AIMS Mathematics, 2020, 5(4): 2923-2943. doi: 10.3934/math.2020189

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order

1 School of Science, China University of Mining and Technology-Beijing, Ding No.11 Xueyuan Road, Haidian District, Beijing, China
2 School of Science, Shandong Jiaotong University, 5001 Haitang Road, Changqing District, Jinan, Shandong Province, China

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.
  Article Metrics


1. S. G. Samko, Fractional integration and differentiation of variable order, Anal. Math., 21 (1995), 213-236.    

2. S. G. Samko, B. Boss, Integration and differentiation to a variable fractional order, Integr. Transforms Spec. Funct., 1 (1993), 277-300.    

3. D. Valério, J. Sá da Costa, Variable-order fractional derivative and their numerical approximations, Signal Process., 91 (2011), 470-483.    

4. J. Yang, H. Yao, B. Wu, An efficient numberical method for variable order fractional functional differential equation, Appl. Math. Lett., 76 (2018), 221-226.    

5. C. M. Chen, F. Liu, V. Anh, et al. Numberical schemes with high spatial accuracy for a variableorder anomalous subdiffusion equation, SIAM J. Sci. Comput., 32 (2012), 1740-1760.

6. H. Sun, W. Chen, H. Wei, et al. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185-192.    

7. A. Razminia, A. F. Dizaji, V. J. Majd, Solution existence for non-autonomous variable-order fractional differential equations, Math. Comput. Model., 55 (2012), 1106-1117.    

8. A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114.    

9. X. Li, B. Wu, A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., 43 (2015), 108-113.    

10. D. Tavares, R. Almeida, D. F. M. Torres, Caputo derivatives of fractional variable order: Numerical approximations, Commun. Nonlinear Sci. Numer. Simul., 35 (2016), 69-87.    

11. Y. Jia, M. Xu, Y. Z. Lin, A numberical solution for variable order fractional functional differential equations, Appl. Math. Lett., 64 (2017), 125-130.    

12. Y. Kian, E. Sorsi, M. Yamamoto, On time-fractional diffusion equations with space-dependent variable order, Ann. Henri Poincaré, 19 (2018), 3855-3881.    

13. J. Vanterler da C. Sousa, E. Capelas de Oliverira, Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation, Comput. Appl. Math., 37 (2018), 5375-5394.    

14. J. F. Gómez-Aguilar, Analytical and numerical solutions of nonlinear alcoholism model via variable-order fractional differential equations, Phys. A, 494 (2018), 52-57.    

15. W. Malesza, M. Macias, D. Sierociuk, Analysitical solution of fractional variable order differential equations, J. Comput. Appl. Math., 348 (2019), 214-236.    

16. S. Zhang, The uniqueness result of solutions to initial value problem of differential equations of variable-order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 112 (2018), 407-423.

17. S. Umarov, S. Steinber, Variable order differential equations and diffusion processes with changing modes, Available from:

18. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.

19. M. Dreher, A. Jüngel, Compact families of piecewise constant functions in Lp(0, T; B), Nonlinear Anal., 75 (2012), 3072-3077.

20. A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203-474.

21. N. Li, C. Wang, New existence results of positive solution for a class of nonlinear fractional differential equations, Acta Math. Sci., 33 (2013), 847-854.    

22. 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.    

23. J. Rong, C. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Differ. Equations., 2152015 (2015), 1-10.

24. M. Jleli, B. Samet, Lyapunov-type inequalities for fractional boundaryvalue problems, Electron. J. Differ. Equations, 2015 (2015), 1-11.

25. A. Chidouh, D. F. M. Torre, A generalized Lyapunov's inequality for a fractional boundary value problem, J. Comput. Appl. Math., 312 (2017), 192-197.    

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved