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A new approach to fixed point theorems in compact 2-generalized metric spaces

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Abstract

In this paper, we present a new approach to solve fixed point theorems in compact 2-generalized metric spaces. The approach is to show that the restriction of a given 2-generalized metric to the closure of a certain orbit is a metric. By using the new approach and constructing a counterexample, we completely answer the open question about the role of the continuity of the 2-generalized metric posed in Kadelburg and Radenovic (TWMS J Pure Appl Math 5:3–13, 2014).

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References

  1. Branciari, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 57(1–2), 31–37 (2000)

    MathSciNet  Google Scholar 

  2. Das, P., Dey, L.K.: Fixed point of contractive mappings in generalized metric spaces. Math. Slovaca 59(4), 499–504 (2009)

    Article  MathSciNet  Google Scholar 

  3. Dung, N.V., Hang, V.T.L.: The metrization of \(\nu \)-generalized metric spaces. Rev. R. Acad. Cienc. Exact. Fís. Nat. Ser. A Mat. RACSAM 112(4), 1295–1303 (2018)

    Article  MathSciNet  Google Scholar 

  4. Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)

    Article  MathSciNet  Google Scholar 

  5. Erhan, I.M., Karapinar, E., Sekulic, T.: Fixed points of \((\psi,\phi )\) contractions on rectangular metric spaces. Fixed Point Theory Appl. 2012(138), 1–12 (2012)

    MathSciNet  Google Scholar 

  6. Franklin, S.P.: Spaces in which sequences suffice. Fund. Math. 57, 107–115 (1965)

    Article  MathSciNet  Google Scholar 

  7. Kadelburg, Z., Radenovic, S.: On generalized metric spaces: a survey. TWMS J. Pure Appl. Math. 5(1), 3–13 (2014)

    MathSciNet  Google Scholar 

  8. Kikina, L., Kikina, K.: On fixed point of a Ljubomir Ciric quasi-contraction mapping in generalized metric spaces. Publ. Math. Debrecen 83(3), 1–6 (2013)

    Article  MathSciNet  Google Scholar 

  9. Kirk, W., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Cham (2014)

    Book  Google Scholar 

  10. Kumam, P., Dung, N.V.: Remarks on generalized metric spaces in the Branciari’s sense. Sarajevo J. Math. 10(2), 209–219 (2014)

    Article  MathSciNet  Google Scholar 

  11. Samet, B.: Discussion on “A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces” by A. Brancia. Publ. Math. Debrecen 76(4), 493–494 (2010)

    MathSciNet  Google Scholar 

  12. Suzuki, T.: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71, 5313–5317 (2009)

    Article  MathSciNet  Google Scholar 

  13. Suzuki, T.: Generalized metric spaces do not have the compatible topology. Abstr. Appl. Anal. 1–5, 2014 (2014)

    MathSciNet  Google Scholar 

  14. Suzuki, T.: Some metrization problem on \(\nu \)-generalized metric spaces. Rev. R. Acad. Cienc. Exact. Fís. Nat. Ser. A Mat. RACSAM 113(2), 1267–1278 (2019)

    Article  MathSciNet  Google Scholar 

  15. Suzuki, T., Alamri, B., Khan, L.A.: Some notes on fixed point theorems in \(\nu \)-generalized metric spaces. Bull. Kyushu Inst. Tech. Pure Appl. Math. 62, 15–23 (2015)

    MathSciNet  Google Scholar 

  16. Suzuki, T., Alamri, B., Kikkawa, M.: Edelstein’s fixed point theorem in generalized metric spaces. J. Nonlinear Convex Anal. 16, 2301–2309 (2015)

    MathSciNet  Google Scholar 

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Acknowledgements

The author is greatly indebted to anonymous reviewers for their helpful comments concerning the paper.

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  1. Department of Mathematics Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap Province, Vietnam

    Nguyen Van Dung

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  1. Nguyen Van Dung
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Correspondence to Nguyen Van Dung.

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Van Dung, N. A new approach to fixed point theorems in compact 2-generalized metric spaces. RACSAM 115, 30 (2021). https://doi.org/10.1007/s13398-020-00972-7

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