Existence and stability of a nonlinear fractional differential equation involving a $\psi$-Caputo operator

Abstract

This paper is devoted to the study of the existence and interval of existence, uniqueness of solutions and estimates on solutions of the nonlocal Cauchy problem for nonlinear fractional differential equations involving a Caputo type fractional derivative with respect to another function $\psi$. Further, we prove four different types of Ulam stability results of solutions for a given problem. The tools used in this article are the classical technique of Banach fixed point theorem and generalized Gronwall inequality. At the end, illustrative examples are presented.

Keywords

• fractional differential equations
• Existence and stability
• Fixed point theorem
• ψ-fractional derivative and ψ-fractional integral

Kaynakça

1. [1] M.S. Abdo and S.K. Panchal, Fractional integro-differential equations involving ψ-Hilfer fractional derivative, Adv. Appl. Math. Mech., 11(2), (2019), 338-359.
2. [2] M.S. Abdo A.G. Ibrahim and S.K. Panchal, Nonlinear implicit fractional differential equation involving ψ-Caputo fractional derivative, Proc. Jangjeon Math. Soc. (PJMS), 22(3), (2019), 387-400.
3. [3] M.S. Abdo and S.K. Panchal, Fractional Boundary value problem with ψ-Caputo fractional derivative, Proc. Indian Acad. Sci. (Math.Sci.), 129(5), (2019), 65.
4. [4] M.S. Abdo and S.K. Panchal, Fractional integro-differential equations with nonlocal conditions and ψ- Hilfer fractional derivative, Mathematical Modelling and Analysis, 24(4), (2019), 564-584.
5. [5] M.S. Abdo, H.A. Wahash, S.K. Panchal, Ulam-Hyers-Mittag-Leffler stability for a ψ -Hilfer problem with fractional order and infnite delay, Results in Applied Mathematics, 7, (2020), 100-115.
6. [6] O.P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15, (2012), 700-711. [7] A. Ali, K. Shah and F. Jarad, Ulam-Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions, Adv. Diff. Equ., 2019(1),(2019), 1-7.
7. [8] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44, (2017), 460-481.
8. [9] R. Almeida, A.B. Malinowska and M.T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Method Appl. Sci., 41, (2018), 336-352.

9. [10] A. Granas, J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003.
10. [11] R.W. Ibrahim, Generalized Ulam-Hyers stability for fractional di?erential equations, Int. J. Math., 23(5), (2012), 1-9
11. [12] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete & Continuous Dynamical Systems-S, 13(3), (2020), 709.
12. [13] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Math. Stud, 204 Elsevier, Amsterdam, 2006.
13. [14] K.S. Miller, and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley: New York, NY, USA, 1993.
14. [15] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego, CA, USA, 1999.
15. [16] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
16. [17] K. Shah, A. Ali, S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Methods Appl. Sci., 41, (2018), 8329-8343.
17. [18] J.V.C. da Sousa, E.C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator, J. Fixed Point Theory Appl. 20(3), (2018), 96.
18. [19] J.V.C. da Sousa and E.C. de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60, (2018), 72-91.
19. [20] J.V.C. da Sousa and E.C. de Oliveira, a Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, Diff. Equ. Appl., 11, (2019), 87-106.
20. [21] H.A. Wahash, M.S. Abdo, S.K. Panchal, Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative, Journal of Applied Mathematics and Computational Mechanics, 19(1), (2020) 89-101.
21. [22] H.A. Wahash, M.S. Abdo, S.K. Panchal, Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative, Ufa Mathematical Journal, 11( 1), (2019), 3-21.
22. [23] J. Wang, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17, (2012), 2530-2538.