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A Note on the Existence and Controllability Results for Fractional Integrodifferential Inclusions of Order \(r \in (1,2]\) with Impulses

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Abstract

This paper investigates the issue of existence and approximate controllability results for impulsive fractional differential inclusions with delay of order \(r \in (1,2]\) in Banach space. To begin, we analyze existence results for impulsive fractional evolution inclusions with delay using fractional calculations, the r-order cosine family, multivalued maps, and Martelli’s fixed point theorem. The approximate controllability results for impulsive fractional evolution inclusions with delay were then derived using Gronwall’s inequality and the sequence method. Then, we investigate the Sobolev fractional integrodifferential inclusions with finite delay. Moreover, we develop the nonlocal conditions in a given system. Finally, an example is presented to illustrate the main results.

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References

  1. Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Dekker, New York (1980)

    MATH  Google Scholar 

  2. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical Group, England (1993)

    MATH  Google Scholar 

  3. Bazhlekova, E.: Fractional Evolution Equations in Banach Spaces. Eindhoven University of Technology, Eindhoven (2001)

    Google Scholar 

  4. Bohnenblust, H.F., Karlin, S.: On a Theorem of Ville, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, pp. 155–160. Princeton University Press, Princeton (1950)

    Google Scholar 

  5. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Byszewski, L., Akca, H.: On a mild solution of a semilinear functional-differential evolution nonlocal problem. J. Appl. Math. Stoch. Anal. 10(3), 265–271 (1997)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Dineshkumar, C., Nisar, K.S., Udhayakumar, R., Vijayakumar, V.: A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions. Asian J. Control 24(5), 2378–2394 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Shukla, A., Nisar, K.S.: A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order \(r \in (1,2)\) with delay. Chaos Solitons Fractals 153, 111565 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Fernandez, S.B., Nieto, J.J.: Basic control theory for linear fractional differential equations with constant coefficients. Front. Phys. 8, 377 (2020)

    Google Scholar 

  11. Gou, H., Li, Y.: A study on impulsive fractional hybrid evolution equations using sequence method. Comput. Appl. Math. 39(225), 1–31 (2020)

    MathSciNet  MATH  Google Scholar 

  12. He, J.W., Liang, Y., Ahmad, B., Zhou, Y.: Nonlocal fractional evolution inclusions of order \(\alpha \in (1,2)\). Mathematics 209(7), 1–17 (2019)

    Google Scholar 

  13. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis (Theory). Kluwer Academic Publishers, Dordrecht (1997)

    MATH  Google Scholar 

  14. Kalman, R.E.: Controllability of linear systems. Contrib. Differ. Equ. 1, 190–213 (1963)

    MathSciNet  Google Scholar 

  15. Kavitha, K., Nisar, K.S., Shukla, A., Vijayakumar, V., Rezapour, S.: A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems. Adv. Differ. Equ. 2021(467), 1–18 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Kavitha, K., Vijayakumar, V.: A discussion concerning to partial-approximate controllability of Hilfer fractional system with nonlocal conditions via approximating method. Chaos Solitons Fractals 157, 111924 (2022)

    MathSciNet  MATH  Google Scholar 

  17. Kumar, A., Kumar, A., Vats, R.K., Kumar, P.: Approximate controllability of neutral delay integrodifferential inclusion of order \(\alpha \in (1,2)\) with non-instantaneous impulses. Evol. Equ. Control Theory (2021). https://doi.org/10.3934/eect.2021058

    Article  Google Scholar 

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

    MATH  Google Scholar 

  19. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    MATH  Google Scholar 

  20. Lasota, A., Opial, Z.: An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 13, 781–786 (1965)

    MATH  Google Scholar 

  21. Li, X., Liu, X., Tang, M.: Approximate controllability of fractional evolution inclusions with damping. Chaos Solitons Fractals 148, 111073 (2021)

    MathSciNet  MATH  Google Scholar 

  22. Li, X., Liu, Z., Tisdell, C.C.: Approximate controllability of fractional control systems with time delay using the sequence method. Electron. J. Differ. Equ. 2017(272), 1–11 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Li, K.X., Peng, J.G., Gao, J.H.: Controllability of nonlocal fractional differential systems of order \(\alpha \in (1,2]\) in Banach spaces. Rep. Math. Phys. 71, 33–43 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Lightbourne, J.H., Rankin, S.: A partial functional differential equation of Sobolev type. J. Math. Anal. Appl. 93(2), 328–337 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Liu, Z., Li, X.: Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM J. Control Optim. 53(4), 1920–1933 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Martelli, M.: A Rothe’s type theorem for non-compact acyclic-valued map. Boll. Un. Math. Ital. 2, 70–76 (1975)

    MathSciNet  MATH  Google Scholar 

  28. Raja, M. Mohan., Vijayakumar, V.: Optimal control results for Sobolev-type fractional mixed Volterra–Fredholm type integrodifferential equations of order \(1 < r < 2\) with sectorial operators. Optim. Control Appl. Methods 43(5), 1314–1327 (2022)

    MathSciNet  Google Scholar 

  29. Raja, M. Mohan., Vijayakumar, V.: New results concerning to approximate controllability of fractional integrodifferential evolution equations of order \(1<r<2\). Numer. Methods Partial Differ. Equ. 38(3), 509–524 (2022)

    Google Scholar 

  30. Raja, M. Mohan., Vijayakumar, V., Shukla, A., Nisar, K.S., Sakthivel, N., Kaliraj, K.: Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order \(r \in (1,2)\). Optim. Control Appl. Methods 43(4), 996–1019 (2022)

