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Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative

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Abstract

The study of this work ensures the existence results of a class of Atangana–Baleanu (A–B) fractional stochastic differential equations with non-instantaneous impulses and nonlocal conditions. For this investigation, the proposed fractional impulsive stochastic system is transformed into an equivalent fixed point problem. The operator is then analyzed for boundedness, contraction, continuity and equicontinuity. Then Arzela–Ascolli theorem ensures the operator is relatively compact and Krasnoselskii’s fixed point theorem is proved for the existence of the mild solution. At last, to verify the theoretical results an example is given. The obtained result suggest that the proposed method is efficient and proper for dealing with the fractional stochastic problems arising in engineering, technology and physics, and in terms of the A–B fractional derivative.

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References

  1. Podlubny, I. 1999. Fractional differential equations. New York: Academic Press.

    Google Scholar 

  2. Miller, K.S., and B. Ross. 1993. An introduction to the fractional calculus and differential equations. New York: Wiley.

    Google Scholar 

  3. Raja, M.M., and V. Vijayakumar. 2023. Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order \(r \in (1,2)\) via sectorial operator. Fractional Calculus and Applied Analysis 26: 1740–1769.

    Article  MathSciNet  Google Scholar 

  4. Williams, W.K., and V. Vijayakumar. 2023. New discussion on the existence and controllability of fractional evolution inclusion of order \(1 < r < 2\) without compactness. Mathematical Methods in the Applied Sciences 46: 13188–13204.

    Article  MathSciNet  Google Scholar 

  5. Diethelm, K., and N.J. Ford. 2002. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265 (2): 229–248.

    Article  MathSciNet  Google Scholar 

  6. Lakshmikantham, V. 2008. Theory of fractional functional differential equations. Nonlinear Analysis 69: 3337–3343.

    Article  MathSciNet  Google Scholar 

  7. Dhayal, R., J.F. Gomez-Aguilar, and G. Fernandez-Anaya. 2022. Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses. Optimal Control Applications and Methods 43: 386–401.

    Article  MathSciNet  Google Scholar 

  8. Shukla, A., N. Sukavanam, and D.N. Pandey. 2015. Approximate controllability of semilinear fractional control systems of order \(\alpha \in (1,2]\). In 2015 Proceedings of the Conference on Control and its Applications, pp. 175-180.

  9. Atangana, A., and D. Balneau. 2016. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science 20 (2): 763–769.

    Article  Google Scholar 

  10. Atangana, A., and R.T. Alqahtani. 2018. New numerical method and application to Keller–Segel model with fractional order derivative. Chaos Solitons & Fractals 116: 14–21.

    Article  MathSciNet  Google Scholar 

  11. Dhayal, R., Y. Zhao, Q. Zhu, Z. Wang, and M. Karimi. 2024. Approximate controllability of Atangana-Baleanu fractional stochastic differential systems with non-Gaussian process and impulses, Discrete and Continuous Dynamical Systems Series S.

  12. Kumar, A., and D.N. Pandey. 2020. Existence of mild solution of Atangana–Baleanu fractional differential equations with non-instantaneous impulses and with non-local conditions. Chaos Solitons & Fractals 132: 109551.

    Article  MathSciNet  Google Scholar 

  13. Aimene, D., D. Baleanu, and D. Seba. 2019. Controllability of semilinear impulsive Atangana–Baleanu fractional differential equations with delay. Chaos Solitons & Fractals 128: 51–57.

    Article  MathSciNet  Google Scholar 

  14. Mao, X.R. 1997. Stochastic differential equations and applications. Chichester: Horwood.

    Google Scholar 

  15. Oksendal, B. 2002. Stochastic differential equations, 5th ed. Berlin: Springer.

    Google Scholar 

  16. Prato, G.D., and J. Zabczyk. 1992. Stochastic equations in infinite dimensions, vol. 44. Encyclopedia of mathematics and its applications. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  17. Shukla, A., N. Sukavanam, and D.N. Pandey. 2015. Complete controllability of semi-linear stochastic system with delay. Rendiconti del Circolo Matematico di Palermo 64: 209–220.

    Article  MathSciNet  Google Scholar 

  18. Shukla, A., N. Sukavanam, D.N. Pandey, and U. Arora. 2016. Approximate controllability of second order semilinear control system. Circuits, Systems, and Signal Processing 35: 3339–3354.

    Article  MathSciNet  Google Scholar 

  19. Yan, Z., and F. Lu. 2015. Existence results for a new class of fractionl impulsive partial neutral stochastic integro-differential equations with infinite delay. Journal of Applied Analysis and Computation 5 (3): 329–346.

    MathSciNet  Google Scholar 

  20. Chendrayan, D., U. Ramalingam, V. Vijayakumar, and A. Shukla. 2021. A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order \(r\in (1,2)\) with delay. Chaos Solitons & Fractals 153 (5): 1–16.

