AIMS Mathematics, 2019, 4(4): 1101-1113. doi: 10.3934/math.2019.4.1101

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions

Faculty of Sciences and Technology, Department of Mathematics and Informatics, University of Souk Ahras, P. O. Box 1553, Souk Ahras, 41000, Algeria

In this paper, we prove the existence and uniqueness of a positive solution of nonlinear Hadamard fractional differential equations with integral boundary conditions. In the process we employ the Schauder and Banach fixed point theorems and the method of upper and lower solutions to show the existence and uniqueness of a positive solution. Finally, an example is given to illustrate our results.
  Figure/Table
  Supplementary
  Article Metrics

References

1.S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252 (2000), 804-812.    

2.Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.    

3.M. Matar, On existence of positive solution for initial value problem of nonlinear fractional differential equations of order 1 < α ≤ 2, Acta Mathematica Universitatis Comenianae, 84 (2015), 51-57.

4.M. Xu, Z. Han, Positive solutions for integral boundary value problem of two-term fractional differential equations, Bound. Value Probl.,2018 (2018), 100.

5.M. Benchohra, J. E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Studia Universitatis Babes-Bolyai Matematica, 62 (2017), 27-38.    

6.B. Ahmad and S. K. Ntouyas, Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral functional differential equations, Electronic Journal of Differential Equations, 2017 (2017), 1-11.

7.Z. Bai, T. T. Qiu, Existence of positive solution for singular fractional differential equation, Appl. Math. Comput., 215 (2009), 2761-2767.

8.H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity, 21 (2017), 1201-1212.    

9.D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609-625.    

10.E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual.Theo., 2008 (2008), 1-11.

11.C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal-Theor, 74 (2011), 5975-5986.    

12.C. Wang, R. Wang, S. Wang, et al. Positive solution of singular boundary value problem for a nonlinear fractional differential equation, Bound. Value Probl., 2011 (2011), 297026.

13.C. Wang, H. Zhang, S. Wang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative, Discrete Dyn. Nat. Soc., 2012 (2012), 425408.

14.M. Xu, S. Sun, Positivity for integral boundary value problems of fractional differential equations with two nonlinear terms, J. Appl. Math.Comput., 59 (2019), 271-283.    

15.S. Zhang, Existence results of positive solutions to boundary value problem for fractional differential equation, Positivity, 13 (2009), 583-599.    

16.A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.

17.K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.

18.I. Podlubny, Fractional differential equations, Academic Press,San Diego, 1999.

19.A. Jannelli, M. Ruggieri and M. P. Speciale, Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation, Commun. Nonlinear Sci., 70 (2019), 89-101.    

20.A. Jannelli, M. Ruggieri and M. P. Speciale, Exact and numerical solutions of time-fractional advection-diffusion equation with a nonlinear source term by means of the Lie symmetries, Nonlinear Dynamics, 92 (2018), 543-555.    

21.D. R. Smart, Fixed point theorems, Cambridge Uni. Press, Cambridge, 1980.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved