Orthogonal sets: The axiom of choice and proof of a fixed point theorem

Abstract

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L p space.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Agarwal R. P., El-Gebeily M. A., O’Regan D.: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 1–8 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Aguirre Salazar L., Reich S.: A remark on weakly contractive mappings. J. Nonlinear Convex Anal. 16, 767–773 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ya. I. Alber and S. Guerre-Delabrere, Principles of weakly contractive maps in Hilbert spaces. In: New Results in Operator Theory and Its Applications, Oper. Theory Adv. Appl. 98, Birkhäuser, Basel, 1997, 7–22.

  4. 4.

    Amini-Harandi A., Emami H.: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, 2238–2242 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Banach S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  6. 6.

    Berinde V.: Approximating fixed points of weak \({\varphi}\)-contractions. Fixed Point Theory 4, 131–142 (2003)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bhaskar T. G., Lakshmikantham V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Boyd D. W., Wong J. S.: On nonlinear contractions. Proc. Amer. Math. Soc. 20, 458–464 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ćirić Lj. B.: A generalization of Banach’s contraction principle. Proc. Amer. Math. Soc. 45, 267–273 (1974)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    M. Eshaghi Gordji, H. Baghani and G. H. Kim, Common fixed point theorems for \({(\psi,\varphi)}\)-weak nonlinear contraction in partially ordered sets. Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-62, 12 pages.

  11. 11.

    M. Eshaghi Gordji, M. Ramezani, M. De La Sen and Y. J. Cho, On orthogonal sets and Banach fixed point theorem. Fixed Point Theory, to appear.

  12. 12.

    Harjani J., Sadarangani K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Lakshmikantham V., Ćirić Lj. B.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Nieto J. J., Pouso R. L., Rodríguez-López R.: Fixed point theorems in ordered abstract sets. Proc. Amer. Math. Soc. 135, 2505–2517 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Nieto J. J., Rodríguez-López R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Nieto J. J., Rodríguez-López R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. (Engl. Ser.) 23, 2205–2212 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Ran A. C. M., Reurings M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132, 1435–1443 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Reich S.: Some remarks concerning contraction mappings. Canad. Math. Bull. 14, 121–124 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Rhoades B. E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-94, 6 pages.

  21. 21.

    Zhang Q., Song Y.: Fixed point theory for generalized \({\varphi}\)-weak contractions. Appl. Math. Lett. 22, 75–78 (2009)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Madjid Eshaghi Gordji.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baghani, H., Eshaghi Gordji, M. & Ramezani, M. Orthogonal sets: The axiom of choice and proof of a fixed point theorem. J. Fixed Point Theory Appl. 18, 465–477 (2016). https://doi.org/10.1007/s11784-016-0297-9

Download citation

Mathematics Subject Classification

  • 34A12
  • 65R10
  • 65R20

Keywords

  • The axiom of choice
  • fixed point
  • orthogonal set
  • Picard operator