Generalized Caputo fractional differential equations with nonlocal conditions

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Abstract

This paper investigates the existence and uniqueness of solutions for fractional differential equations with nonlocal conditions. The results are derived using measure theory, Monch’s fixed point theorem, and semigroup theory. An illustrative example is also presented, involving a nonlocal fractional evolution problem in $L^{2}((0, \pi), \mathbb{R})$. This example demonstrates the effect of the ψ-Caputo derivative on the mild solution in comparison with the standard Caputo derivative, highlighting the influence of non-uniform time scaling and memory effects.

Keywords

semigroup / Mönch’s fixed point theorem / ψ-fractional derivative / evolution equations

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References

[1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations , North-Holland Mathematics Studies, Elsevier, 2006.

[2] I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications , Elsevier, 1998.

[3] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations , John Wiley, New York, 1993.

[4] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. https://doi.org/10.1016/j.camwa.2009.06.026

[5] C. Derbazi, H. Hammouche, Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions, Arab. J. Math., 9 (2020), 531-544. https://doi.org/10.1007/s40065-020-00288-9

[6] Z. Cui, Z. Zhou, Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions, Fixed Point Theory Algorithms Sci. Eng., 2023 (2023). https://doi.org/10.1186/s13663-022-00738-3

[7] Y. H. Liang, K. J. Wang, Dynamics of the new exact wave solutions to the local fractional Vakhnenko-Parkes equation, Fractals, 34 (2026), 2550102. https://doi.org/10.1142/S0218348X25501026

[8] K. J. Wang, The fractal active low-pass filter within the local fractional derivative on the Cantor set, COMPEL, 42 (2023), 1396-1407. https://doi.org/10.1108/COMPEL-09-2022-0326

[9] K. Diethelm, A. D. Freed, On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity, in Scientific Computing in Chemical Engineering II, Springer, Berlin, Heidelberg, (1999), 217-224.

[10] M. Alqhtani, L. Sadek, K. M. Saad, The Mittag-Leffler-Caputo-Fabrizio fractional derivative and its numerical approach, Symmetry, 17 (2025), 800. https://doi.org/10.3390/sym17050800

[11] L. Sadek, A. S. Bataineh, E. M. Sadek, I. Hashim, A general definition of the fractal derivative: Theory and applications, AIMS Math., 10 (2025), 15390-15409. https://doi.org/10.3934/math.2025690

[12] L. Sadek, F. Jarad, The general Caputo-Katugampola fractional derivative and numerical approach for solving the fractional differential equations, Alexandria Eng. J., 121 (2025), 539-557. https://doi.org/10.1016/j.aej.2025.02.065

[13] W. G. Glöckle, T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68 (1995), 46-53. https://doi.org/10.1016/S0006-3495(95)80157-8

[14] Y. Zhou, Basic Theory of Fractional Differential Equations , World Scientific, 2023. https://doi.org/10.1142/13289

[15] Y. Zhou, J. W. He, New results on controllability of fractional evolution systems with order α ∈ (1, 2), Evol. Equations Control Theory, 10 (2021), 491-509. https://doi.org/10.3934/eect.2020077

[16] 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 (2018), 336-352. https://doi.org/10.1002/mma.4624

[17] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst. Ser. S, 13 (2019), 709-722. https://doi.org/10.3934/dcdss.2019040

[18] F. Mainardi, P. Paradisi, R. Gorenflo, Probability distributions generated by fractional diffusion equations, preprint, arXiv:0704.0320. https://doi.org/10.48550/arXiv.0704.0320

[19] J. Banas, Measure of noncompactness in Banach space, Comment. Math. Univ. Carolin., 21 (1980), 131-143.

[20] A. A. Elaiw, M. Manigandan, M. Awadalla, K. Abuasbeh, Mönch’s fixed point theorem in investigating the existence of a solution to a system of sequential fractional differential equations, AIMS Math., 8 (2023), 2591-2610. https://doi.org/10.3934/math.2023133