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Positive solutions to boundary value problems for fractional differential equations with time delays

Funds:  Supported by University Science Research Key Project of Anhui Province (KJ2018A0565), Fostering Master’s Degree Empowerment Point Project of Hefei University (2018xs03), Major Project of Humanities and Social Sciences of Anhui Province (SK2019ZD55), Operational Research High-Level Teaching Team of Anhui Province (2018jxtd049).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.02.004
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  • Author Bio:

    HU Xiulin, female, born in 1981, master/associate Prof. Research field:Functional differential equations.E-mail: huxlxy@hfuu.edu.cn

  • Corresponding author: ZHANG Yongjun
  • Received Date: 19 December 2018
  • Accepted Date: 05 May 2019
  • Rev Recd Date: 05 May 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The existence of positive solutions for a class of

    Abstract

    The existence of positive solutions for a class of
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    • References(21)
  • [1]
    PODLUBNY I. Fractional Differential Equations[M]. New York: Academic Press, 1999.
    [2]
    KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations[M]. Amsterdam: Elsevier B V, 2006.
    [3]
    DIETHELM K. The Analysis of Fractional Differential Equations[M]. Berlin: Springer, 2010.
    [4]
    SAMKO S G, KILBAS A A, MARICHEV O I. Fractional Integral and Derivatives: Theory and Applications[M]. Yverdon, Switzerland: Gordon and Breach, 1993.
    [5]
    HALE J K, LUNEL S M V. Introduction to Functional Differential Equations[M]. New York: Springer-Verlag, 1993.
    [6]
    SU X. Solutions to boundary value problem of fractional order on unbounded domains in a Banach space[J]. Nonlinear Anal, 2011, 74: 2844-2852.
    [7]
    CABADA A, WANG G. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions[J]. J Math Anal Appl, 2012, 389: 403-411.
    [8]
    ZHANG L, WANG G, AHMAD B, et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space[J]. J Compt Appl Math, 2013, 249: 51-56.
    [9]
    XIE W,XIAO J, LUO Z. Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions[J]. Appl Math Lett, 2015, 41: 46-51.
    [10]
    AHMAD B, NIETO J J. Sequential fractional differential equations with three-point boundary conditions[J]. Computers and Mathematics with Applications, 2012, 64: 3045-3052.
    [11]
    ZHANG X, LIU L, WU Y. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term[J]. Math and Comput Modelling, 2012, 55: 1263-1274.
    [12]
    CUI Y. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Appl Math Lett, 2016, 51: 48-54.
    [13]
    AGARWAL R P, ZHOU Y, WANG J, et al. Fractional functional differential equations with causal operators in Banach spaces[J]. Math Compt Modelling, 2011, 54: 1440-1452.
    [14]
    LI X Y, LIU S, JIANG W. Positive solutions for boundary value problem of nonlinear fractional functional differential equations[J]. Appl Math Comput, 2011, 217: 9278-9285.
    [15]
    SU X. Positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity[J]. Computers and Mathematics with Applications, 2012, 64: 3425-3435.
    [16]
    DONG Xiaoyu, BAI Zhanbing, ZHANG Wei. Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives[J]. Journal of Shandong University of Science and Technology (Natural Science), 2016, 35(3): 85-91. (in Chinese)
    [17]
    LIU X, JIA M, GE W. The method of lower and upper solutions for mixed fractional differential four-point boundary value problem with p-Laplacian operator[J]. Appl Math Lett, 2017, 65: 56-62.
    [18]
    MIN D, LIU L, WU Y. Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions[J]. Bound Value Prob, 2018, 2018: 23.
    [19]
    BATARFI H, LOSADA J, NIETO J J, et al. Three-point boundary value problems for conformable fractional differential equations[J]. Journal of Function Spaces, 2015, 2015: 706383.
    [20]
    KRASNOSEL’SKII M A. Positive Solutions of Operator Equations[M]. Groningen, Netherlands: Noordhoff Publishing, 1964.
    [21]
    GUO Dajun, SUN Jingxian. Nonlinear Integral Equations[M]. Jinan: Shandong Science and Technology Press, 1987. (in Chinese))
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    [1]
    PODLUBNY I. Fractional Differential Equations[M]. New York: Academic Press, 1999.
    [2]
    KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations[M]. Amsterdam: Elsevier B V, 2006.
    [3]
    DIETHELM K. The Analysis of Fractional Differential Equations[M]. Berlin: Springer, 2010.
    [4]
    SAMKO S G, KILBAS A A, MARICHEV O I. Fractional Integral and Derivatives: Theory and Applications[M]. Yverdon, Switzerland: Gordon and Breach, 1993.
    [5]
    HALE J K, LUNEL S M V. Introduction to Functional Differential Equations[M]. New York: Springer-Verlag, 1993.
    [6]
    SU X. Solutions to boundary value problem of fractional order on unbounded domains in a Banach space[J]. Nonlinear Anal, 2011, 74: 2844-2852.
    [7]
    CABADA A, WANG G. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions[J]. J Math Anal Appl, 2012, 389: 403-411.
    [8]
    ZHANG L, WANG G, AHMAD B, et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space[J]. J Compt Appl Math, 2013, 249: 51-56.
    [9]
    XIE W,XIAO J, LUO Z. Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions[J]. Appl Math Lett, 2015, 41: 46-51.
    [10]
    AHMAD B, NIETO J J. Sequential fractional differential equations with three-point boundary conditions[J]. Computers and Mathematics with Applications, 2012, 64: 3045-3052.
    [11]
    ZHANG X, LIU L, WU Y. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term[J]. Math and Comput Modelling, 2012, 55: 1263-1274.
    [12]
    CUI Y. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Appl Math Lett, 2016, 51: 48-54.
    [13]
    AGARWAL R P, ZHOU Y, WANG J, et al. Fractional functional differential equations with causal operators in Banach spaces[J]. Math Compt Modelling, 2011, 54: 1440-1452.
    [14]
    LI X Y, LIU S, JIANG W. Positive solutions for boundary value problem of nonlinear fractional functional differential equations[J]. Appl Math Comput, 2011, 217: 9278-9285.
    [15]
    SU X. Positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity[J]. Computers and Mathematics with Applications, 2012, 64: 3425-3435.
    [16]
    DONG Xiaoyu, BAI Zhanbing, ZHANG Wei. Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives[J]. Journal of Shandong University of Science and Technology (Natural Science), 2016, 35(3): 85-91. (in Chinese)
    [17]
    LIU X, JIA M, GE W. The method of lower and upper solutions for mixed fractional differential four-point boundary value problem with p-Laplacian operator[J]. Appl Math Lett, 2017, 65: 56-62.
    [18]
    MIN D, LIU L, WU Y. Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions[J]. Bound Value Prob, 2018, 2018: 23.
    [19]
    BATARFI H, LOSADA J, NIETO J J, et al. Three-point boundary value problems for conformable fractional differential equations[J]. Journal of Function Spaces, 2015, 2015: 706383.
    [20]
    KRASNOSEL’SKII M A. Positive Solutions of Operator Equations[M]. Groningen, Netherlands: Noordhoff Publishing, 1964.
    [21]
    GUO Dajun, SUN Jingxian. Nonlinear Integral Equations[M]. Jinan: Shandong Science and Technology Press, 1987. (in Chinese))
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