International Journal of Nonlinear Analysis and Applications
Nonlinear implicit Caputo-Fabrizio fractional hybrid differential equations
Document Type : Review articles
Author
Department of Mathematics, Higher Normal School of Technological Education- Skikda, Algeria
Abstract
The present paper is mainly concerned with the existence, uniqueness, interval of existence and continuous dependence of solutions on initial conditions for nonlinear implicit Caputo-Fabrizio fractional hybrid differential equations. The results are obtained by using fractional calculus, contraction principle theorem and the Gronwall's inequality to show the estimate of the solutions. An example is given to illustrate the effectiveness of our main results.
Keywords
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[10] A. Boudaoui and A. Slama, On coupled systems of fractional impulsive differential equations by using a new Caputo- Fabrizio fractional derivative, Math. Moravica 24 (2020), no. 2, 1–19.
[11] N. Derdar, Mixed nonlocal boundary value problem for implicit fractional differential equation involving both retarded and advanced arguments, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 2697—2708.
[12] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, Victoria University of Technology, Victoria 8001, Australia, 2002.
[13] Eiman, K. Shah, M. Sarwar and D. Baleanu, Study on Krasnoselskii’s fixed point theorem for Caputo-Fabrizio fractional differential equations, Adv. Differ. Equ. 178 (2020), 9 pages.
[14] A. Granas and J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003.
[15] R. Gul, M. Sarwar, K. Shah and T. Abdeljawad, Qualitative analysis of implicit Dirichlet boundary value problem for Caputo-Fabrizio fractional differential equations, J. Function Spaces 2020 (2020), Article ID 4714032, 9 pages.
[16] R. Hilfer, Applications of Fractional calculus in Physics, World Scientific, Singapore, 2000.
[17] M.A.E. Herzallah, D. Baleanu, On Fractional Order Hybrid Differential Equations, Abstr. Appl. Anal. 2014 (2014), Article ID 389386, 1-8.
[18] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
[19] A. Lachouri, A. Ardjouni and A. Djoudi, Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations, Math. Moravica 24 (2020), no. 1, 109-122.
[20] J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), no. 2, 87-92.
[21] N. Nyamoradi and A. Razani, Existence to fractional critical equation with Hardy-Littlewood-Sobolev nonlinearities, Acta Math. Scientia 41 (2021), 1321–1332.
[22] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[23] I. Podlubny, Fractional Differential Equations, Acadmic press, New York, USA, 1993.
[24] A. Razani and F. Behboudi, Weak solutions for some fractional singular (p, q)-Laplacian nonlocal problems with Hardy potential, Rendi. Circ. Mat. Palermo Series 2 (2022), https://doi.org/10.1007/s12215-022-00768-1
[25] S. Sun, Y. Zhao, Z. Han and Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4961–4967.
[26] S. Toprakseven, The existence and uniqueness of initial-boundary value problems of the fractional Caputo- Fabrizio differential equations, Univer. J. Math. Appl. 2 (2019), no. 2, 100–106.
[27] D. Vivek, O. Baghani and K. Kanagarajan, Existence results for hybrid fractional differential equations with Hilfer fractional derivative, Caspian J. Math. Sci. 9 (2020), no. 2, 294–304.
[28] S. Zhanga, L. Hub and S. Sunc, The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative, J. Nonlinear Sci. Appl. 11 (2018), no. 3, 428–436.
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Volume 14, Issue 1
January 2023Pages 2091-2099
- Receive Date: 05 November 2021
- Revise Date: 23 December 2022
- Accept Date: 29 December 2022