Finite-time stability of mild solutions for Caputo-Katugampola fractional differential equations

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Abstract

This paper focuses on studying the existence and robust finite-time stability (FTS) for a more generalized class of nonlinear Caputo-Katugampola fractional differential equations with perturbation. The existence of mild solutions is proved by using the Sadovskii fixed-point theorem (SFPT). Then, sufficient conditions for robust FTS are obtained by applying the Jensen inequality and Hölder inequality. Finally, our theoretical results are supported by the research on the Lasota- Wazewska red blood cell (LWRBC) model.

Keywords

Caputo-Katugampola fractional differential equation / finite-time stability / Sadovskii fixed-point theorem / Jensen inequality / Grönwall inequality

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References

[1] L. Plociniczak, H. Okrasińska, Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative, Physica D Nonlinear Phenom., 261 (2013), 85-91. https://doi.org/10.1016/j.physd.2013.07.006

[2] A. M. Nagy, Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method, Appl. Math. Comput., 310 (2017), 139-148. https://doi.org/10.1016/j.amc.2017.04.021

[3] A. M. Nagy, N. H. Sweilam, A. A. El-Sayed, New operational matrix for solving multiterm variable order fractional differential equations, J. Comput. Nonlinear Dyn., 13 (2018), 011001. https://doi.org/10.1115/1.4037922

[4] H. Wang, C. Weng, Z. Song, J. Cai, Research on the law of spatial fractional calculus diffusion equation in the evolution of chaotic economic system, Chaos, Solitons Fractals, 131 (2020), 109462. https://doi.org/10.1016/j.chaos.2019.109462

[5] Y. Guan, Z. Cui, W. Zhou, Reconstruction in off-axis digital holography based on hybrid clustering and the fractional Fourier transform, Opt. Laser Technol., 186 (2025), 112622. https://doi.org/10.1016/j.optlastec.2025.112622

[6] A. Ben Makhlouf, O. Naifar, On the Barbalat lemma extension for the generalized conformable fractional integrals: Application to adaptive observer design, Asian J. Control, 25 (2023), 563-569. https://doi.org/10.1002/asjc.2797

[7] Z. Yakoub, O. Naifar, D. Ivanov, Unbiased identification of fractional order system with unknown time-delay using bias compensation method, Mathematics, 10 (2022), 3028. https://doi.org/10.3390/math10163028

[8] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999.

[9] A. M. Yang, Y. Han, J. Li, W. X. Liu, On steady heat flow problem involving yang-srivastava-machado fractional derivative without singular kernel, Therm. Sci., 20 (2016), 717-721. https://doi.org/10.2298/TSCI16S3717Y

[10] Y. N. Wang, J. N. Zhou, Y. K. Li, Fractional sobolev’s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales, Adv. Math. Phys., 2016 (2016), 9636491. https://doi.org/10.1155/2016/9636491

[11] J. F. Jiang, H. T. Chen, J. L. G. Guirao, D. Q. Cao, Existence of the solution and stability for a class of variable fractional order differential systems, Chaos, Solitons Fractals, 128 (2019), 269-274. https://doi.org/10.1016/j.chaos.2019.07.052

[12] C. Anusha, C. Ravichandran, F. Aljuaydi, K. S. Nisar, D. Chalishajar, Stability and solutions of adjoint nonlinear impulsive neutral mixed integral equations in dynamic systems, J. Appl. Math. Comput., 71 (2025), 7521-7541. https://doi.org/10.1007/s12190-025-02579-w

[13] D. Chalishajar, D. Kasinathan, R. Kasinathan, R. Kasinathan, Viscoelastic Kelvin-Voigt model on Ulam-Hyer’s stability and T-controllability for a coupled integro fractional stochastic systems with integral boundary conditions via integral contractors, Chaos, Solitons Fractals, 191 (2025), 115785. https://doi.org/10.1016/j.chaos.2024.115785

[14] D. Chalishajar, D. Kasinathan, R. Kasinathan, R. Kasinathan, Ulam-Hyers-Rassias stability of Hilfer fractional stochastic impulsive differential equations with non-local condition via Time-changed Brownian motion followed by the currency options pricing model, Chaos, Solitons Fractals, 197 (2025), 116468. https://doi.org/10.1016/j.chaos.2025.116468

