Article ID: MTJPAM-D-24-00053

Title: An investigation on a certain type of psi-Caputo fractional differential equations with deviating arguments

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-24-00053; Volume 6 / Issue 3 / Year 2024, Pages 339-349

Document Type: Research Paper

Author(s): ‪Mohammed Dahmane ^a , Mohammed Derhab ^b

^aDynamic Systems and Applications Laboratory, Department of Mathematics, Faculty of Sciences, University Abou-Bekr Belkaid Tlemcen, Algeria

^bDynamic Systems and Applications Laboratory, Department of Mathematics, Faculty of Sciences, University Abou-Bekr Belkaid Tlemcen, Algeria

Received: 16 April 2024, Accepted: 30 November 2024, Published: 2 March 2025

Corresponding Author: Mohammed Derhab (Email address: derhab@yahoo.fr)

Full Text: PDF

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Abstract

Solution uniqueness and existence of a class of psi-Caputo fractional differential equations with deviating arguments is the main subject of this paper. Furthermore, we illustrate the practical use of our result with multiple examples.

Keywords: Psi-Caputo fractional derivative, deviating, Banach’s fixed point theorem, existence, uniqueness

References:

1. R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44, 460–481, 2017.
2. R. Almeida, Functional differential equations involving the ψ-Caputo fractional derivative, Fractal. Fract. 4 , 29, 2020; https://doi.org/10.3390/fractalfract4020029.
3. 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. Methods Appl. Sci. 41 (1), 336–352, 2018.
4. M. Aydin, N. I. Mahmudov, H. Aktuğlu, E. Baytunç and M. S. Atamert, On a study of the representation of solutions of a Ψ-Caputo fractional differential equations with a single delay, Electron. Res. Arch. 30 (3), 1016–1034, 2022.
5. S. Belarbi, Z. Dahmani and M. Sarikaya, A sequential fractional differential problem of pantograph type: Existence uniqueness and illustrations, Turk. J. Math. 46 (SI1), 563–586, 2022.
6. A. Chrysovergis, Some remarks on Talenti’s semigroup, Canad. Math. Bull. 14 (2), 147–150, 1971.
7. C. Derbazi, Z. Baitiche, M. Benchohra and A. Cabada, Initial value problem for nonlinear fractional differential equations with ψ-Caputo derivative via monotone iterative technique, Axioms 9, 2020; Aricle ID: 57, https://doi.org/10.3390/axioms9020057.
8. M. Derhab, On a class of Caputo modified fractional differential equations with advanced arguments, Jordan J. Math. Stat. 17, 493–510, 2024.
9. G. Derfel, P. J. Grabner and R. F. Tichy, On the asymptotic behaviour of the zeros of the solutions of a functional-differential equation with rescaling, In: Operator Theory: Advances and Applications (Ed. by D. Alpay and B. Kirstein), Birkhauser, Cham. (Volume 263), 281–295, 2018; https://doi.org/10.1007/978-3-319-68849-7_10.
10. A. Erdélyi, An integral equation involving Legendre functions, J. Soc. Indust. Appl. Math. 12 (1), 15–30, 1964.
11. L. Fox, D. F. Mayers, J. R. Ockendon and A. B. Tayler, On a functional differential equation, IMA J. Appl. Math. 8 (3), 271–307, 1971.
12. A. A. Kilbas and S. A. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Calc. Appl. Anal. 7 (3), 297-321, 2004.
13. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies (Volume 204), Elsevier Science B.V., Amsterdam, 2006.
14. K. Mahler, On a special functional equation, J. London Math. Soc. s1-15 (2), 115–123, 1940.
15. G. R. Morris, A. Feldstein and E. W. Bowen, The Phragmén–Lindelöf principle and a class of functional differential equations, In: Ordinary Differential Equations (Ed. by L. Weiss), 1971 NRL-MRC Conference, Academic Press, New York, 513–540, 1972.
16. J. J. Nieto, M. Alghanmi, B. Ahmad, A. Alsaedi and B. Alharbi, On fractional integrals and derivatives of a function with respect to another function, Fractals 31, 2023; Article ID: 2340066.
17. J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A 322 (1551), 447–468, 1971.
18. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
19. 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), 173–190, 2022.
20. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Theory and Applications Gordon and Breach, Yverdon, 1993.
21. H. Sebbagh and M. Derhab, The Adomian decomposition method for solving a class of fractional nonhomogeneous multi-pantograph equations with initial conditions, Comm. Appl. Nonlinear Anal. 28 (1), 1–30, 2021.
22. F. A. Shelkovnikov, The generalized Cauchy formula (in Russian), Uspekhi Mat. Nauk 6 (3), 157–159, 1951.
23. V. Spiridonov, Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A 52 (3), 1909–1935, 1995.
24. J. A. Tenreiro Machado, Fractional calculus: Models, algorithms, technology, Discontinuity, Nonlinearity, and Complexity 4 (4), 383–389, 2015.
25. H. A. Wahash, M. S. Abdo and S. K. Panchal, Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative, Ufa Math. J. 11 (4), 151–170, 2019.
26. H. A. Wahash, M. S. Abdo, A. M. Saeed and S. K. Panchal, Singular fractional differential equations with ψ-Caputo operator and modified Picard’s iterative method, Appl. Math. E-Notes 20, 215–229, 2020.

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