On $k$-Generalized $\psi$-Hilfer Boundary Value Problems with Retardation and Anticipation

Abstract

In this paper, we prove some existence and uniqueness results for a class of boundary valued problems for implicit nonlinear $k$-generalized $\psi$-Hilfer fractional differential equations involving both retarded and advanced arguments. Further, examples are given to illustrate the viability of our results.

Keywords

• $\psi$-Hilfer fractional derivative
• $k$-generalized $\psi$-Hilfer fractional derivative
• Cauchy-type problem
• retarded arguments
• advanced arguments
• existence
• uniqueness

References

1. [1] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Differential and Integral Equations: Existence and Stability, Walter de Gruyter, London, 2018.
2. [2] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.
3. [3] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.
4. [4] A. Almalahi and K. Panchal, Existence results of ψ-Hilfer integro-differential equations with fractional order in Banach space, Ann. Univ. Paedagog. Crac. Stud. Math, 19 (2020), 171-192.
5. [5] M. Benchohra, S. Bouriah and J. Henderson, Nonlinear implicit Hadamard's fractional differential equations with retarded and advanced arguments, Azerbaijan J. Math. 8 (2018), 72-85.
6. [6] M. Benchohra, J.E. Lazreg and G.M. N'Guérékata, Nonlinear implicit Hadamard's fractional differential equations on Banach space with retarded and advanced arguments, Intern. J. Evol. Equ., 10 (2015), 283-295.
7. [7] M. Benchohra, J. Henderson and S.K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York, 2006.
8. [8] Y.M. Chu, M.U. Awan, S. Talib, M.A. Noor and K.I. Noor, Generalizations of Hermite-Hadamard like inequalities involving χ κ -Hilfer fractional integrals, Adv. Difference Equ. 2020 (2020), 594.

9. [9] C. Derbazi, Z. Baitiche and M. Benchohra, Cauchy problem with ψ-Caputo fractional derivative in Banach spaces, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 349-360.
10. [10] R. Diaz and C. Teruel, q,k-Generalized gamma and beta functions, J. Nonlinear Math. Phys. 12 (2005), 118-134.
11. [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
12. [12] J.P. Kharade and K.D. Kucche, On the impulsive implicit ψ-Hilfer fractional differential equations with delay, Math. Meth. Appl. Sci. 43 (2020), 1938-1952.
13. [13] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Mathematics Studies, Amsterdam, 2006.
14. [14] S. Krim, S. Abbas, M. Benchohra, and E. Karapinar, Terminal value problem for implicit Katugampola fractional differ- ential equations in b-metric spaces, J. Funct. Spaces. Volume 2021, Article ID 5535178, 7 pages.
15. [15] J.E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Math. 19 (1) (2021), 363-372.
16. [16] K. Liu, J. Wang and D. O'Regan, Ulam-Hyers-Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equa- tions, Adv Difference Equ. 2019 (2019), 50.
17. [17] S. Mubeen and G. M. Habibullah, k-Fractional Integrals and Application, Int. J. Contemp. Math. Sci. 7 (2012), 89-94.
18. [18] J.E. Nápoles Valdés, Generalized fractional Hilfer integral and derivative, Contr. Math. 2 (2020), 55-60.
19. [19] S. Rashid, M. Aslam Noor, K. Inayat Noor and Y.M. Chu, Ostrowski type inequalities in the sense of generalized K- fractional integral operator for exponentially convex functions, AIMS Math. 5 (2020), 2629-2645.
20. [20] A. Salim, M. Benchohra, J.R. Graef and J.E. Lazreg, Boundary value problem for fractional generalized Hilfer-type fractional derivative with non-instantaneous impulses, Fractal Fract. 5 (2021), 1-21. https://dx.doi.org/10.3390/fractalfract5010001
21. [21] A. Salim, M. Benchohra, E. Karapinar and J.E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Di?er. Equ. 2020 (2020), 21 pp. https://doi.org/10.1186/s13662-020-03063-4
22. [22] A. Salim, M. Benchohra, J.E. Lazreg and J. Henderson, Nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses in Banach spaces, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 332-348. https://doi.org/10.31197/atnaa.825294
23. [23] A. Salim, M. Benchohra , J.E. Lazreg and G. N'Guérékata, Boundary Value Problem for Nonlinear Implicit General- ized Hilfer-Type Fractional Differential Equations with Impulses, Abstract and Applied Analysis. 2021 (2021), 17pp. https://doi.org/10.1155/2021/5592010
24. [24] A. Salim, M. Benchohra, J.E. Lazreg, J.J. Nieto and Y. Zhou, Nonlocal Initial Value Problem for Hybrid Generalized Hilfer- type Fractional Implicit Differential Equations, Nonauton. Dyn. Syst. 8 (2021), 87-100. https://doi.org/10.1515/msds-2020- 0127
25. [25] J.V.C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, Di?er. Equ. Appl. 11 (2019), 87-106.
26. [26] J.V.C. Sousa and E. Capelas de Oliveira, Fractional order pseudo-parabolic partial differential equation: Ulam-Hyers stability, Bull. Braz. Math. Soc. 50 (2019), 481-496.
27. [27] J.V.C. Sousa and E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72-91.