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On two questions of A. Petruşel and G. Petruşel in b-metric fixed point theory

Article

Abstract

In this paper, we study two questions of A. Petruşel and G. Petruşel in b-metric fixed point theory [12]. The main results of the paper are as follows.
  1. (1)

    Using the b-metric metrization theorem [9], fixed point results in the setting of b-metric spaces proved in [10, 11, 12] and some others may be seen as consequences of Ran–Reurings fixed point theorem in the classical metric spaces [13, Theorem 2.1]. This gives a partial answer to the question in [12, Remark 3.(2)].

     
  2. (2)

    Using the product space of two JS-metric spaces, main results of [12] and some others in the setting of b-metric spaces can be extended to the setting of JS-spaces. This answers the question in [12, Open question on page 1809].

     

Keywords

b-metric JS-metric fixed point metrization 

Mathematics Subject Classification

47H10 54H25 

Notes

Acknowledgements

The authors are greatly indebted to anonymous reviewers for their helpful comments to revise the paper.

References

  1. 1.
    Czerwik, S.: Nonlinear set-valued contraction mappings in \(b\)-metric spaces. Atti. Sem. Math. Fis. Univ. Modena 46, 263–276 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Dung, N. V., An, T. V., Hang V. T. L.: Remarks on Frink’s metrization technique and applications. Fixed Point Theory, 1–22, (2017). (accepted paper) Google Scholar
  3. 3.
    Dung, N.V., Hang, V.T.L.: On relaxations of contraction constants and Caristi’s theorem in \(b\)-metric spaces. J. Fixed Point Theory Appl. 18(2), 267–284 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dung, N .V., Hang, V .T .L.: Remarks on cyclic contractions in \(b\)-metric spaces and applications to integral equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(1), 247–255 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Jachymski, J., Matkowski, J., Światkowski, T.: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1(2), 125–134 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Jleli, M., Samet, B.: A generalized metric space and related fixed point theorems. Fixed Point Theory Appl. 2015(61), 1–14 (2015)MathSciNetMATHGoogle Scholar
  7. 7.
    Nieto, J.J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22(3), 223–239 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Miculescu, R., Mihail, A.: New fixed point theorems for set-valued contractions in \(b\)-metric spaces. J. Fixed Point Theory Appl. 19(3), 2153–2163 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Paluszyński, M., Stempak, K.: On quasi-metric and metric spaces. Proc. Amer. Math. Soc. 137(12), 4307–4312 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Petruşel, A., Petruşel, G., Samet, B., Yao, J.-C.: Coupled fixed point theorems for symmetric multi-valued contractions in \(b\)-metric space with applications to systems of integral inclusions. J. Nonlinear Convex Anal. 17, 1265–1282 (2016)MathSciNetMATHGoogle Scholar
  11. 11.
    Petruşel, A., Petruşel, G., Samet, B., Yao, J.-C.: Coupled fixed point theorems for symmetric contractions in \(b\)-metric spaces with applications to operator equation systems. Fixed Point Theory 17(2), 457–475 (2016)MathSciNetMATHGoogle Scholar
  12. 12.
    Petruşel, A., Petruşel, G.: A study of a general system of operator equations in \(b\)-metric spaces via the vector approach in fixed point theory. J. Fixed Point Theory Appl. 19(3), 1793–1814 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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(5), 1435–1444 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Senapati, T., Dey, L.K.: A new approach of couple fixed point results on \(JS\)-metric spaces. 1–17 (2016). arXiv:1606.05970

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Nonlinear Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Journal of ScienceDong Thap UniversityCao Lanh CityVietnam
  4. 4.Faculty of Mathematics Teacher EducationDong Thap UniversityCao Lanh CityVietnam

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