Differential Equations and Dynamical Systems

, Volume 27, Issue 4, pp 395–411 | Cite as

Existence of Solutions for a Coupled Fractional Differential Equations with Infinitely Many Points Boundary Conditions at Resonance on an Unbounded Domain

  • Fu-Dong Ge
  • Hua-Cheng ZhouEmail author
  • Chun-Hai Kou
Original Research


This paper investigates the existence of solutions for infinitely many points boundary value problems at resonance with \(\mathrm{dimker}\,L=2\) regarding fractional differential equations on an unbounded domain. By the well-known coincidence degree theory of Mawhin, it is shown that the considered system under certainly conditions admits at least one solution. An example to illustrate the applicability of the given conditions is also given.


Fractional differential equations Infinitely many points Resonance Coincidence degree theory Unbounded domain 

Mathematics Subject Classification

34A08 34B10 34B40 



This work is supported by Chinese Universities Scientific Fund (No. CUSF-DH-D-2014061) and the Natural Science Foundation of Shanghai (No. 15ZR1400800).


  1. 1.
    Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential. Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  2. 2.
    Burton, T.A., Zhang, B.: Fractional equations and generalizations of Schaefer’s and Krasnoselskiis fixed point theorems. Nonlinear Anal. 75, 6485–6495 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  4. 4.
    Ge, F.D., Zhou, H.C.: Existence of solutions for fractional differential equations with three-point boundary conditions at resonance in \(\mathbb{R}^n\). J. Qual. Theory Differ. Equ. 68, 1–18 (2014)Google Scholar
  5. 5.
    Jiang, W.H.: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal. Real World Appl. 13, 2285–2292 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kosmatov, N.: A multi-point boundary value problem with two critical conditions. Nonlinear Anal. 65, 622–633 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kosmatov, N.: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 68, 2158–2171 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kosmatov, N.: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 135, 1–10 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V, Netherlands (2006)zbMATHGoogle Scholar
  10. 10.
    Kou, C.H., Zhou, H.C., Yan, Y.: Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal. 74, 5975–5986 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Liu, Y.J., Ahmad, B., Agarwal, R.P.: Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half-line. Adv. Differ. Equ. 46, 2–19 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lakshmikantham, V., Leela, S., Vasundhara, J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)zbMATHGoogle Scholar
  13. 13.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  14. 14.
    Mawhin, J.: NSFCBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1979)Google Scholar
  15. 15.
    Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)CrossRefGoogle Scholar
  16. 16.
    Su, X.W.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Su, X.W.: Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 74, 2844–2852 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Su, X.W., Zhang, S.Q.: Unbounded solutions to a boundary value problem of fractional order on the half-line. Comput. Math. Appl. 61, 1079–1087 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yang, A.J., Ge, W.G.: Positive solutions for boundary value problems of N-dimension nonlinear fractional differential system. Bound. Value Probl., 437–453, (2008) Article ID437453Google Scholar
  20. 20.
    Zhang, Y.H., Bai, Z.B.: Existence of solutions for nonlinear fractional three-point boundary value problems at resonance. J. Appl. Math. Comput. 36, 417–440 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, Y.H., Bai, Z.B., Feng, T.T.: Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl. 61, 1032–1047 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhou, H.C., Kou, C.H., Xie, F.: Existence of solutions for fractional differential equations with multi-point boundary conditions at resonance on a half-line. Electron. J. Qual. Theory Differ. Equ. 27 (2011)Google Scholar
  23. 23.
    Zhao, X.K., Ge, W.G.: Unbounded solutions for fractional bounded value problems on the infinite interval. Acta Appl. Math. 109, 495–505 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2015

Authors and Affiliations

  1. 1.College of Information Science and TechnologyDonghua UniversityShanghaiChina
  2. 2.Academy of Mathematics and Systems ScienceAcademia SinicaBeijingChina
  3. 3.Department of Applied MathematicsDonghua UniversityShanghaiChina

Personalised recommendations