    Google Scholar 

  21. Dineshkumar, C., K.S. Nisar, R. Udhayakumar, and V. Vijayakumar. 2022. A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions. Asian Journal of Control 24: 2378–2394.

    Article  MathSciNet  Google Scholar 

  22. Dhayal, R., M. Malik, and Q. Zhu. 2024. Optimal controls of impulsive fractional stochastic differential systems driven by Rosenblatt process with state-dependent delay. Asian Journal of Control 26: 162–174.

    Article  MathSciNet  Google Scholar 

  23. Sathiyaraj, T., J. Wang, and P. Balasubramaniam. 2021. Controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systems. Applied Mathematics & Optimization 84 (3): 2527–2554.

    Article  MathSciNet  Google Scholar 

  24. Revathi, P., R. Sakthivel, Y. Ren, and S.M. Anthoni. 2014. Existence of almost automorphic mild solutions to non-autonomous neutral stochastic differential equations. Applied Mathematics and Computation 230: 639–649.

    Article  MathSciNet  Google Scholar 

  25. Dhayal, R., M. Malik, S. Abbas, and A. Debbouche. 2020. Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses. Mathematical Methods in the Applied Sciences 43 (7): 4107–4124.

    MathSciNet  Google Scholar 

  26. Sakthivel, R., P. Revathi, and Y. Ren. 2013. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Analysis 81: 70–86.

    Article  MathSciNet  Google Scholar 

  27. Dineshkumar, C., V. Vijayakumar, R. Udhayakumar, K.S. Nisar, and A. Shukla. 2023. Results on approximate controllability for fractional stochastic delay differential systems of order \(r \in (1,2)\). Stochastics and Dynamics 23: 2350047.

    Article  MathSciNet  Google Scholar 

  28. Hernandez, E., and D. O’Regan. 2013. On a new class of abstract impulsive differential equations. Proceedings of the American Mathematical Society 141 (5): 1641–1649.

    Article  MathSciNet  Google Scholar 

  29. Guendouzi, T., and O. Benzatout. 2014. Existence of mild solutions for impulsive fractional stochastic differential inclusions with state-dependent delay. Chinese Journal of Mathematics 2014: 981714.

    Article  MathSciNet  Google Scholar 

  30. Kasinathan, R., R. Kasinathan, V. Sandrasekaran, and R. Dhayal. 2024. Solvability and optimal control for secondorder stochastic differential systems under the influence of delay and impulses. Stochastics. https://doi.org/10.1080/17442508.2024.2352072.

  31. Bao, H., and J. Cao. 2017. Existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delay. Advances in Difference Equations 2017: 66.

    Article  MathSciNet  Google Scholar 

  32. Abouagwa, M., F. Cheng, and J. Li. 2020. Impulsive stochastic fractional differential equations driven by fractional Brownian motion. Advances in Difference Equations 2020: 7.

    Article  MathSciNet  Google Scholar 

  33. Pierri, M., D. O’Regan, and V. Rolnik. 2013. Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Applied Mathematics and Computation 219: 6743–6749.

    Article  MathSciNet  Google Scholar 

  34. Dhayal, R., J.F. Gomez-Aguilar, and E. Perez-Careta. 2024. Stability and controllability of Caputo fractional stochastic differential systems driven by Rosenblatt process with impulses. International Journal of Dynamics and Control 12: 1626–1639.

    Article  MathSciNet  Google Scholar 

  35. Boudaoui, A., T. Caraballo, and A. Ouahab. 2017. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems 22 (5): 2521–2541.

    Article  MathSciNet  Google Scholar 

  36. Arjunan, M.M., T. Abdeljawad, K. Velusamy, and A. Yousef. 2021. On a new class of Atangana–Baleanu fractional Volterra–Fredholm integro-differential inclusions with non-instantaneous impulses. Chaos Solitons & Fractals 148: 111075.

    Article  MathSciNet  Google Scholar 

  37. Boufoussi, B., and S. Hajji. 2012. Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Statistics & Probability Letters 82: 1549–1558.

    Article  MathSciNet  Google Scholar 

  38. Sakthivel, R., P. Revathi, and Y. Ren. 2013. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Analysis: Theory, Methods & Applications 81: 70–86.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are very thankful to the anonymous reviewers and associate editor for their constructive comments and suggestions which help us to improve the manuscript.

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Authors

Contributions

Rajesh Dhayal: conceptualization, methodology, formal analysis, writing original draft; Mohd Nadeem: supervision, writing review and editing, formal analysis. All authors read and approved the final manuscript.

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Correspondence to Rajesh Dhayal.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Communicated by S. Ponnusamy.

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Dhayal, R., Nadeem, M. Existence results of a nonlocal impulsive fractional stochastic differential systems with Atangana–Baleanu derivative. J Anal (2024). https://doi.org/10.1007/s41478-024-00793-6

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  • DOI: https://doi.org/10.1007/s41478-024-00793-6

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