5): 413–424.

    Article  MathSciNet  MATH  Google Scholar 

  18. Mahmudov NI. Controllability of linear stochastic systems. IEEE Trans Autom Contr 2001;46(5):724–731.

    Article  MathSciNet  MATH  Google Scholar 

  19. Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. NewYork: Wiley; 1993.

    MATH  Google Scholar 

  20. Muthukumar P, Balasubramaniam P. Approximate controllability of second order damped Mckean-vlasov stochastic evolution equations. Comput Math Appl 2010; 60(10):2788–2796.

    Article  MathSciNet  MATH  Google Scholar 

  21. Muthukumar P, Rajivganthi C. Approximate controllability of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. J Syst Sci Complex 2015;28(5):1033–1048.

    Article  MathSciNet  MATH  Google Scholar 

  22. Podlubny I. Fractional Differential Equations. New York: Academic Press; 1999.

    MATH  Google Scholar 

  23. Rajivganthi C, Thiagu K, Muthukumar P, Balasubramaniam P. Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps. Applications Math 2015;60(4):395–419.

    Article  MathSciNet  MATH  Google Scholar 

  24. Ren Y, Sakthivel R. Existence,uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps. J Math Phys 2012;53(7):073517.

    Article  MathSciNet  MATH  Google Scholar 

  25. Sadovskii BN. On a fixed point principle. Funct Anal Appl 1967;1(2):151–153.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sakthivel R. Approximate controllability of second-order stochastic differential equations with impulsive effects. Modern Phys Lett B 2010;24(14):1559–1572.

    Article  MathSciNet  MATH  Google Scholar 

  27. Sakthivel R, Ganesh R, Suganya S. Approximate controllability of fractional neutral stochastic systems with infinite delay. Rep Math Phys 2012;70(3):291–311.

    Article  MathSciNet  MATH  Google Scholar 

  28. Sakthivel R, Ganesh R, Ren Y, Anthoni SM. Approximate controllability of nonlinear fractional dynamical systems. Commun Nonlinear Sci Numer Simul 2013;18 (2):3498–3508.

    Article  MathSciNet  MATH  Google Scholar 

  29. Sakthivel R, Ren Y. Complete controllability of stochastic evolution equations with jumps. Rep Math Phys 2011;68(2):163–174.

    Article  MathSciNet  MATH  Google Scholar 

  30. Sakthivel R, Ren Y. Exponential stability of second-order stochastic evolution equations with poisson jumps. Commun Nonlinear Sci Numer Simul 2012;17(12):4517–4523.

    Article  MathSciNet  MATH  Google Scholar 

  31. Sakthivel R, Revathi P, Marshal Anthoni S. Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations. Nonlinear Anal 2012;75(7):3339–3347.

    Article  MathSciNet  MATH  Google Scholar 

  32. Sukavanam N, Tafesse S. Approximate controllability of a delayed non-linear control system with growing non-linear term. Nonlinear Anal TMA 2011;74(18):6868–6875.

    Article  MATH  Google Scholar 

  33. Shu XB, Wang Q. The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1<α<2. Comput Math Appl 2012;64(6):2100–2110.

    Article  MathSciNet  MATH  Google Scholar 

  34. Tan J, Wong H, Guo Y. Existence and uniqueness of solutions to neutral stochastic functional differential equations with Poisson jumps. Abstr Appl Anal 2012. Article ID 371239.

  35. Vijayakumar V, Selvakumar A, Murugesu R. Controllability for a class of integro-differential equations with unbounded delay. Comput Math Appi 2014;232: 303–312.

    Article  MathSciNet  Google Scholar 

  36. Yan Z, Lu F. On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay. Applicable Anal 2014;94(6):1235–1258.

    Article  MathSciNet  MATH  Google Scholar 

  37. Yue C. Second-order neutral impulsive stochastic evolution equations with infinite delay. Advances in Difference Equations 2014;2014(1):1–13.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to express their sincere thanks to the Editor in Chief, Associate Editors, and anonymous reviewers for helpful comments and suggestions to improve the quality of this manuscript. This work was supported by the National Board for Higher Mathematics, Mumbai, India under the grant no: 2/48(5)/2013/NBHM (R.P.)/RD-II/688 dated 16 January 2014.

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Authors and Affiliations

  1. Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India

    P. Muthukumar & K. Thiagu

  2. Department of Mathematics, Kandaswamy Kandar’s College, Velur - 638182, Tamilnadu, India

    K. Thiagu

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  1. P. Muthukumar
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  2. K. Thiagu
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Correspondence to P. Muthukumar.

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Muthukumar, P., Thiagu, K. Existence of Solutions and Approximate Controllability of Fractional Nonlocal Neutral Impulsive Stochastic Differential Equations of Order 1 < q < 2 with Infinite Delay and Poisson Jumps. J Dyn Control Syst 23, 213–235 (2017). https://doi.org/10.1007/s10883-015-9309-0

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  • DOI: https://doi.org/10.1007/s10883-015-9309-0

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