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AIMS Mathematics

2026, Volume 11, Issue 3: 8202-8224. doi: 10.3934/math.2026337
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Research article

Lyapunov type inequalities for coupled fractional differential equations under multi-point boundary conditions

  • College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
  • Published: 27 March 2026

  • MSC : 34A08, 34B15, 34B27

  • In this paper, we derive Lyapunov type inequalities for a coupled system of Caputo fractional differential equations with boundary values. By constructing the associated Green's function and discussing its relevant properties, we provide two distinct proofs based on matrix spectral analysis and Perov's fixed point theorem, respectively. These analyses derive explicit necessary conditions on the coefficient functions for the existence of nontrivial solutions, extending the classical Lyapunov inequality theory to a coupled system framework involving differential and boundary coupling. The findings of this study are therefore valuable, offering new perspectives that enrich the literature.

    Citation: Shuangqiao Chen, Zhanbing Bai, Suiming Shang. Lyapunov type inequalities for coupled fractional differential equations under multi-point boundary conditions[J]. AIMS Mathematics, 2026, 11(3): 8202-8224. doi: 10.3934/math.2026337

    Related Papers:

  • In this paper, we derive Lyapunov type inequalities for a coupled system of Caputo fractional differential equations with boundary values. By constructing the associated Green's function and discussing its relevant properties, we provide two distinct proofs based on matrix spectral analysis and Perov's fixed point theorem, respectively. These analyses derive explicit necessary conditions on the coefficient functions for the existence of nontrivial solutions, extending the classical Lyapunov inequality theory to a coupled system framework involving differential and boundary coupling. The findings of this study are therefore valuable, offering new perspectives that enrich the literature.



