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

Keywords coupled system of Riemann-Liouville type fractional differential equations; three-point boundary value condition; positive solution; Schauder’s fixed point theorem

Citation: Md. Asaduzzaman, Md. Zulfikar Ali. Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880

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.    

 

Reader Comments

your name: *   your email: *  

© 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

Copyright © AIMS Press All Rights Reserved