Existence and ulam stability of k-generalized ψ-Hilfer fractional problem

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Jamal Eddine Lazreg
Mouffak Benchohra
Abdelkrim Salim
https://orcid.org/0000-0003-2795-6224

Abstract

In this paper, we prove existence, uniqueness stability results for a class of initial value problem for fractional differential equations involving generalized ψ-Hilfer fractional derivative. The result is based on the Banach contraction mapping principle. In addition, two examples are given to illustrate our results.

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How to Cite
[1]
Lazreg, J.E., Benchohra, M. and Salim, A. 2022. Existence and ulam stability of k-generalized ψ-Hilfer fractional problem . Journal of Innovative Applied Mathematics and Computational Sciences. 2, 2 (May 2022), 1–13.
Section
Conference paper (ICMA'2021)

References

S. Abbas, M. Benchohra, J. R. Graef and J. Henderson, Implicit Differential and Integral Equations: Existence and stability, Walter de Gruyter, London, 2018.

S. Abbas, M. Benchohra and G. M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.

S. Abbas, M. Benchohra and G. M. N’Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.

B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham, 2017.

Y. M. Chu, M. U. Awan, S. Talib, M. A. Noor and K. I. Noor, Generalizations of Hermite-Hadamard like inequalities involving ck-Hilfer fractional integrals, Adv. Difference Equ. 2020:594 (2020), 1–15.

R. Diaz and C. Teruel, q, k-Generalized gamma and beta functions, J. Nonlinear Math. Phys 12 (2005), 118–134.

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

A. A. Kilbas, H. M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Amsterdam, 2006.

K. D. Kucche, A. D. Mali, On the nonlinear (k, ψ)-Hilfer fractional differential equations, Chaos Solitons Fractals, 152:111335 (2021) 1–14.

J. E. Lazreg, S. Abbas, M. Benchohra and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Math. 19 (2021), 363-372.

S. Mubeen and G. M. Habibullah, k-Fractional Integrals and Application, Int. J. Contemp. Math. Sciences, 7 (2012), 89-94.

S. Naz and M. N. Naeem, On the Generalization of k-Fractional Hilfer-Katugampola Derivative with Cauchy Problem, Turk. J. Math. 45 (2021), 110-124.

S. Rashid, M. Aslam Noor, K. Inayat Noor, Y. M. Chu,Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Mathematics 5(3) (2020), 2629–2645.

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(1) (2021), 1–21.

A. Salim, M. Benchohra, J. R. Graef and J. E. Lazreg, Initial value problem for hybrid ψ-Hilfer fractional implicit differential equations, J. Fixed Point Theory Appl. 24:7 (2022), 1–14.

A. Salim, M. Benchohra, E. Karapinar and J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Differ. Equ. 2020:601 (2020), 1–21.

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(4) (2020), 332-348.

A. Salim, M. Benchohra, J. E. Lazreg and J. Henderson, On k-Generalized ψ-Hilfer Boundary Value Problems with Retardation and Anticipation, Adv. Theory Nonlinear Anal. Appl. 6(2) (2022), 173-190.

A. Salim, M. Benchohra, J. E. Lazreg and E. Karapinar, On k-Generalized ψ-Hilfer Impulsive Boundary Value Problem with Retarded and Advanced Arguments, J. Math. Ext. 15 (2021), 1–39.

A. Salim, M. Benchohra, J. E. Lazreg and G. N’Guérékata, Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses, Abstr. Appl. Anal. 2021:5592010 (2021), 1–17.

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.

A. Salim, J. E. Lazreg, B. Ahmad, M. Benchohra and J. J. Nieto, A Study on k-Generalized ψ-Hilfer Derivative Operator, (2021) accepted.

J. V. da C. Sousa, G. S. F. Frederico and E. C. de Oliveira, ψ-Hilfer pseudo-fractional operator: new results about fractional calculus, Comput. Appl. Math. 39(4):254 (2020), 1–37.

J. V. da C. Sousa and E. C. de Oliveira, On the y-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72-91.

J. V. da C. Sousa, M. A. P. Pulido and E. C. de Oliveira, Existence and Regularity of Weak Solutions for ψ-Hilfer Fractional Boundary Value Problem, Mediterr. J. Math. 18:147 (2021), 1–15.

Y. Zhou, J. R. Wang and L. Zhang Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2017.