Skip to main content
SpringerLink
Log in

Unique Solutions for Fractional q-Difference Boundary Value Problems Via a Fixed Point Method

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, by applying the cone theory in ordered Banach spaces associated with the characters of increasing \(\varphi -(h,e)\)-concave operators, we investigate the existence and uniqueness of nontrivial solutions for a nonlinear fractional q-difference equation boundary value problem. The main results show that we can construct an iterative scheme approximating the unique nontrivial solution. Relying on an example, we show the efficiency and applicability of the main result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahmad, B., Ntouyas, S.K., Tariboon, J.: Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities, Trends in Abstract and Applied Analysis, vol. 4. World Scientific Publishing Co. Pte. Ltd., Hackensack (2016)

    Book  MATH  Google Scholar 

  3. Ahmad, B., Ntouyas, S.K., Tariboon, J., Alsaedi, A., Alsulami, H.H.: Impulsive fractional \(q\)-integro-difference equations with separated boundary conditions. Appl. Math. Comput. 281, 199–213 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 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. Roumanie 59, 119–134 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Ahmad, B., Ntouyas, S.K., Purnaras, I.K.: Existence results for nonlocal boundary value problems of nonliner fractional q-difference equations. Adv. Differ. Equ. 140, 1–15 (2012)

    Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Al-Salam, W.A.: Some fractional \(q\)-integrals and \(q\)-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  8. Annaby, M.H., Mansour, Z.S.: \(q\)-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  9. Chen, Q., Debbouche, A., Luo, Z., Wang, J.: Impulsive fractional differential equations with Riemann–Liouville derivative and iterative learning control. Chaos Solitons Fractals 102, 111–118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferreira, R.A.C.: Nontrivial solutions for fractional \(q\)-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 70, 1–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferreira, R.A.C.: Positive solutions for a class of boundary value problems with fractional \(q\)-differences. Comput. Math. Appl. 61, 367–373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khodabakhshi, N., Vaezpour, S.M.: Existence and uniqueness of positive solution for a class of boundary value problems with fractional \(q\)-differences. J. Nonlinear Conv. Anal. 16, 375–384 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Li, X.H., Han, Z.L., Li, X.C.: Boundary value problems of fractional \(q\)-difference Schr\(\acute{o}\)inger equations. Appl. Math. Lett. 46, 100–105 (2015)

    Article  MathSciNet  Google Scholar 

  14. Li, X., Han, Z., Sun, S.: Existence of positive solutions of nonlinear fractional \(q\)-difference equation with parameter. Adv. Differ. Equ. 1, 205–215 (2013)

    MathSciNet  Google Scholar 

  15. Li, X., Han, Z., Sun, S., Zhao, P.: Existence of solutions for fractional \(q\)-difference equation with mixed nonlinear boundary conditions. Adv. Differ. Equ. 326, 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Li, M.M., Wang, J.R.: Finite time stability of fractional delay differential equations. Appl. Math. Lett. 64, 170–176 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Y.: Existence of positive solutions for boundary value problem of nonlinear fractional \(q\)-difference equation. Appl. Math. 4, 1450–1454 (2013)

    Article  Google Scholar 

  18. Miao, F., Liang, S.: Uniqueness of positive solutions for fractional \(q\)-difference boundary-value problems with \(p\)-Laplacian operator. Electron. J. Differ. Equ. 174, 1–11 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Rajković, P.M., Marinković, S.D., Stanković, M.S.: Fractional integrals and derivatives in \(q\)-calculus. Appl. Anal. Disc. Math. 1, 311–323 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shu, X., Shi, Y.: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465–476 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Shu, X., Xu, F., Shi, Y.: S-asymptotically \(\omega \)-positive periodic solutions for a class of neutral fractional differential equations. Appl. Math. Comput. 270, 768–776 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Wang, J., Fečkan, M., Zhou, Y.: Center stable manifold for planar fractional damped equations. Appl. Math. Comput. 296, 257–269 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Yang, W.: Positive solution for fractional \(q\)-difference boundary value problems with \(\Phi \)-Laplacian operator. Bull. Malays. Math. Sci. Soc. 36, 1195–1203 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Yang, W.: Positive solutions for nonlinear semipositone fractional \(q\)-difference system with coupled integral boundary conditions. Appl. Math. Comput. 244, 702–725 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Zhai, C., Ren, J.: Positive and negative solutions of a boundary value problem for a fractional \(q\)-difference equation. Adv. Differ. Equ. 82, 1–13 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Zhai, C., Wang, F.: Properties of positive solutions for the operator equation \(Ax=\lambda x\) and applications to fractional differential equations with integral boundary conditions. Adv. Differ. Equ. 366, 1–10 (2015)

    MathSciNet  Google Scholar 

  27. Zhai, C., Wang, L.: \(\varphi -(h, e)\)-concave operators and applications. J. Math. Anal. Appl. 454, 571–584 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhai, C., Wang, L.: Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions. Nonlinear Anal. Model. Control 22(4), 566–577 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper was supported financially by the Youth Science Foundation of China (11201272), Shanxi Province Science Foundation (2015011005) and 131 Talents Project of Shanxi Province (2015).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Jing Ren & Chengbo Zhai

Authors
  1. Jing Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chengbo Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chengbo Zhai.

Additional information

Communicated by Shangjiang Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ren, J., Zhai, C. Unique Solutions for Fractional q-Difference Boundary Value Problems Via a Fixed Point Method. Bull. Malays. Math. Sci. Soc. 42, 1507–1521 (2019). https://doi.org/10.1007/s40840-017-0560-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0560-2

Keywords

Mathematics Subject Classification

Search

Navigation