Document Type : Research Paper

**Author**

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

**Abstract**

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.

**Keywords**

- [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.

October 2021

Pages 1001-1012

**Receive Date:**03 March 2020**Revise Date:**23 June 2020**Accept Date:**24 June 2020**First Publish Date:**01 October 2021