Research article

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

  • Received: 04 November 2019 Accepted: 10 February 2020 Published: 19 March 2020
  • MSC : 26A33, 34B15

  • 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.

    Citation: Shuqin Zhang, Lei Hu. The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order[J]. AIMS Mathematics, 2020, 5(4): 2923-2943. doi: 10.3934/math.2020189

    Related Papers:

  • 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.


    加载中


    [1] S. G. Samko, Fractional integration and differentiation of variable order, Anal. Math., 21 (1995), 213-236. doi: 10.1007/BF01911126
    [2] S. G. Samko, B. Boss, Integration and differentiation to a variable fractional order, Integr. Transforms Spec. Funct., 1 (1993), 277-300. doi: 10.1080/10652469308819027
    [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. doi: 10.1016/j.aml.2017.08.020
    [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. doi: 10.1140/epjst/e2011-01390-6
    [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. doi: 10.1016/j.mcm.2011.09.034
    [8] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043
    [9] X. Li, B. Wu, A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., 43 (2015), 108-113. doi: 10.1016/j.aml.2014.12.012
    [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. doi: 10.1016/j.cnsns.2015.10.027
    [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. doi: 10.1016/j.aml.2016.08.018
    [12] Y. Kian, E. Sorsi, M. Yamamoto, On time-fractional diffusion equations with space-dependent variable order, Ann. Henri Poincaré, 19 (2018), 3855-3881. doi: 10.1007/s00023-018-0734-y
    [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. doi: 10.1016/j.cam.2018.08.035
    [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: http://arXiv.org/abs/0903.2524v1.
    [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. doi: 10.1016/j.na.2011.12.004
    [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. doi: 10.1016/S0252-9602(13)60044-2
    [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. doi: 10.1016/j.jmaa.2013.11.025
    [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. doi: 10.1016/j.cam.2016.03.035
  • Reader Comments
  • © 2020 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3857) PDF downloads(380) Cited by(22)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog