AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880

Research article

Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations

1 Department of Mathematics, Islamic University, Kushtia-7003, Bangladesh
2 Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh

In this paper, we investigate the existence criteria of at least one positive solution to the three-point boundary value problems with coupled system of Riemann-Liouville type nonlinear fractional order differential equations. The analysis of this study is based on the well-known Schauder’s fixed point theorem. Some new existence and multiplicity results for coupled system of Riemann-Liouville type nonlinear fractional order differential equation with three-point boundary value conditions are obtained.
  Figure/Table
  Supplementary
  Article Metrics

References

1. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, 2010.

2. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.

3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 of North-Holland Mathematics Studies, Elsevier Science Limited, 2006.

4. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

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

6. N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765-771.    

7. Q. Sun, H. Ji and Y. Cui, Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions, J. Funct. Space. Appl., 2018 (2018), 1-6.

8. W. Ma, S. Meng and Y. Cui, Resonant Integral Boundary Value Problems for Caputo Fractional Differential Equations, Math. Probl. Eng., 2018 (2018), 1-8.

9. Y. Cu, W. Ma, Q. Sun, et al. New uniqueness results for boundary value problem of fractional differential equation, Nonlinear Anal-Model, 23 (2018), 31-39.

10. X. Han and X. Yang, Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters, Bound. Value Probl., 2017 (2017), 78.

11. Y. Cui, Q. Sun and X. Su, Monotone iterative technique for nonlinear boundary value problems of fractional order p∈ (2 ,3], Adv. Differ. Equ-NY, 2017 (2017), 248.

12. T. Qi, Y. Liu and Y. Cui, Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions, J. Funct. Space. Appl., 2017 (2017), 1-9.

13. T. Qi, Y. Liu and Y. Zou, Existence result for a class of coupled fractional differential systems with integral boundary value conditions, J. Nonlinear Sci. Appl., 10 (2017), 4034-4045.    

14. T. Bashiri, S. M. Vaezpour and C. Park, A coupled fixed point theorem and application to fractional hybrid differential problems, Fixed Point Theory and Applications, 2016 (2016), 23.

15. B. Zhu, L. Liu, and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.    

16. Y. Wang, L. Liu, X. Zhang, et al. Positive solutions of an abstract fractional semi-positone differential system model for bioprocesses of HIV infection, Appl. Math. Comput., 258 (2015), 312-324.

17. D. Luo and Z. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Differ. Equ-NY, 2019 (2019), 155.

18. D. Luo, and Z. Luo, Uniqueness and Novel Finite-Time Stability of Solutions for a Class of Nonlinear Fractional Delay Difference Systems, Discrete Dyn. Nat. Soc., 2018 (2018), 1-7.

19. P. Agarwal, M. Chand, D. Baleanu, et al. On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function, Adv. Differ. Equ-NY, 2018 (2018), 249.

20. P. Agarwal, M. Chand, J. Choi, et al. Certain fractional integrals and image formulas of generalized k-Bessel function, Communications of the Korean Mathematical Society, 33 (2018), 423-436.

21. P. Agarwal, A.A. El-Sayed, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Physica A, 500 (2018), 40-49.    

22. K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti-periodic boundary conditions, Differ. Equ. Appl., 7 (2015), 245-262.

23. M. Hao and C. Zhai, Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order, J. Nonlinear Sci. Appl., 7 (2014), 131-137.    

24. Y. Cui, Y. Zou, Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point boundary value problems, Abstr. Appl. Anal., 2014 (2014), 1-6.

25. M. J. Li, Y. L. Liu, Existence and uniqueness of positive solutions for a coupled system of nonlinear fractional differential equations, Open Journal of Applied Sciences, 3 (2013), 53-61.

26. C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62 (2011), 1251-1268.    

27. C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055.    

28. X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69.    

29. D. R. Dunninger and H. Y. Wang, Existence and multiplicity of positive solutions for elliptic systems, Nonlinear Anal-Theor, 29 (1997), 1051-1060.    

30. J. Leray, J. Schauder, Topologie et equations fonctionels, Ann. Sci. École Norm. Sup., 51 (1934), 45-78.    

31. M. Fréchet, Sur quelques points du calculfonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1-74.    

© 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