[15] X. L. Fu, X. B. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267. https://doi.org/10.1016/j.jmaa.2006.01.048

[16] C. Li, W. Wang, H. Zeng, X. Zhang, Novel stability and stabilization criteria for T-S fuzzy systems with time-varying delay based on fuzzy line integral, Commun. Nonlinear Sci., 156 (2026), 109663. https://doi.org/10.1016/j.cnsns.2026.109663

[17] F. F. Du, J. G. Lu, New criterion for finite-time stability of fractional delay systems, Appl. Math. Lett., 104 (2020), 106248. https://doi.org/10.1016/j.aml.2020.106248

[18] N. Cheng, W. Wang, H. B. Zeng, X. Liu, X. M. Zhang, Novel exponential-weighted integral inequality for exponential stability analysis of time-varying delay systems, Appl. Math. Lett., 172 (2026), 109730. https://doi.org/10.1016/j.aml.2025.109730

[19] Q. Lu, X. Wu, J. She, F. Guo, L. Yu, Disturbance rejection for systems with uncertainties based on fixed-time equivalent-input-disturbance approach, IEEE-CAA J. Automatic., 11 (2024), 2384-2395. https://doi.org/10.1109/JAS.2024.124650

[20] Y. K. Li, Y. L. Wang, B. Li, Existence and finite-time stability of a unique almost periodic positive solution for fractional-order Lasota-Wazewska red blood cell models, Int. J. Biomath., 13 (2020), 2050013. https://doi.org/10.1142/S1793524520500138

[21] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. https://doi.org/10.1016/j.amc.2011.03.062

[22] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.

[23] R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dyn., 11 (2016), 061017. https://doi.org/10.1115/1.4034432

[24] S. D. Zeng, D. Baleanu, Y. R. Bai, G. C. Wu, Fractional differential equations of Caputo-Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554. https://doi.org/10.1016/j.amc.2017.07.003

[25] Y. Adjabi, F. Jarad, T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457-5473. https://doi.org/10.2298/FIL1717457A

[26] M. X. Dai, Y. R. Jiang, J. S. Du, G. J. Tang, Finite time stability of Caputo-Katugampola fractional order time delay projection neural networks, Neural Process. Lett., 54 (2022), 4851-4867. https://doi.org/10.1007/s11063-022-10838-1

[27] L. Cao, H. Kong, S. D. Zeng, Maximum principles for time-fractional Caputo-Katugampola diffusion equations, J. Nonlinear Sci. Appl., 10 (2017), 2257-2267. https://doi.org/10.22436/jnsa.010.04.75

[28] A. Jmal, A. Ben Makhlouf, A. M. Nagy, O. Naifar, Finite-time stability for Caputo-Katugampola fractional-order time-delayed neural networks, Neural Process. Lett., 50 (2019), 607-621. https://doi.org/10.1007/s11063-019-10060-6

[29] A. B. Makhlouf, A. M. Nagy, Finite-time stability of linear Caputo-Katugampola fractional order time delay systems, Asian. J. Control., 22 (2020), 297-306. https://doi.org/10.1002/asjc.1880

[30] B. N. Sadovskii, A fixed-point principle, Funct. Anal. Appl., 1 (1967), 151-153. https://doi.org/10.1007/BF01076087

[31] F. F. Du, B. G. Jia, Finite-time stability of a class of nonlinear fractional delay difference systems, Appl. Math. Lett., 98 (2019), 233-239. https://doi.org/10.1016/j.aml.2019.06.017

[32] S. Ferraoun, Z. Dahmani, Gronwall type inequalities: New fractional integral results with some applications on hybrid differential equations, Int. J. Nonlinear Anal. Appl., 12 (2021), 799-809. https://doi.org/10.22075/ijnaa.2020.19595.2089

[33] C. Corduneanu, Principles of Differential and Integral Equations , Allyn & Bacon, Boston, 1971.

[34] T. T. H. Nguyen, N. T. Nguyen, M. N. Tran, Global fractional Halanay inequalities approach to finite-time stability of nonlinear fractional order delay systems, J. Math. Anal. Appl., 525 (2023), 127145. https://doi.org/10.1016/j.jmaa.2023.127145

[35] J. F. Tian, Properties of generalized hölder’s inequalities, J. Math. Inequal., 9 (2015), 473-480. https://doi.org/10.7153/jmi-09-40