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

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, England (1993)
• 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. 155–160. Princeton University Press, Princeton (1950)
• 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)
• 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)
• 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 fractional stochastic differential inclusions. Asian J. Control 24(5), 2378–2394 (2022)
• 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 ∈ (1, 2) with delay. Chaos Solitons Fractals 153, 111565 (2021)
• 10. Fernandez, S.B., Nieto, J.J.: Basic control theory for linear fractional differential equations with constant coefficients. Front. Phys. 8, 377 (2020)
• 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 Hilfer fractional delay integro-differential systems. Adv. Differ. Equ. 2021(467), 1–18 (2021)
• 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)
• 17. Kumar, A., Kumar, A., Vats, R.K., Kumar, P.: Approximate controllability of neutral delay integrodifferential inclusion of order α ∈ (1, 2) with non-instantaneous impulses. Evol. Equ. Control Theory (2021). https://doi.org/10.3934/eect.2021058
• 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 map. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 13, 781–786 (1965)
• 21. Li, X., Liu, X., Tang, M.: Approximate controllability of fractional evolution inclusions with damping. Chaos Solitons Fractals 148, 111073 (2021)
• 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)
• 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. Math. Phys. 71, 33–43 (2013)
• 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 Optim. 53(4), 1920–1933 (2015)
• 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 equations of order 1 < r < 2 with sectorial operators. Optim. Control Appl. Methods 43(5), 1314–1327 (2022)
• 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)
• 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 ∈ (1, 2). Optim. Control Appl. Methods 43(4), 996–1019 (2022)
• 31. Raja, M. Mohan., Vijayakumar, V.: Existence results for Caputo fractional mixed Volterra–Fredholmtype integrodifferential inclusions of order r ∈ (1, 2) with sectorial operators. Chaos Solitons Fractals 159, 112127 (2022)
• 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)
• 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)
• 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)
• 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)
• 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. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6662
• 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)
• 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)
• 40. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of semilinear fractional control systems of order α ∈ (1, 2] with infinite delay. Mediterr. J. Math. 13, 2539–2550 (2016)
• 41. Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of fractional semilinear stochastic system of order α ∈ (1, 2]. J. Dyn. Control Syst. 23, 679–691 (2017)
• 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)
• 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)
• 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. Int. J. Control 91(10), 2366–2386 (2018)
• 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)
• 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)
• 48. Vijayakumar, V., Murugesu, R.: Controllability for a class of second order evolution differential inclusions without compactness. Appl. Anal. 98(7), 1367–1385 (2019)
• 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)
• 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)
• 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)
• 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
• 53. Yan, Z., Jia, X.: Optimal controls for fractional stochastic functional differential equations of order α ∈ (1, 2]. Bull. Malays. Math. Soc. Ser. 41, 1581–1606 (2018)
• 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)
• 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), 491–509 (2021)
• 59. Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 21(4), 551–565 (1983)