Research article Special Issues

The existence of solutions of Hadamard fractional differential equations with integral and discrete boundary conditions on infinite interval

  • Received: 13 January 2024 Revised: 19 February 2024 Accepted: 11 March 2024 Published: 21 March 2024
  • In this article, the properties of solutions of Hadamard fractional differential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids.

    Citation: Jinheng Liu, Kemei Zhang, Xue-Jun Xie. The existence of solutions of Hadamard fractional differential equations with integral and discrete boundary conditions on infinite interval[J]. Electronic Research Archive, 2024, 32(4): 2286-2309. doi: 10.3934/era.2024104

    Related Papers:

  • In this article, the properties of solutions of Hadamard fractional differential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids.



    加载中


    [1] M. Nategh, A novel approach to an impulsive feedback control with and without memory involvement, J. Differ. Equations, 263 (2017), 2661–2671. https://doi.org/10.1016/j.jde.2017.04.008 doi: 10.1016/j.jde.2017.04.008
    [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
    [3] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge: Cambridge Academic Publishers, 2009.
    [4] S. G. Samko, A. A. Kilbas, O. I. Maricev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
    [5] I. Podlubny, Fractional Differential Equations, MDPI, 1999. https://doi.org/10.3390/books978-3-03921-733-5
    [6] X. Q. Zhang, Q. Y. Zhong, Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions, Fract. Calc. Appl. Anal., 20 (2017), 1471–1484. https://doi.org/10.1515/fca-2017-0077 doi: 10.1515/fca-2017-0077
    [7] G. T. Wang, A. Cabada, L. H. Zhang, An integral boundary value problem for nonlinear differential equations of fractional order on an unbounded domain, J. Integr. Equations Appl., 26 (2014), 117–129. https://doi.org/10.1216/jie-2014-26-1-117 doi: 10.1216/jie-2014-26-1-117
    [8] J. K. He, M. Jia, X. P. Liu, H. Chen, Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity, Adv. Differ. Equations, 2018 (2018), 49. https://doi.org/10.1186/s13662-018-1465-6 doi: 10.1186/s13662-018-1465-6
    [9] F. A. McRae, Monotone iterative technique and existence results for fractional differential equations, Nonlinear Anal. Theory Methods Appl., 71 (2009), 6093–6096. https://doi.org/10.1016/j.na.2009.05.074 doi: 10.1016/j.na.2009.05.074
    [10] C. Z. Hu, B. Liu, S. F. Xie, Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments, Appl. Math. Comput., 222 (2013), 72–81. https://doi.org/10.1016/j.amc.2013.07.048 doi: 10.1016/j.amc.2013.07.048
    [11] J. Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, J. Math. Pures Appl., 8 (1892), 101–186.
    [12] F. Mainard, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010. https://doi.org/10.1142/9781848163300
    [13] H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering. J. Nonlinear Sci. Appl., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
    [14] B. Ahmad, A. Alsaedi, B. S. Alghamdi, Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real World Appl., 9 (2008), 1727–1740. https://doi.org/10.1016/j.nonrwa.2007.05.005 doi: 10.1016/j.nonrwa.2007.05.005
    [15] R. $\breve{\rm{C}}$iegis, A. Bugajev, Numerical approximation of one model of the bacterial self-organization, J. Appl. Math. Comput., 17 (2012), 253–270. https://doi.org/10.15388/NA.17.3.14054 doi: 10.15388/NA.17.3.14054
    [16] S. H. Liang, S. Y. Shi, Existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval, J. Appl. Math. Comput., 38 (2012), 687–707. https://doi.org/10.1007/s12190-011-0505-0 doi: 10.1007/s12190-011-0505-0
    [17] X. K. Zhao, W. G. Ge, Unbounded solutions for a fractional boundary value problem on the infinite interval, Acta Appl. Math., 109 (2010), 495–505. https://doi.org/10.1007/s10440-008-9329-9 doi: 10.1007/s10440-008-9329-9
    [18] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86–90.
    [19] X. A. Hao, H. Sun, L. S. Liu, Existence results for fractional integral boundary value problem involving fractional derivatives on an infinite interval, Math. Methods Appl. Sci., 41 (2018), 6984–6996. https://doi.org/10.1002/mma.5210 doi: 10.1002/mma.5210
    [20] X. C. Li, X. P. Liu, M. Jia, Y. Li, S. Zhang, Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval, Math. Methods Appl. Sci., 36 (2017), 1892–1904. https://doi.org/10.1002/mma.4106 doi: 10.1002/mma.4106
    [21] W. Zhang, W. B. Liu, Existence, uniqueness, and multiplicity results on positive solutions for a class of Hadamard-type fractional boundary value problem on an infinite interval, Math. Methods Appl. Sci., 43 (2020), 2251–2275. https://doi.org/10.1002/mma.6038 doi: 10.1002/mma.6038
    [22] T. S. Cerdik, F. Y. Deren, New results for higher-order hadamard-type fractional differential equations on the half-line, Math. Methods Appl. Sci., 45 (2022), 2315–2330. https://doi.org/10.1002/mma.7926 doi: 10.1002/mma.7926
    [23] B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer Cham, 2017. https://doi.org/10.1007/978-3-319-52141-1
    [24] G. T. Wang, K. Pei, R. P. Agarwal, L. H. Zhang, B. Ahmad, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343 (2018), 230–239. https://doi.org/10.1016/j.cam.2018.04.062 doi: 10.1016/j.cam.2018.04.062
  • Reader Comments
  • © 2024 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(847) PDF downloads(98) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog