Search Advanced

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 4437-4454. doi: 10.3934/mbe.2023206
Previous Article Next Article
Research article Special Issues

Existence results of fractional differential equations with nonlocal double-integral boundary conditions

  • School of Mathematics and Statistics, Heze University, Heze 274000, China

  • Academic Editor: Feliz Minhós
  • Received: 24 October 2022 Revised: 15 December 2022 Accepted: 16 December 2022 Published: 26 December 2022

  • This article presents the existence outcomes concerning a family of singular nonlinear differential equations containing Caputo's fractional derivatives with nonlocal double integral boundary conditions. According to the nature of Caputo's fractional calculus, the problem is converted into an equivalent integral equation, while two standard fixed theorems are employed to prove its uniqueness and existence results. An example is presented at the end of this paper to illustrate our obtained results.

    Citation: Debao Yan. Existence results of fractional differential equations with nonlocal double-integral boundary conditions[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4437-4454. doi: 10.3934/mbe.2023206

    Related Papers:

  • This article presents the existence outcomes concerning a family of singular nonlinear differential equations containing Caputo's fractional derivatives with nonlocal double integral boundary conditions. According to the nature of Caputo's fractional calculus, the problem is converted into an equivalent integral equation, while two standard fixed theorems are employed to prove its uniqueness and existence results. An example is presented at the end of this paper to illustrate our obtained results.