    MathSciNet  Google Scholar 

  31. Raja, M. Mohan., Vijayakumar, V.: Existence results for Caputo fractional mixed Volterra–Fredholm-type integrodifferential inclusions of order \(r \in (1,2)\) with sectorial operators. Chaos Solitons Fractals 159, 112127 (2022)

    MathSciNet  Google Scholar 

  32. Raja, M. Mohan., Vijayakumar, V., Shukla, A., Nisar, K.S., Baskonus, Haci Mehmet: On the approximate controllability results for fractional integrodifferential systems of order \(1 < r < 2\) with sectorial operators. J. Comput. Appl. Math. 415, 114492 (2022)

    MathSciNet  MATH  Google Scholar 

  33. Raja, M. Mohan., Vijayakumar, V., Le Huynh, Nhat, Udhayakumar, R., Nisar, K.S.: Results on the approximate controllability of fractional hemivariational inequalities of order \(1<r<2\). Adv. Differ. Equ. 237, 1–25 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Mophou, G.M., N’Guerekata, G.M.: On integral solutions of some nonlocal fractional differential equations with nondense domain. Nonlinear Anal. 71(10), 4668–4675 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Mophou, G.M., N’Guerekata, G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79(2), 322–335 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Papageorgiou, N.: Boundary value problems for evolution inclusions. Comment. Math. Univ. Carol. 29, 355–363 (1988)

    MathSciNet  MATH  Google Scholar 

  37. Patel, R., Shukla, A., Jadon, S.S.: Existence and optimal control problem for semilinear fractional order \((1,2]\) control system. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6662

    Article  Google Scholar 

  38. Podlubny, I.: Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Method of Their Sand Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  39. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear system with state delay using sequence method. J. Frankl. Inst. 352, 5380–5392 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear fractional control systems of order \(\alpha \in (1,2]\) with infinite delay. Mediterr. J. Math. 13, 2539–2550 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of fractional semilinear stochastic system of order \(\alpha \in (1,2]\). J. Dyn. Control Syst. 23, 679–691 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Singh, A., Shukla, A., Vijayakumar, V., Udhayakumar, R.: Asymptotic stability of fractional order \((1,2]\) stochastic delay differential equations in Banach spaces. Chaos Solitons Fractals 150, 1–9 (2021). (111095)

    MathSciNet  MATH  Google Scholar 

  43. Sivasankaran, S., Mallika Arjunan, M., Vijayakumar, V.: Existence of global solutions for second order impulsive abstract partial differential equations. Nonlinear Anal. Theory Methods Appl. 74(17), 6747–6757 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Hung. 32, 75–96 (1978)

    MathSciNet  MATH  Google Scholar 

  45. Vijayakumar, V.: Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay. Int. J. Control 91(10), 2366–2386 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Vijayakumar, V.: Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. IMA J. Math. Control Inf. 35(1), 297–314 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Vijayakumar, V.: Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type. Results Math. 73(1), 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  48. Vijayakumar, V., Murugesu, R.: Controllability for a class of second order evolution differential inclusions without compactness. Appl. Anal. 98(7), 1367–1385 (2019)

    MathSciNet  MATH  Google Scholar 

  49. Vijayakumar, V., Ravichandran, C., Murugesu, R.: Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in Banach spaces. Dyn. Contin. Discrete Impuls. Syst. 20(4–5b), 485–502 (2013)

    MathSciNet  MATH  Google Scholar 

  50. Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154, 292–302 (2012)

    MathSciNet  MATH  Google Scholar 

  51. Wang, J., Ibrahim, Ahmed G.: Existence and controllability results for nonlocal fractional impulsive differential inclusions in Banach Spaces. J. Funct. Spaces Appl. 2013, 1–16 (2013)

    MathSciNet  Google Scholar 

  52. Williams, W.K., Vijayakumar, V.: Discussion on the controllability results for fractional neutral impulsive Atangana–Baleanu delay integro-differential systems. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7754

    Article  Google Scholar 

  53. Yan, Z., Jia, X.: Optimal controls for fractional stochastic functional differential equations of order \(\alpha \in (1,2]\). Bull. Malays. Math. Soc. Ser. 41, 1581–1606 (2018)

    MathSciNet  MATH  Google Scholar 

  54. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)

    MathSciNet  MATH  Google Scholar 

  55. Zeidler, E.: Nonlinear Functional Analysis and Its Application II/A. Springer, New York (1990)

    MATH  Google Scholar 

  56. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    MATH  Google Scholar 

  57. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier, New York (2015)

    Google Scholar 

  58. Zhou, Y., He, J.W.: New results on controllability of fractional evolution systems with order \(\alpha \in (1,2)\). Evol. Equ. Control Theory 10(3), 491–509 (2021)

    MathSciNet  MATH  Google Scholar 

  59. Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 21(4), 551–565 (1983)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research of J. J. Nieto has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER), project MTM2016-75140-P; and by Xunta de Galicia under grant ED431C 2019/02.

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamilnadu, 632 014, India

    M. Mohan Raja & V. Vijayakumar

  2. Department of Applied Science, Rajkiya Engineering College, Kannauj, Kannauj, 209732, India

    Anurag Shukla

  3. Department of Statistics, Mathematical Analysis and Optimization, Institute of Mathematics, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Juan J. Nieto

  4. Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia

    Kottakkaran Sooppy Nisar

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  1. M. Mohan Raja
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Mohan Raja, M., Shukla, A., Nieto, J.J. et al. A Note on the Existence and Controllability Results for Fractional Integrodifferential Inclusions of Order \(r \in (1,2]\) with Impulses. Qual. Theory Dyn. Syst. 21, 150 (2022). https://doi.org/10.1007/s12346-022-00681-z

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