Positive Solutions for a Class of Fractional Boundary Value Problem with q-Derivatives

  • Furi GuoEmail author
  • Shugui Kang


This work is concerned with the existence and uniqueness of positive solutions to a nonlinear boundary value problem with fractional q-derivative:
$$\begin{aligned}&D_q^\alpha u(t)+f(t,u(t),u(t))+g(t,u(t))=0, \quad 0<t<1 ,\\&u(0)=0 ,\quad u(1)=\beta u(\eta ), \end{aligned}$$
where \(D_q^\alpha \) is the fractional q-derivative of Riemann–Liouville type, \(0<q<1\)\(1<\alpha \le 2\)\(0<\eta <1,\)  \(0<\beta \eta ^{\alpha -1}<1.\) By virtue of fixed-point theorems for mixed monotone operator, some existence and uniqueness results of positive solutions are obtained. An example to illustrate our results is given.


Positive solution mixed monotone operator fractional q-difference equation boundary value problem 

Mathematics Subject Classification

34B18 33D05 39A13 



This work is supported by the National Nature Science Foundation under Grant (11271235).


  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Acdemic Press, New York (1974)zbMATHGoogle Scholar
  2. 2.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calcus and Fractional Differential Equation. Wiley, New York (1993)Google Scholar
  3. 3.
    Glockle, W.G., Nonnenmacher, T.F.: A fractional calcus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)CrossRefGoogle Scholar
  4. 4.
    Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engnineering. Academic Press, New York (1999)Google Scholar
  5. 5.
    Field, C., Joshi, N., Nijhoff, F.: q-Difference equations of KdV type and Chazy-type second-degree difference equations. J. Phys. Math. Theor. 41, 1–13 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Abreu, L.: Sampling theory associated with \(q\)-difference equations of the Sturm–Liouville type. J. Phys. A 38(48), 10311–10319 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jackson, F.: On \(q\)-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)CrossRefGoogle Scholar
  8. 8.
    Jackson, F.: On \(q\)-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)zbMATHGoogle Scholar
  9. 9.
    Rajković, P., Marinković, S., Stankovicć, M.: Fractional integrals and derivatives in \(q\)-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)CrossRefGoogle Scholar
  11. 11.
    Al-Salam, W.A.: Some fractional \(q\)-integrals and \(q\)-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1966)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Agarwal, R.P.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ma, K., Sun, S., Han, Z.: Existence of solutions of boundary value problems for singular fractional \(q\)-difference equations. J. Appl. Math. Comput. 54(1–2), 23–40 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferreira, R.A.C.: Nontrivial solutions for fractional \(q\)-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ferreira, R.A.C.: Positive solutions for a class of boundary value problems with fractional \(q\)-differences. Comput. Math. Appl. 61(2), 367–373 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    El-Shahed, M., Al-Askar, F.: Positive solution for boundary value problem of nonlinear fractional q-difference equation. ISRN Math. Anal. 2011, 1–12 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Darzi, R., Agheli, B.: Existence results to positive solution of fractional BVP with \(q\)-derivatives. J. Appl. Math. Comput. 55(1–2), 353–367 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ahmad, B., Ntouyas, S.K., Purnaras, I.K.: Existence results for nonlocal boundary value problems of nonliner fractional \(q\)-difference equations. Adv. Differ. Equ. 2012, 140 (2012)CrossRefGoogle Scholar
  19. 19.
    Graef, J.R., Kong, L.J.: Positive solutions for a class of higher order boundary value problems with fractional \(q\)-derivatives. Comput. Math. Appl. 218, 9682–9689 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Almeida, R., Martins, N.: Existence results for fractional \(q\)-difference equations of order \(\alpha \in ]2,3[\) with three-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 19, 1675–1685 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)zbMATHGoogle Scholar
  22. 22.
    Yang, W.: Positive solution for fractional \(q\)-difference boundary value problems with \(\Phi \)-laplacian operator. Bull. Malays. Math. Sci. Soc. 36, 1195–1203 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ahmad, B., Etemad, S., Ettefagh, M., Rezapour, S.: On the existence of solutions for fractional \(q\)-difference inclusions with \(q\)-antiperiodic boundary conditions. Bull. Math. Soc. Sci. Math. Roum. 59, 119–134 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Agarwal, R.P., Ahmad, B., Alsaedi, A., Al-Hutami, H.: Existence theory for q-antiperiodic boundary value problems of sequential q-fractional integro-differential equations. Abstract. Appl. Anal. 2014, Article ID 174156 (2014)Google Scholar
  25. 25.
    Wang, J.R., Zhang, Y.R.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39, 85–90 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhai, C.B., Yan, W.P., Yang, C.: A sum operator method for the existence and uniqueness of positive solution to Riemann–Liouville fractional differential equation boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 18, 858–866 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhai, C.B., Ren, J.: The unique solution for a fractional \(q\)-difference equation with three-point boundary conditions. Indagat. Math. New Ser. 29, 948–961 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ren, J., Zhai, C.B.: A fractional \(q\)-difference equation with integral boundary conditions and comparison theorem. Int. J. Nonlinear Sci. Numer. Simul. 18(7–8), 575–583 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhai, C.B., Ren, J.: Positive and negative solutions of a boundary value problem for a fractional q-difference equation. Adv. Differ. Equ. 2017, Article ID 82 (2017)Google Scholar
  30. 30.
    Zhai, C.B., Hao, M.R.: Fixed point theorems for mixed monotone operattors with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. 75, 2542–2551 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhai, C.B., Yang, C., Zhang, X.Q.: Positive solutions for nonlinear operator equations and several classes of applications. Math. Z. 266, 43–63 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Neamaty, A., Yadollahzadeh, M., Darzi, R.: Existence of solution for a nonlocal boundary value problem with fractional \(q\)-derivatives. J. Fract. Calc. Appl. 6(2), 18–27 (2015)MathSciNetGoogle Scholar
  33. 33.
    Guo, D., Laksmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11(5), 623–632 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceShanxi Datong UniversityDatongPeople’s Republic of China

Personalised recommendations