A new sequential proportional fractional derivative of hybrid di?erential equations with nonlocal hybrid condition

Year 2023, Volume: 7 Issue: 1, 148 - 161, 31.03.2023

Hamıd Beddanı , Beddani Moustafa , Saada Hamouda

https://doi.org/10.31197/atnaa.1122002

Abstract

In this paper, we study the existence of solutions for a new problem of hybrid differential
equations with nonlocal integro multi point boundary conditions by using the proportional fractional
derivative. The presented results are obtained by using hybrid fixed point theorems for three Dhage
operators. The application of theoretical conclusions is demonstrated through an example.

Keywords

Hybrid fixed point theorems;, Proportional fractional derivative ;, Existence of solution.

References

• [1] M.I. Abbas, M.A. Ragusa, On the Hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry 2021, 13, 264. https://doi.org/10.3390/sym13020264
• [2] M.I. Abbas, Existence results and the Ulam stability for fractional differential equations with hybrid proportional-Caputo derivatives. J. Nonlinear Funct. Anal. 2020, 2020, 1-14.
• [3] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit differential and integral equations: existence and stability, Walter de Gruyter, London, 2018.
• [4] S. Abbas, M. Benchohra and G. M. N'Guérékata, Advanced fractional differential and integral equations, Nova Science Publishers, New York, 2014.
• [5] S. Abbas, M. Benchohra and G. M. N'Guérékata, Topics in fractional differential equations, Springer-Verlag, New York, 2012.
• [6] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solutions of fractional differential equations via geraghty type hybrid contractions, Appl. Comput. Math., V.20, N.2, 2021,313-333.
• [7] H. Afshari and E. Karapinar, A solution of the fractional differential equations in the setting of b-metric space. Carpathian Math. Publ. 13 (2021), 764-774. https://doi.org/10.15330/cmp.13.3.764-774.
• [8] H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces, Adv. Di?erence Equ. 2020, 616. https://doi.org/10.1186/s13662-020-03076-z.
• [9] G.A. Anastassiou, Generalized fractional calculus: New advancements and applications, Springer International Publishing, Switzerland, 2021.
• [10] H. Beddani and Z. Dahmani, Solvability for nonlinear differential problem of Langevin type via ϕ-Caputo approch, Eur. J. Math. Appl. (2021) 1:11, DOI: 10.28919/ejma.2021.1.11
• [11] H. Beddani and M. Beddani, Solvability for a differential systems via ϕ-Caputo approach. J. Sci. Arts. 56(3)2021 [12] A. Bharucha-Reid, Random integral equations, Academic Press, New York, 1972.
• [13] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1, 73-85.
• [14] B.C. Dhage, Fixed point theorems in ordered Banach algebras and applications. Panamer. Math. J. 1999, 9, 93-102.
• [15] B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J. 44, 145-155 (2004).
• [16] B.C. Dhage, On a fixed point theorem in banach algebras with applications, Applied Mathematics Letters, 18, (2005) p: 273-280.
• [17] B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, differential eEquations and applications, Vol 2, Number 4 (2010), p: 465-486.
• [18] S. Etemad, S. Rezapour, M. E. Samei, On fractional hybrid and non-hybrid multi-term integrodi?erential inclusions with three-point integral hybrid boundary conditions, Adv. Differ. Equ., 2020 (2020), 161. doi: 10.1186/s13662-020-02627-8.
• [19] S. Ferraoun, and Z. Dahmani, Existence and stability of solutions of a class of hybrid fractional differential equations involv- ing RL-operator. J. Interdisciplinary Math., vol 23 no 4(2020), 885-903. https://doi.org/10.1080/09720502.2020.1727617
• [20] F. Jarad, M.A. Alqudah, T. Abdeljawad, On more general forms of proportional fractional operators. Open Math. 2020, 18, 167-176.
• [21] F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function. Adv. Differ. Equ. 2020, 2020, 303
• [22] A. Keten, M. Yavuz, D. Baleanu, Nonlocal Cauchy problem via a fractional operator involving power kernel in banach spaces. Fractal Fract. 2019, 3, 27.
• [23] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam, 2006.
• [24] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
• [25] S.G. Samko, A. A. Kilbas and O. I. Mariche, Fractional integrals and derivatives, translated from the 1987 Russian original. Yverdon: Gordon and Breach, (1993).
• [26] M. Yavuz, European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels. Numer. Methods Partial. Differ. Equ. 2021, 37.
• [27] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), 1312-1324. doi: 10.1016/j.camwa.2011.03.041.