On the existence and uniqueness of positive solutions for a p-Laplacian fractional boundary value problem with an integral boundary condition with a parameter

Document Type : Research Paper


Department of Mathematics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran.


The aim of this work is to prove the existence and uniqueness of the positive solutions for a fractional boundary value problem by a parameterized integral boundary condition with p-Laplacian operator. By using iteration sequence, the existence of two solutions is proved. Also by applying a fixed point theorem on solid cone, the result for the uniqueness of a positive solution to the problem is obtained. Two examples are given to confirm our results.


  • [1]   Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differ- ential equation., J. Math. Anal. App., 31(2) (2005), 495–505.
  • [2]   Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Non. Anal.: Theory, Meth. Appl., 72 (2) (2010), 916924.
  • [3]   A. Cabada and Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Applied Mathematics and Computation, 228 (2014), 251257.
  • [4]   Y. Cui, W. Ma, X. Wang, and X. Su, Uniqueness theorem of differential system with coupled integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 2018(9) (2018).
  • [5]   Qi Ge and C. Hou, Positive solution for a class ofp-laplacian fractional q-differenceequations involving the integral boundary condition, Mathematica Aeterna, 5 (2015), 927–944.
  • [6]   D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, volume 5, Academic Press, Orlando, Fla, USA, 1088.
  • [7]   Z. Han, H. Zhang, and C. Zhang, Positive solutions for eigenvalue problems of fractional dif- ferential equation withgeneralizedp-laplacian, Appl. Math. Comput., 257 (2015), 526536.
  • [8]   X. Hao, Positive solution for singular fractional differential equations involving derivatives, Advances in Difference Equations, 2016(139) (2016).
  • [9]   X. Hao, H. Wang, L. Liu, and Y. Cui, Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator, Boundary Value Problems, 2017 (182), (2017).
  • [10] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, 2000.
  • [11] H. Khan, C. Tunc, W.  Chen,  and  A.  Khan,  Existence theorems  and Hyers-Ulam stability  for a class of hybrid fractional differential equations with P-laplacian operator, J. Appl. Anal.Comput., 8(4) (2018), 1211-1226.
  • [12] H. Khan, C. Tunc, and A. Khan, Stability results and existence theorems for nonlinear delay- fractional differential equations with φ∗p-operator,   J.  Appl.  Anal.Comput.,  10(2)  (2020),  584- 597.
  • [13] H. Khan, C. Tunc ,and A. Khan, Green function’s properties and existence theorems for nonlinear singular-delay-fractional differential equations, Discr. contin. Dyn. Sys. Series S, DOI:10.3934/dcdss.2020139.
  • [14] R. Ali Khan and A. Khan, Existence and uniqueness of solutions for p-laplacian fractional or-der boundary value problems, Comput. Methods Differ. Equ., 2(4) (2014), 205–215.
  • [15] Y. Li and G. Li, Positive solutions of p-laplacian fractional differential equations with integral boundary value conditions, J. Nonlinear Sci. Appl., 9 (2016), 717–726.
  • [16] S. Liang and J. Zhang, Existence and uniqueness of positive solutions for integral boundary problems of nonlinearfractional differential equations withp-laplacian operator, Rocky Mt. J. Math., 44(3) (2014),953974.
  • [17] X. Liu, M. Jia, and W. Ge, Multiple solutions of ap-laplacian model involving a fractional derivative, Adv. Differ. Equ., 2013(5) (2013).
  • [18] L. Liu, F. Sun, X. Zhang, and Y. Wu, Bifurcation analysis for a singular differential system with two parameters via to topological degree theory, Lithuanian Association of Nonlinear Analysts. Nonlinear Analysis: Modelling and Control, 22(1) (2017), 3150.
  • [19] N. I. Mahmudov and S. Unul, Existence of solutions of fractional boundary value problems withp-laplacian, Bound. ValueProbl., 99 (2015).
  • [20] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  • [21] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentia- tion and Integration to Arbitrary Order of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974.
  • [22] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Aca- demic Press, San Diego, Calif, USA, 1999.
  • [23] S. G. Samko, A. A. Kilbas, and O. I. Marichev, The Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, UK, 1993.
  • [24] F. Yan, M. Zuo, and X. Hao, Positive solution for a fractional singular boundary value problem with -Laplacian operator, Boundary Value Problems, 2018(51) (2018).
  • [25] C. Yuan, Multiple positive solutions for (n1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual.  Theory  Differ.  Equ.,  36 (2010).
  • [26] L. Zhang, W. Zhang, X. Liu, and M. Jia, Positive solutions of fractionalp-laplacianequations with integral boundary valueand two parameters, Mathematica Aeterna, 2020(2) (2020).
  • [27]  X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, The eigenvalue for a class of singular p-laplacian fractional differential equations involving the riemann-stieltjes integral boundary condition, Appl. Math. Comput., 235(1) (2014), 412–422.