Existence and Uniqueness of Solutions for Nonlinear Fractional Differential Equations with $\mho$-Caputo Fractional Differential Equations

Year 2024, , 187 - 198, 31.12.2024

Abduljawad Anwar

https://doi.org/10.33434/cams.1556314

Abstract

This paper examines the existence, uniqueness, and Ulam-Hyers stability of solutions to nonlinear $\mho$-fractional differential equations with boundary conditions with a $\mho$-Caputo fractional derivative. The acquired results for the suggested problem are validated using a novel technique and minimum assumptions about the function $f$. The analysis reduces the problem to a similar integral equation and uses Banach and Sadovskii fixed point theorems to reach the desired findings. Finally, the inquiry is demonstrated by illustrative example to validate the theoretical findings.

Keywords

Banach Contraction mapping, Caputo fractional derivative, $\mho$-Caputo fractional derivative, Fixed point theorem, Fractional differential equations, Stability analysis, Ulam-Hyers stability

References

• [1] A. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59(3) (2010), 1095-1100.
• [2] R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci., 41(1) (2018), 336-352.
• [3] R. Agarwal, S. Hristova, D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal., 19(2) (2016), 290-318. https://doi.org/10.1515/fca-2016-0017.
• [4] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [5] L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442.
• [6] A. P. Agarwal, S. K. Ntouyas, B. Ahmad, A. K. Alzahrani, Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, Adv. Differential Equations, 2016(92) (2016), 1-15.
• [7] S. Zhang, Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26(2) (2006), 220-228.
• [8] M. Benchohra, S. Hamani, S. K. Ntouyas, Existence for differential equations with fractional order, Surveys Math. Appl., 3 (2008), 1-12.
• [9] M. Aydin, N. I. Mahmudov, f-Caputo type time-delay Langevin equations with two general fractional orders, Math. Methods Appl. Sci., 46(8) (2023), 9187-9204.
• [10] M. Benchohra, J. E. Lazreg, Existence and Ulam-stability for nonlinear implicit fractional differential equations with Hadamard derivative, Studia Univ. Babes¸-Bolyai Math., 62(1) (2017), 27-38.
• [11] S. Song, Y. Cu, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance, Bound. Value Probl., 2020 (2020), Article 23.
• [12] A. Khan, K. Shah, Y. Li, T. S. Khan, Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations, J. Funct. Spaces, 2017 (2017), Article 3046013.
• [13] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.
• [14] P. Rabinowitz, A Collection of Mathematical Problems, S. M. Ulam, Interscience, New York, 1960.
• [15] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27(4) (1941), 222-224.
• [16] A. K. Anwar, S. A. Murad, Existence and Ulam stability of solutions for Caputo-Hadamard fractional differential equations, General Letters Math., 12(2) (2022), 85-95. https://doi.org/10.31559/glm2022.12.2.5.
• [17] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63 (2011), 1-10.
• [18] M. Khanehgir, R. Allahyari, M. Mursaleen, H. A. Kayvanloo, On infinite systems of Caputo fractional differential inclusions with boundary conditions for convex-compact multivalued mappings, Alexandria Engineering J., 59(5) (2020), 3233-3238.
• [19] P. M. Mohammadi Babak, M. M. Parvanah Vahid, Existence of solutions for some f-Caputo fractional differential inclusions via Wardowski-Mizoguchi-Takahashi multi-valued contractions, Filomat, 37(12) (2023), 3777-3789.
• [20] H. Amiri Kayvanloo, H. Mehravaran, M. Mursaleen, R. Allahyari, A. Allahyari, Solvability of infinite systems of Caputo- Hadamard fractional differential equations in the triple sequence space c3(D), J. Pseudo-Differ. Oper. Appl., 15(2) (2024), Article 26.
• [21] B. Mohammadi, V. Parvaneh, M. Mursaleen, Existence of solutions for some nonlinear g-Caputo fractional-order differential equations based on Wardowski-Mizoguchi-Takahashi contractions, J. Inequal. Appl., 2024 (1) (2024), Article 105.
• [22] Q. Dai, R. Gao, Z. Li, et al., Stability of Ulam-Hyers and Ulam-Hyers-Rassias for a class of fractional differential equations, Adv. Differential Equations, 2020 (2020), Article ID 103.
• [23] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Boston, 2006.
• [24] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional integro-differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63 (2011), 1-10.