Skip to main content
SpringerLink
Log in

Existence of positive solutions in the space of Lipschitz functions for a fractional boundary problem with nonlocal boundary condition

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear fractional boundary value problem:

$$\begin{aligned} \left. \begin{array}{ll}D^{\alpha }_{0^+} u(t)+f(t,u(t),(Hu)(t))=0,&{} 0<t<1,\\ u(0)=u'(0)=0,\ \ u'(1)=\beta u(\xi ), \end{array} \right\} \end{aligned}$$

where \(2<\alpha \le 3\), \(0<\xi < 1\), \(0\le \beta \xi ^{\alpha -1}<(\alpha -1)\), H is an operator (not necessarily linear) applying \(\mathcal {C}[0,1]\) into itself and \(D^{\alpha }_{0^+}\) denotes the standard Riemann–Liouville fractional derivative of order \(\alpha \). Our solutions are placed in the space of Lipschitz functions and the main tools used in the study are a sufficient condition for the relative compactness in Hölder spaces and the Schauder fixed point theorem. Moreover, we present one example illustrating our results.

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. Gaul, L., Klein, P., Kemple, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991)

    Article  Google Scholar 

  2. Metzler, R., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)

    Article  Google Scholar 

  3. F. Mainardi, Fractional calculus. In: Carpinteri, A., Mainardi, F. (eds) Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences, vol. 378, Springer, Vienna (1997)

  4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of fractional differential equations, vol. 204. Elsevier, New York (2006)

    Book  Google Scholar 

  5. Podlubny, I.: Fractional Differential Equations, vol. 198. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  6. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon & Breach, New York (1993)

    MATH  Google Scholar 

  7. G. Molica Bisci, V. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its applications, vol. 162, Cambridge University Press, Cambridge (2016)

  8. Graef, J.R., Kong, L., Yang, B.: Positive solutions for a fractional boundary value problem. Appl. Math. Lett. 56, 49–55 (2016)

    Article  MathSciNet  Google Scholar 

  9. Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal. Model. Control 22, 99–114 (2017)

    Article  MathSciNet  Google Scholar 

  10. Zhang, X., Zhong, Q.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 20, 1471–1484 (2017)

    Article  MathSciNet  Google Scholar 

  11. J. Chen, X.H. Tang, Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory, Abstr. Appl. Anal. 2012, (201), https://doi.org/10.1155/2012/648635.

  12. Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)

    Article  MathSciNet  Google Scholar 

  13. Cabada, A., Kisela, T.: Existence of positive periodic solutions of some nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 50, 51–67 (2017)

    Article  MathSciNet  Google Scholar 

  14. Zhang, X., Liu, L., Wu, Y.: The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526–8536 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Wang, Y., Liu, L., Zhang, X., Wu, Y.: Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Appl. Math. Comput. 258, 312–324 (2015)

    MathSciNet  Google Scholar 

  16. Xu, J., Wei, Z., Dong, W.: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 25, 590–593 (2012)

    Article  MathSciNet  Google Scholar 

  17. Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)

    Article  MathSciNet  Google Scholar 

  18. Goodrich, C.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)

    Article  MathSciNet  Google Scholar 

  19. Cabrera, I., Harjani, J., Sadarangani, K.: Existence and uniqueness of solutions for a boundary value problem of fractional type with nonlocal integral boundary conditions in Hölder spaces. Mediterr. J. Math. 15, 98 (2018)

    Article  Google Scholar 

  20. Ersoy, M.T., Furkan, H.: On the existence of the solutions of a Fredholm integral equation with a modified argument in Hölder spaces. Symmetry 10, 522 (2018)

    Article  Google Scholar 

  21. Caballero, J., Darwish, M.A., Sadarangani, K.: Solvability of a quadractic integral equation of Fredholm type in Hölder spaces. Electron. J. Differ. Equ. 31, 1–10 (2014)

    MATH  Google Scholar 

  22. Caballero, J., Darwish, M.A., Sadarangani, K.: Positive solutions in the space of Lipschitz functions with integral boundary conditions. Mediterr. J. Math. 14, 201 (2017)

    Article  Google Scholar 

  23. Banaś, J., Nalepa, R.: On the space of functions with growths tempered by a modulus of continuity and its applications. J. Funct. Spaces Appl. 2013, (2013). https://doi.org/10.1155/2013/820437

  24. Cabrera, I., Sadarangani, K., Samet, B.: Hartman-Wintner-type inequalities for a class of nonlocal fractional boundary value problems. Math. Methods Appl. Sci. 40, 129–136 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors were partially supported by project MTM 2016–79436–P.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017, Las Palmas de Gran Canaria, Spain

    J. Caballero, B. López & K. Sadarangani

Authors
  1. J. Caballero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Sadarangani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. López.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caballero, J., López, B. & Sadarangani, K. Existence of positive solutions in the space of Lipschitz functions for a fractional boundary problem with nonlocal boundary condition. J. Fixed Point Theory Appl. 23, 27 (2021). https://doi.org/10.1007/s11784-021-00864-2

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11784-021-00864-2

Keywords

Mathematics Subject Classification

Search

Navigation