Research article

Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral

  • Received: 01 April 2020 Accepted: 12 May 2020 Published: 01 June 2020
  • MSC : 34B18, 34B15, 26A33

  • In this article, we study a class of p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. By two fixed point theorems of a sum operator in partial ordering Banach spaces, we get the existence and uniqueness of positive solutions for addressed problem. Moreover, we can make iterative sequences to approximate the unique positive solution. In addition, two examples are given to illustrate the main results.

    Citation: Chengbo Zhai, Yuanyuan Ma, Hongyu Li. Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral[J]. AIMS Mathematics, 2020, 5(5): 4754-4769. doi: 10.3934/math.2020304

    Related Papers:

  • In this article, we study a class of p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. By two fixed point theorems of a sum operator in partial ordering Banach spaces, we get the existence and uniqueness of positive solutions for addressed problem. Moreover, we can make iterative sequences to approximate the unique positive solution. In addition, two examples are given to illustrate the main results.


    加载中


    [1] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
    [2] L. Yang, H. Chen, Unique positive solutions for fractional differential equation boundary value problems, Appl. Math. Lett., 23 (2010), 1095-1098. doi: 10.1016/j.aml.2010.04.042
    [3] Y. Zhao, S. Sun, Z. Han, et al. The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 2086-2097. doi: 10.1016/j.cnsns.2010.08.017
    [4] L. Zhang, B. Ahmad, G. Wang, et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. App. Math., 249 (2013), 51-56. doi: 10.1016/j.cam.2013.02.010
    [5] H. Lu, Z. Han, S. Sun, Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian, Adv. Differ. Equ., 2013 (2013), 30.
    [6] C. Zhai, L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2820-2827. doi: 10.1016/j.cnsns.2014.01.003
    [7] X. Zhang, L. Liu, B. Wiwatanapataphee, et al. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition, Appl. Math. Comput., 235 (2014), 412-422.
    [8] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26-33. doi: 10.1016/j.aml.2014.05.002
    [9] H. Wang, L. Zhang, The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems, Bound. Value Probl., 2015 (2015), 203.
    [10] C. Yang, Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter, Hacet. J. Math. Stat., 44 (2015), 665-673.
    [11] B. Ahmad, S. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Soliton. Fract., 83 (2016), 234-241. doi: 10.1016/j.chaos.2015.12.014
    [12] M. Zuo, X. Hao, L. Liu, et al. Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions, Bound. Value Probl., 2017 (2017), 161.
    [13] L. Hu, S. Zhang, Existence results for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions, Bound. Value Probl., 2017 (2017), 88.
    [14] X. Hao, H. Wang, L. Liu, et al. Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator, Bound. Value. Probl., 2017 (2017), 182.
    [15] X. Li, X. Liu, M. Jia, et al. Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval, Math. Method Appl. Sci., 40 (2017), 1892-1904.
    [16] Y. Zou, G. He, On the uniqueness of solutions for a class of fractional differential equations, Appl. Math. Lett., 74 (2017), 68-73. doi: 10.1016/j.aml.2017.05.011
    [17] C. Yang, C. Zhai, L. Zhang, Local uniqueness of positive solutions for a coupled system of fractional differential equations with integral boundary conditions, Adv. Diff. Equa., 2017 (2017), 282.
    [18] C. Yang, Positive solutions for a class of integral boundary value condition of fractional differential equations with a parameter, J. Nonlinear Sci. Appl., 10 (2017), 2710-2718. doi: 10.22436/jnsa.010.05.37
    [19] C. Zhai, R. Jiang, Unique solutions for a new coupled system of fractional differential equations, Adv. Differ. Equ., 2018 (2018), 1.
    [20] C. Zhai, P. Li, H. Li, Single upper-solution or lower-solution method for Langevin equations with two fractional orders, Adv. Differ. Equ., 2018 (2018), 360.
    [21] J. Wu, X. Zhang, L. Liu, et al. Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation, Math. Model. Anal., 23 (2018), 611-626. doi: 10.3846/mma.2018.037
    [22] C. Zhai, L. Wei, The unique positive solution for fractional integro-differential equations on infinite intervals, ScienceAsia, 44 (2018), 118-124. doi: 10.2306/scienceasia1513-1874.2018.44.118
    [23] C. Zhai, P. Li, Nonnegative solutions of initial value problems for Langevin equations involving two fractional orders, Mediterr, J. Math., 15 (2018), 164.
    [24] C. Zhai, J. Ren, The unique solution for a fractional q-difference equation with three-point boundary conditions, Indagat. Math. New Ser., 29 (2018), 948-961. doi: 10.1016/j.indag.2018.02.002