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    [1] A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse., 2 (1907), 27–247.
    [2] R. P. Agarwal, M. Bohner, A. Ozbekler, Lyapunov Inequalities and Applications, Berlin: Springer, 2021. https://doi.org/10.1007/978-3-030-69029-8
    [3] P. L. De Napoli, J. P. Pinasco, Lyapunov type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995–2018. https://doi.org/10.1016/j.jfa.2016.01.006 doi: 10.1016/j.jfa.2016.01.006
    [4] P. Hartman, A. Wintner, On an oscillation criterion of Liapounoff, Amer. J. Math., 73 (1951), 885–890. https://doi.org/10.2307/2372122 doi: 10.2307/2372122
    [5] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), 978–984. https://doi.org/10.2478/s13540-013-0060-5 doi: 10.2478/s13540-013-0060-5
    [6] R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058–1063. https://doi.org/10.1016/j.jmaa.2013.11.025 doi: 10.1016/j.jmaa.2013.11.025
    [7] X. Meng, M. Stynes, Green's functions, positive solutions, and a Lyapunov inequality for a Caputo fractional-derivative boundary value problem, Fract. Calc. Appl. Anal., 22 (2019), 750–766. https://doi.org/10.1515/fca-2019-0041 doi: 10.1515/fca-2019-0041
    [8] W. Zhang, J. Zhang, J. Ni, Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative, J. Inequal. Appl., 2022 (2022), 1–14. https://doi.org/10.1186/s13660-022-02783-3 doi: 10.1186/s13660-022-02783-3
    [9] Y. Wang, Y. Wu, Z. Cao, Lyapunov-type inequalities for differential equation with Caputo-Hadamard fractional derivative under multi-point boundary conditions, J. Inequal. Appl., 2021 (2021), 1–12. https://doi.org/10.1186/s13660-021-02610-1 doi: 10.1186/s13660-021-02610-1
    [10] N. M. Dien, J. J. Nieto, Lyapunov-type inequalities for a nonlinear sequential fractional BVP in the frame of generalized Hilfer derivatives, Math. Inequal. Appl., 25 (2022), 851–867. https://doi.org/10.7153/mia-2022-25-54 doi: 10.7153/mia-2022-25-54
    [11] W. Zhang, J. Zhang, J. Ni, New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative, AIMS Math., 7 (2022), 1074–1094. https://doi.org/10.3934/math.2022064 doi: 10.3934/math.2022064
    [12] B. Lupińska, E. Schmeidel, Analysis of some Katugampola fractional differential equations with fractional boundary conditions, Math. Biosci. Eng., 18 (2021), 7269–7279. https://doi.org/10.3934/mbe.2021359 doi: 10.3934/mbe.2021359
    [13] B. Lupińska, Existence and nonexistence results for fractional mixed boundary value problems via a Lyapunov-type inequality, Period. Math. Hung., 88 (2024), 118–126. https://doi.org/10.1007/s10998-023-00542-5 doi: 10.1007/s10998-023-00542-5
    [14] S. Toprakseven, A Lyapunov-type inequality for a class of higher-order fractional boundary value problems, J. Math. Inequal., 17 (2023), 435–445. https://doi.org/10.7153/jmi-2023-17-29 doi: 10.7153/jmi-2023-17-29
    [15] L. Wang, H. Dong, G. Chen, New Lyapunov-type inequalities for fractional differential equations with bi-ordinal Psi-Hilfer fractional derivative involving multi-point boundary conditions, Eng. Lett., 31 (2023), 602–609. https://www.engineeringletters.com/issues-v31/issue-2/EL-31-2-20.pdf
    [16] S. N. Srivastava, S. Pati, S. Padhi, A. Domoshnitsky, Lyapunov inequality for a Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions, Math. Methods Appl. Sci., 46 (2023), 13110–13123. https://doi.org/10.1002/mma.9238 doi: 10.1002/mma.9238
    [17] S. T. M. Thabet, I. Kedim, An investigation of a new Lyapunov-type inequality for Katugampola-Hilfer fractional BVP with nonlocal and integral boundary conditions, J. Inequal. Appl., 2023 (2023), 1–22. https://doi.org/10.1186/s13660-023-03070-5 doi: 10.1186/s13660-023-03070-5
    [18] L. Ma, G. Yang, Hadamard type inequalities via fractional calculus in the space of exp-convex functions and applications, Electron. J. Differ. Equ., 2021 (2021), 1–18. https://doi.org/10.58997/ejde.2021.33 doi: 10.58997/ejde.2021.33
    [19] D. Breaz, Ç. Yildiz, L. I. Cotîrlă, G. Rahman, B. Yergöz, New Hadamard type inequalities for modified h-convex functions, Fractal. Fract., 7 (2023), 1–12. https://doi.org/10.3390/fractalfract7030216 doi: 10.3390/fractalfract7030216
    [20] A. Lakhdari, W. Saleh, Multiplicative fractional Hermite–Hadamard-type inequalities in G-calculus, Mathematics., 13 (2025), 1–24. https://doi.org/10.3390/math13213426 doi: 10.3390/math13213426
    [21] B. Benaissa, N. Azzouz, H. Budak, Hermite-Hadamard type inequalities for new conditions on h-convex functions via $\psi$-Hilfer integral operators, Anal. Math. Phys., 14 (2024), 1–20. https://doi.org/10.1007/s13324-024-00893-3 doi: 10.1007/s13324-024-00893-3
    [22] O. Abdelhamid, J. R. Graef, A. Ouahab, Existence and uniqueness of solutions of nonlinear fractional stochastic differential systems with nonlocal functional boundary conditions, Stoch. Anal. Appl., 41 (2023), 713–733. https://doi.org/10.1080/07362994.2022.2078839 doi: 10.1080/07362994.2022.2078839
    [23] F. Erkan, N. A. Hamal, S. K. Ntouyas, J. Tariboon, Existence and uniqueness analysis for (k, $\psi$)-Hilfer and (k, $\psi$)-Caputo sequential fractional differential equations and inclusions with non-separated boundary conditions, Fractal. Fract., 9 (2025), 1–26. https://doi.org/10.3390/fractalfract9070437 doi: 10.3390/fractalfract9070437
    [24] K. Diethelm, S. Hashemishahraki, H. D. Thai, H. T. Tuan, A constructive approach for investigating the stability of incommensurate fractional differential systems, J. Math. Anal. Appl., 540 (2024), 1–25. https://doi.org/10.1016/j.jmaa.2024.128642 doi: 10.1016/j.jmaa.2024.128642
    [25] L. Ma, Y. Shu, Center manifold reduction for Caputo–Hadamard fractional differential system, J. Comput. Nonlinear Dyn., 20 (2025), 1–11. https://doi.org/10.1115/1.4068728 doi: 10.1115/1.4068728
    [26] M. Jleli, D. O'Regan, B. Samet, Lyapunov-type inequalities for coupled systems of nonlinear fractional differential equations via a fixed point approach, J. Fix. Point. Theory. A., 21 (2019), 21–45. https://doi.org/10.1007/s11784-019-0683-1 doi: 10.1007/s11784-019-0683-1
    [27] Y. Zou, Y. Cui, Lyapunov-type inequalities for systems of Riemann-Liouville fractional differential equations with multi-point coupled boundary conditions, Fractal. Fract., 7 (2023), 1–15. https://doi.org/10.3390/fractalfract7060454 doi: 10.3390/fractalfract7060454
    [28] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006. https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204
    [29] D. Ma, Z. Yang, Lyapunov-type inequality and solution for a fractional differential equation, J. Inequal. Appl., 2020 (2020), 1–18. https://doi.org/10.1186/s13660-020-02448-z doi: 10.1186/s13660-020-02448-z
    [30] G. Debreu, I. N. Herstein, Nonnegative square matrices, Econometrica, 21 (1953), 597–607. https://doi.org/10.2307/1907925 doi: 10.2307/1907925
    [31] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Priblizhen. Met. Reshen. Differ. Uravn., 2 (1964), 115–134.
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  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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