Positive solution of Hilfer fractional differential equations with integral boundary conditions
DOI:
https://doi.org/10.24193/subbmath.2021.4.09Keywords:
Fractional differential equations, positive solution, upper and lower solutions, fixed point theorem, existence and uniqueness.Abstract
In this article, we have interested the study of the existence and uniqueness of positive solutions of the first-order nonlinear Hilfer fractional differential equation...
... are fractional ope- 0+ 0+ rators in the Hilfer, Riemann-Liouville concepts, respectively. In this approach, we transform the given fractional differential equation into an equivalent integral equation. Then we establish sufficient conditions and employ the Schauder fixed point theorem and the method of upper and lower solutions to obtain the existence of a positive solution of a given problem. We also use the Banach contraction principle theorem to show the existence of a unique positive solution. The result of existence obtained by structure the upper and lower control functions of the nonlinear term is without any monotonous conditions. Finally, an example is presented to show the effectiveness of our main results.
Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 34A12, 47H10.
References
Abdo, M.S., Wahash, H.A., Panchal, S.K., Positive solution of a fractional differential equation with integral boundary conditions, Journal of Applied Mathematics and Computational Mechanics, 17(2018), no. 3.
Almalahi, M.A., Abdo, M.S., Panchal, S.K., Existence and Ulam-Hyers-Mittag-Leffler stability results of ψ-Hilfer nonlocal Cauchy problem, Rend. Circ. Mat. Palermo, II. Ser (2020).
Ardjouni, A., Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions, AIMS Mathematics, 4(2019), 1101-1113.
Ardjouni, A., Djoudi, A., Existence and uniqueness of positive solutions for first-order nonlinear Liouville–Caputo fractional differential equations, S˜ao Paulo Journal of Mathematical Sciences, (2019), 1-10.
Boulares, H., Ardjouni, A., Laskri, Y., Positive solutions for nonlinear fractional differential equations, Positivity, 21(3)(2017), 1201-1212.
Furati, K.M., Kassim, M.D., Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Applic., 64(2012), 1616-1626.
Hilfer R., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River 27 Edge, NJ, Singapore, 2000.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 207, 2006.
Long, T., Li, C., He, J., Existence of positive solutions for period BVPs with Hilfer derivative, Journal of Applied Mathematics and Computing, 60(2019), no. 1-2, 223-236.
Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, New York, John Wiley, 1993.
Nan, L.I., Changyou, W.A.N.G., New existence results of positive solution for a class of nonlinear fractional differential equations, Acta Mathematica Scientia, 33(2013), no. 3, 847-854.
Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 198, 1998.
Wang, F., Liu, L., Kong, D., Wu, Y., Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with mixed-type boundary value conditions, Nonlinear Analysis: Modelling and Control, 24(2019), no. 1, 73-94.
Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 252(2000), no. 2, 804-812.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