    [25] C. Zhai, W. Wang, H. Li, A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions, J. Inequa. Appl., 2018 (2018), 207.
    [26] B. Ahmad, Y. Alruwaily, A. Alsaedi, et al. Existence and stability results for a fractional order differential equation with non-conjugate Riemann-Stieltjes integro-multipoint boundary conditions, Mathematics, 7 (2019), 249.
    [27] G. Wang, K. Pei, Y. Chen, Stability analysis of nonlinear Hadamard fractional differential system, J. Franklin. I., 356 (2019), 6538-6546. doi: 10.1016/j.jfranklin.2018.12.033
    [28] J. Wang, A. Zada, H. Waheed, Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Math. Meth. Appl. Sci., 42 (2019), 6706-6732. doi: 10.1002/mma.5773
    [29] K. Zhang, Z. Fu, Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity, J. Funct. Spaces, 2019 (2019), 9046472.
    [30] J. Jiang, D. O'Regan, J. Xu, et al., Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, J. Inequa. Appl., 2019 (2019), 204.
    [31] S. Meng, Y. Cui, Multiplicity results to a conformable fractional differential equations involving integral boundary condition, Complexity, 2019 (2019), 8402347.
    [32] W. Wang, Properties of Green's function and the existence of different types of solutions for nonlinear fractional BVP with a parameter in integral boundary conditions, Bound. Value Probl., 2019 (2019), 76.
    [33] H. Zhang, Y. Li, J. Xu, Positive solutions for a system of fractional integral boundary value problems involving Hadamard-type fractional derivatives, Complexity, 2019 (2019), 11.
    [34] C. Zhai, J. Ren, A coupled system of fractional differential equations on the half-line, Bound. Value Probl., 2019 (2019), 117.
    [35] J. Ren, C. Zhai, Nonlocal q-fractional boundary value problem with Stieltjes integral conditions, Nonlinear Anal. Model. Control, 24 (2019), 582-602.
    [36] J. Ren, C. Zhai, Unique solutions for fractional q-difference boundary value problems via a fixed point method, Bull. Malay. Math. Sci. Soc., 42 (2019), 1507-1521. doi: 10.1007/s40840-017-0560-2
    [37] C. Zhai, W. Wang, Properties of positive solutions for m-point fractional differential equations on an infinite interval, RACSAM, 113 (2019), 1289-1298. doi: 10.1007/s13398-018-0548-2
    [38] K. Liu, J. Wang, Y. Zhou, et al. Hyers-Ulam stability and existence of solutions for fractional differential equations with Mittag-Leffler kernel, Chaos, Soliton. Fract., 132 (2020), 109534.
    [39] C. Zhai, W. Wang, Solutions for a system of Hadamard fractional differential equations with integral conditions, Numer, Func. Anal. Opt., 41 (2020), 209-229. doi: 10.1080/01630563.2019.1620771
    [40] X. Liu, M. Jia, Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives, Appl. Math. Comput., 353 (2019), 230-242.
    [41] X. Liu, M. Jia, The method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian, Adv. Difer. Equ., 2018 (2018), 28.
    [42] J. He, X. Zhang, L. Liu, et al. A singular fractional Kelvin-Voigt model involving a nonlinear operator and their convergence properties, Bound. Value Probl., 2019 (2019), 112.
    [43] J. He, X. Zhang, L. Liu, et al. Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions, Bound. Value Probl., 2018 (2018), 189.
    [44] T. Ren, S. Li, X. Zhang, et al. Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes, Bound. Value Probl., 2017 (2017), 118.
    [45] X. Zhang, L. Liu, Y. Wu, et al. Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion, Appl. Math. Letters, 66 (2017), 1-8. doi: 10.1016/j.aml.2016.10.015
    [46] X. Zhang, C. Mao, L. Liu, et al. Exact iterative solution for an abstract fractional dynamic system model for bioprocess, Qual. Theory Dyn. Syst., 16 (2017), 205-222. doi: 10.1007/s12346-015-0162-z
    [47] X. Zhang, L. Liu, Y. Wu, et al. The spectral analysis for a singular fractional differential equation with a signed measure, Appl. Math. Comput., 257 (2015), 252-263.
    [48] X. Zhang, Y. Wu, L. Caccetta, Nonlocal fractional order differential equations with changing-sign singular perturbation, Appl. Math. Model., 39 (2015), 6543-16552. doi: 10.1016/j.apm.2015.02.005
    [49] X. Zhang, L. Liu, Y. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Model., 55 (2012), 1263-1274. doi: 10.1016/j.mcm.2011.10.006
    [50] L. S. Leibenson, General problem of the movement of a compressible flfluid in a porous medium, Izv. Akad. Nauk. Kirg. SSSR, 9 (1983), 7-10 (in Russian).
    [51] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press, New York, 1988.
    [52] C. Zhai, D. R. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl., 375 (2011), 388-400. doi: 10.1016/j.jmaa.2010.09.017
    [53] C. Yang, C. Zhai, M. Hao, Uniqueness of positive solutions for several classes of sum operator equations and applications, J. Inequal. Appl., 2014 (2014), 58.
  • 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(2924) PDF downloads(277) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog