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

• V. Vijayakumar ^[3] ; M. Mohan Raja ^[3] ; Anurag Shukla ^[4] ; Juan J. Nieto ^[1] ; Kottakkaran Sooppy Nisar ^[2]
  1. [1] Universidade de Santiago de Compostela

     Universidade de Santiago de Compostela

     Santiago de Compostela, España

     
  2. [2] Prince Sattam Bin Abdulaziz University

     Prince Sattam Bin Abdulaziz University

     Arabia Saudí

     
  3. [3] Vellore Institute of Technology
  4. [4] Rajkiya Engineering College

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • This paper investigates the issue of existence and approximate controllability results for impulsive fractional differential inclusions with delay of orderr ∈ (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.

• Referencias bibliográficas
  • 1. Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Dekker, New York (1980)
  • 2. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical Group,...
  • 3. Bazhlekova, E.: Fractional Evolution Equations in Banach Spaces. Eindhoven University of Technology, Eindhoven (2001)
  • 4. Bohnenblust, H.F., Karlin, S.: On a Theorem of Ville, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, pp....
  • 5. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal....
  • 6. Byszewski, L., Akca, H.: On a mild solution of a semilinear functional-differential evolution nonlocal problem. J. Appl. Math. Stoch. Anal....
  • 7. Deimling, K.: Multivalued Differential Equations. Walter de Gruyter, Berlin (1992)
  • 8. Dineshkumar, C., Nisar, K.S., Udhayakumar, R., Vijayakumar, V.: A discussion on approximate controllability of Sobolev-type Hilfer neutral...
  • 9. Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Shukla, A., Nisar, K.S.: A note on approximate controllability for nonlocal fractional...
  • 10. Fernandez, S.B., Nieto, J.J.: Basic control theory for linear fractional differential equations with constant coefficients. Front. Phys....
  • 11. Gou, H., Li, Y.: A study on impulsive fractional hybrid evolution equations using sequence method. Comput. Appl. Math. 39(225), 1–31 (2020)
  • 12. He, J.W., Liang, Y., Ahmad, B., Zhou, Y.: Nonlocal fractional evolution inclusions of order α ∈ (1, 2). Mathematics 209(7), 1–17 (2019)
  • 13. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis (Theory). Kluwer Academic Publishers, Dordrecht (1997)
  • 14. Kalman, R.E.: Controllability of linear systems. Contrib. Differ. Equ. 1, 190–213 (1963)
  • 15. Kavitha, K., Nisar, K.S., Shukla, A., Vijayakumar, V., Rezapour, S.: A discussion concerning the existence results for the Sobolev-type...
  • 16. Kavitha, K., Vijayakumar, V.: A discussion concerning to partial-approximate controllability of Hilfer fractional system with nonlocal...
  • 17. Kumar, A., Kumar, A., Vats, R.K., Kumar, P.: Approximate controllability of neutral delay integrodifferential inclusion of order α ∈ (1,...
  • 18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
  • 19. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)
  • 20. Lasota, A., Opial, Z.: An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued...
  • 21. Li, X., Liu, X., Tang, M.: Approximate controllability of fractional evolution inclusions with damping. Chaos Solitons Fractals 148, 111073...
  • 22. Li, X., Liu, Z., Tisdell, C.C.: Approximate controllability of fractional control systems with time delay using the sequence method. Electron....
  • 23. Li, K.X., Peng, J.G., Gao, J.H.: Controllability of nonlocal fractional differential systems of order α ∈ (1, 2] in Banach spaces. Rep....
  • 24. Lightbourne, J.H., Rankin, S.: A partial functional differential equation of Sobolev type. J. Math. Anal. Appl. 93(2), 328–337 (1983)
  • 25. Liu, Z., Li, X.: Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM J. Control...
  • 26. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
  • 27. Martelli, M.: A Rothe’s type theorem for non-compact acyclic-valued map. Boll. Un. Math. Ital. 2, 70–76 (1975)
  • 28. Raja, M. Mohan., Vijayakumar, V.: Optimal control results for Sobolev-type fractional mixed Volterra– Fredholm type integrodifferential...
  • 29. Raja, M. Mohan., Vijayakumar, V.: New results concerning to approximate controllability of fractional integrodifferential evolution equations...
  • 30. Raja, M. Mohan., Vijayakumar, V., Shukla, A., Nisar, K.S., Sakthivel, N., Kaliraj, K.: Optimal control and approximate controllability...
  • 31. Raja, M. Mohan., Vijayakumar, V.: Existence results for Caputo fractional mixed Volterra–Fredholmtype integrodifferential inclusions of...
  • 32. Raja, M. Mohan., Vijayakumar, V., Shukla, A., Nisar, K.S., Baskonus, Haci Mehmet: On the approximate controllability results for fractional...
  • 33. Raja, M. Mohan., Vijayakumar, V., Le Huynh, Nhat, Udhayakumar, R., Nisar, K.S.: Results on the approximate controllability of fractional...
  • 34. Mophou, G.M., N’Guerekata, G.M.: On integral solutions of some nonlocal fractional differential equations with nondense domain. Nonlinear...
  • 35. Mophou, G.M., N’Guerekata, G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup...
  • 36. Papageorgiou, N.: Boundary value problems for evolution inclusions. Comment. Math. Univ. Carol. 29, 355–363 (1988)
  • 37. Patel, R., Shukla, A., Jadon, S.S.: Existence and optimal control problem for semilinear fractional order (1, 2] control system. Math....
  • 38. Podlubny, I.: Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Method...
  • 39. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear system with state delay using sequence method. J. Frankl....
  • 40. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear fractional control systems of order α ∈ (1, 2] with...
  • 41. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of fractional semilinear stochastic system of order α ∈ (1, 2]. J....
  • 42. Singh, A., Shukla, A., Vijayakumar, V., Udhayakumar, R.: Asymptotic stability of fractional order (1, 2]stochastic delay differential...
  • 43. Sivasankaran, S., Mallika Arjunan, M., Vijayakumar, V.: Existence of global solutions for second order impulsive abstract partial differential...
  • 44. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Hung. 32, 75–96 (1978)
  • 45. Vijayakumar, V.: Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay....
  • 46. Vijayakumar, V.: Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert...
  • 47. Vijayakumar, V.: Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type....
  • 48. Vijayakumar, V., Murugesu, R.: Controllability for a class of second order evolution differential inclusions without compactness. Appl....
  • 49. Vijayakumar, V., Ravichandran, C.,Murugesu, R.: Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential...
  • 50. Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim....
  • 51. Wang, J., Ibrahim, Ahmed G.: Existence and controllability results for nonlocal fractional impulsive differential inclusions in Banach...
  • 52. Williams, W.K., Vijayakumar, V.: Discussion on the controllability results for fractional neutral impulsive Atangana–Baleanu delay integro-differential...
  • 53. Yan, Z., Jia, X.: Optimal controls for fractional stochastic functional differential equations of order α ∈ (1, 2]. Bull. Malays. Math....
  • 54. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal....
  • 55. Zeidler, E.: Nonlinear Functional Analysis and Its Application II/A. Springer, New York (1990)
  • 56. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)
  • 57. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier, New York (2015)
  • 58. Zhou, Y., He, J.W.: New results on controllability of fractional evolution systems with order α ∈ (1, 2). Evol. Equ. Control Theory 10(3),...
  • 59. Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 21(4), 551–565 (1983)