The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved.
[1] H. Afshari, V. Roomi, and S. Kalantari, The existence of solutions of the inclusion problems involving Caputo and Hadamard fractional derivatives by applying some new contractions, Journal of Nonlinear and Convex Analysis, 23(6) (2022), 1213–1229.
[2] H. Afshari, M. S. Abdo, and M. Nosrati Sahlan, Some new existence results for boundary value problems involving -Caputo fractional derivative, TWMS J. App. and Eng. Math., 13(1) (2023), 246–255.
[3] A. Ahmadkhanlu, Existence and uniqueness results for a class of fractional differential equations with an integral fractional boundary condition, Filomat, 31(5) (2017), 1241–1249.
[4] B. Ahmad and Sotiris K. Ntouyas, On hadamard fractional integro-differential boundary value problems, Journal of Applied Mathematics and Computing, 47(1-2) (2014), 119131.
[5] C. Arzela, Sulle funzioni di linee, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat, 5(5) (1895), 55–74.
[6] G. Ascoli, Le curve limite di una varieta data di curve, Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat, 18(3) (1884), 521–586.
[7] A. A. Kilbas, B. Bonilla, and J . J. Trujillo, Existence and uniqueness theorems for nonlinear fractional differential equations, Demonstratio Mathematica, 33(3) (2000).
[8] A. A. Kilbas, H. M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations. Number 204 in North-Holland mathematics studies. Elsevier, Amsterdam ; Boston, 1st ed edition, 2006.
[9] A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order:methods results and problem I, Applicable Analysis, 78(1-2) (2001), 153–192.
[10] A. A. Kilbas and J. J. Trujillo, Differential Equations of Fractional Order: Methods, Results and Problems. II, Applicable Analysis, 81(2) (2002), 435–493.
[11] K. S Miller and B. Ros, An Introduction to the Fractional Calculus and Fractional Differential Equations, Number 204. John Wiley & Sonsr, New York, USA, 1993.
[12] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in science and engineering. Academic Press, San Diego, 198 (1999).
[13] Xiaojie Xu, Daqing Jiang, and Chengjun Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, 71(10) (2009), 4676–4688.
[14] K. Zhang and Z. Fu, Solutions for a class of hadamard fractional boundary value problems with sign-changing nonlinearity, Journal of Function Spaces, 2019 (2019).
Ahmadkhanlu, A., & Jamshidzadeh, S. (2024). Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem. Computational Methods for Differential Equations, 12(4), 741-748. doi: 10.22034/cmde.2023.51601.2150
MLA
Asghar Ahmadkhanlu; Shabnam Jamshidzadeh. "Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem". Computational Methods for Differential Equations, 12, 4, 2024, 741-748. doi: 10.22034/cmde.2023.51601.2150
HARVARD
Ahmadkhanlu, A., Jamshidzadeh, S. (2024). 'Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem', Computational Methods for Differential Equations, 12(4), pp. 741-748. doi: 10.22034/cmde.2023.51601.2150
VANCOUVER
Ahmadkhanlu, A., Jamshidzadeh, S. Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem. Computational Methods for Differential Equations, 2024; 12(4): 741-748. doi: 10.22034/cmde.2023.51601.2150
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