    加载中


    [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 198 (1999), 1–340. https://doi.org/10.1016/s0076-5392(99)x8001-5
    [2] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. https://doi.org/10.1142/9069
    [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, Elsevier Science B. V., Amsterdam, 204 (2006). https://doi.org/10.1016/S0304-0208(06)80001-0
    [4] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. https://doi:10.1007/978-3-642-18101-6-2
    [5] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
    [6] A. A. Kilbas, J. J. Trujiuo, Differential equations of fractional order: methods, results and problems Ⅰ, Appl. Anal., 78 (2001), 153–192. https://doi.org/10.1080/00036810108840931 doi: 10.1080/00036810108840931
    [7] A. A. Kilbas, J. J. Trujiuo, Differential equations of fractional order: methods, results and problems Ⅱ, Appl. Anal., 81 (2002), 435–493. https://doi.org/10.1080/0003681021000022032 doi: 10.1080/0003681021000022032
    [8] D. Delbosco, Fractional calculus and function spaces, J. Fract. Calc., 6 (1994), 45–53.
    [9] K. Diethelm, N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
    [10] V. Laksmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
    [11] V. Laksmikantham, A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11 (2007), 395–402. Available from: http://www.acadsol.eu/en/articles/11/3/4.pdf.
    [12] E. Demirci, N. Ozalp, A method for solving differential equations of fractional, J. Comput. Appl. Math., 236 (2012), 2754–2762. https://doi.org/10.1016/j.cam.2012.01.005 doi: 10.1016/j.cam.2012.01.005
    [13] K. Sayevand, A. Golbabai, A. Yidirim, Analysis of differential equations of fractional order, Appl. Math. Modell., 36 (2012), 4356–4364. https://doi.org/10.1016/j.apm.2011.11.061 doi: 10.1016/j.apm.2011.11.061
    [14] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 70 (2009), 2521–2529. https://doi.org/10.1016/j.na.2008.03.037 doi: 10.1016/j.na.2008.03.037
    [15] B. Ahmad, N. Alghamdi, A. Alsaedi, S. K. Ntouyas, A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions, Fract. Calc. Appl. Anal., 22 (2019), 601–616. https://doi.org/10.1515/fca-2019-0034 doi: 10.1515/fca-2019-0034
    [16] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609–625. http://dx.doi.org/10.1006/jmaa.1996.0456 doi: 10.1006/jmaa.1996.0456
    [17] X. Zheng, H. Wang, A hidden-memory variable-order time-fractional optimal control model: analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. https://doi.org/10.1137/20M1344962 doi: 10.1137/20M1344962
    [18] X. Zheng, H. Wang, An optimal-order numerical approximation to variable-order space-fractional diffusion equation uniform or graded meshes, SIAM J. Numer. Anal., 58 (2020), 330–352. https://doi.org/10.1137/19M1245621 doi: 10.1137/19M1245621
    [19] J. Li, X. Su, K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simul., 205 (2023), 340–347. https://doi.org/10.1016/j.matcom.2022.10.005 doi: 10.1016/j.matcom.2022.10.005
    [20] X. Zheng, H. Wang, An error estimate of a numerical approximation to a hidden-memory variable-order space-time Fractional diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492–2514. https://doi.org/10.1137/20M132420X doi: 10.1137/20M132420X
    [21] T. Qiu, Z. Bai, Existence of positive solutions for singular fractional differential equations, Electron. J. Differ. Equations, 146 (2008), 1–9. Available from: https://ejde.math.txstate.edu/index.html.
    [22] R. P. Agarwal, D. O'regan, S. Stank, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57–68. https://doi.org/10.1016/j.jmaa.2010.04.034 doi: 10.1016/j.jmaa.2010.04.034
    [23] X. Zhang, Q. Zhong, Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations, Bound. Value Probl., 65 (2016). https://doi.org/10.1186/s13661-016-0572-0 doi: 10.1186/s13661-016-0572-0
    [24] Y. Wang, Existence and multiplicity of positive solutions for a class of singular fractional nonlocal boundary value problems, Bound. Value Probl., 92 (2019). https://doi.org/10.1186/s13661-019-1205-1 doi: 10.1186/s13661-019-1205-1
    [25] D. Yan, Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with singular boundary value conditions, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/6692620 doi: 10.1155/2021/6692620
    [26] Y. Wang, L. Liu, Y. Wu, Existence and uniqueness of a positive solution to singular fractional differential equations, Bound. Value Probl., 81 (2012). https://doi.org/10.1186/1687-2770-2012-81 doi: 10.1186/1687-2770-2012-81
    [27] T. Wang, Z.Hao, Existence and uniqueness of positive solutions for singular nonlinear fractional differential equation via mixed monotone operator method, J. Funct. Space, 2020 (2020). https://doi.org/10.1155/2020/2354927 doi: 10.1155/2020/2354927
    [28] L. Guo, X. Zhang, Existence of positive solutions for the singular fractional differential equations, J. Appl. Math. Comput., 44 (2014), 215–228. https://doi.org/10.1007/s12190-013-0689-6 doi: 10.1007/s12190-013-0689-6
    [29] A. Cabada, Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput., 228 (2014), 251–257. https://doi.org/10.1016/j.amc.2013.11.057 doi: 10.1016/j.amc.2013.11.057
    [30] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403–411. https://doi.org/10.1016/j.jmaa.2011.11.065 doi: 10.1016/j.jmaa.2011.11.065
    [31] T. Wang, F. Xie, Existence and uniqueness of fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl., 1 (2008), 206–212. http://doi.org/10.22436/jnsa.001.04.02 doi: 10.22436/jnsa.001.04.02
    [32] Y. Qiao, Z. Zhou, Existence and uniqueness of positive solutions for a fractional differential equation with integral boundary conditions, Adv. Differ. Equations, 31 (2016). https://doi.org/10.1186/s13662-016-0772-z doi: 10.1186/s13662-016-0772-z
    [33] B. Ahmad, J. J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. Value Probl., 2009 (2009). https://doi.org10.1155/2009/708576 doi: 10.1155/2009/708576
    [34] S. Hamani, M. Benchohra, J. R. Graef, Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions, Electron. J. Differ. Equations, 20 (2010), 1–16. https://ejde.math.txstate.edu/index.html
    [35] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of Riemann-Liouville fractional differential equation with integral boundary condition, J. Nonlinear Sci., 14 (2012), 410–415. Available from: http://www.internonlinearscience.org/bookseries.aspx?jouid=53&journals=Volume.
    [36] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, J. Nonlinear Sci. Appl., 7 (2014), 246–254. Available from: https://www.emis.de/journals/TJNSA/includes/files/articles/.
    [37] S. Padhi, J. R. Graef, S. Pati, Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions, Fract. Calc. Appl. Anal., 21 (2018), 716–745. https://doi.org/10.1515/fca-2018-0038 doi: 10.1515/fca-2018-0038
    [38] M. A. Darwish, S. K. Ntouyas, Existence results for first order boundary value problems for fractional differential equations with four-point integral boundary conditions, Miskolc Math. Notes, 15 (2014), 51–61. http://doi.org/10.18514/MMN.2014.511 doi: 10.18514/MMN.2014.511
    [39] B. Ahmad, R. P. Agarwal, Some new versions of fractional boundary value problems with slit-strips conditions, Bound. Value Probl., 175 (2014). https://doi.org/10.1186/s13661-014-0175-6 doi: 10.1186/s13661-014-0175-6
    [40] Y. He, Existence and multiplicity of positive solutions for singular fractional differential equations with integral boundary value conditions, Adv. Differ. Equations, 31 (2016). https://doi.org/10.1186/s13662-015-0729-7 doi: 10.1186/s13662-015-0729-7
    [41] S. Vong, Positive solutions of singular fractional differential equation with integral boundary conditions, Math. Comput. Model., 57 (2013), 1053–1059. https://doi.org/10.1016/j.mcm.2012.06.024 doi: 10.1016/j.mcm.2012.06.024
    [42] D. Min, L. Liu, Y. Wu, Uniqueness of positive solutions for the singular nonlinear fractional differential equations involving integral boundary value conditions, Bound. Value Probl., 23 (2018). https://doi.org/10.1186/s13661-018-0941-y doi: 10.1186/s13661-018-0941-y
    [43] K. Chandran, K. Gopalan, S. T. Zubair, T. Abdeljawad, A fixed point approach to the solution of singular fractional differential equations with integral boundary conditions, Adv. Differ. Equations, 56 (2021). https://doi.org/10.1186/s13662-021-03225-y doi: 10.1186/s13662-021-03225-y
    [44] D. Yan, Solutions for a category of singular nonlinear fractional differential equations subject to integral boundary conditions, Bound. Value Probl., 3 (2022). https://doi.org/10.1186/s13661-022-01585-2 doi: 10.1186/s13661-022-01585-2
    [45] M. A. Krasnoselskii, Two remarks on the method of successive approximations, Usp. Math. Nauk, 10 (1955), 123–127. Available from: https://www.mathnet.ru/links/eaecb7dd6edb854d12cbc2c3a8ed39f4/rm7954.pdf.
    [46] Y. Wang, J. Xu, Sobolev Space (in Chinese), Southeast University Press, 2003.
  • Reader Comments
  • © 2023 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
Mathematical Biosciences and Engineering
2.6 3.9

Metrics

Article views(935) PDF downloads(94) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog