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Modular Cone Metric Spaces

Received: 31 December 2013     Published: 30 January 2014
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Abstract

In this paper a generalization of a modular metric which is also a generalization of cone metric, is introduced and some of its topological properties are studied. Next, a fixed point theorem in this space is proved and finally by an example, it is proved that the fixed point result of the paper "Ch. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mapping in modular metric spaces, Fixed Point Theory Appl. (2011)" is not true.

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 6)
DOI 10.11648/j.pamj.20130206.14
Page(s) 191-196
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Ordered Spaces, Modular Cone Metric, Fixed Point Theorem

References
[1] H. Cakalli, A. Sonmez, C. Genc, On an equivalence of topological vector space valued cone metric spaces and metric spaces.Appl. Math. Lett. 25 (2012) 429-433.
[2] G.Y. Chen, X.X. Huang,X.Q. Yang,Vector Optimization.Springer-Verlag, Berlin, Heidelberg. (2005).
[3] V. Chistyakov, A fixed point theorem for contractions in modular metric spaces.arXiv: 1112.5561v1. (2011)
[4] V. Chistyakov, Metric modulars and their application. Dokl.Math. 73 (1)(2006) 32-35.
[5] V. Chistyakov, Modular metric spaces generated by F-modulars, Folia Math‎.‎14‎ (2008)‎ 3-25.
[6] V. Chistyakov, Modular metric spaces, I: Basic concepts. Nonlinear Anal.72 (2010) 1-14.
[7] K. Deimling,Nonlinear Functional Analysis. Springer-Verlag, Berlin. (1985).
[8] W.S. Du,A note on cone metric fixed point theory and it’s equivalence.Nonlinear Anal. 72. (2010)2259-2261.
[9] S. Jankovic, Z.Golubovic,S. Radenovic,Compatible and weakly compatible mappings in cone metric spaces.Math. Comp.Modelling. 52(2010) 1728-1738.
[10] Ch. Mongkolkeha, W. Sintunavarat,P. Kumam, Fixed point theorems for contraction mapping in modular metric spaces.Fixed Point Theory Appl. (2011)
[11] H. Nakano, Modulared Semi-Ordered Linear Spaces.In Tokyo Math Book Ser, vol. 1. Maruzen Co., Tokyo. (1950).
[12] B. Rzepecki,On fixed point theorems of Maia type.Publications de l’InstitutMathématique,28 (42)(1980) 179-186.
[13] J. Musielak, W. Orlicz, On modular spaces.Studia Math. 18 (1959)591-597.
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  • APA Style

    Saeedeh Shamsi Gamchi, Asadollah Niknam. (2014). Modular Cone Metric Spaces. Pure and Applied Mathematics Journal, 2(6), 191-196. https://doi.org/10.11648/j.pamj.20130206.14

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    ACS Style

    Saeedeh Shamsi Gamchi; Asadollah Niknam. Modular Cone Metric Spaces. Pure Appl. Math. J. 2014, 2(6), 191-196. doi: 10.11648/j.pamj.20130206.14

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    AMA Style

    Saeedeh Shamsi Gamchi, Asadollah Niknam. Modular Cone Metric Spaces. Pure Appl Math J. 2014;2(6):191-196. doi: 10.11648/j.pamj.20130206.14

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  • @article{10.11648/j.pamj.20130206.14,
      author = {Saeedeh Shamsi Gamchi and Asadollah Niknam},
      title = {Modular Cone Metric Spaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {6},
      pages = {191-196},
      doi = {10.11648/j.pamj.20130206.14},
      url = {https://doi.org/10.11648/j.pamj.20130206.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130206.14},
      abstract = {In this paper a generalization of a modular metric which is also a generalization of cone metric, is introduced and some of its topological properties are studied. Next, a fixed point theorem in this space is proved and finally by an example, it is proved that the fixed point result of the paper "Ch. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mapping in modular metric spaces, Fixed Point Theory Appl. (2011)" is not true.},
     year = {2014}
    }
    

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  • TY  - JOUR
    T1  - Modular Cone Metric Spaces
    AU  - Saeedeh Shamsi Gamchi
    AU  - Asadollah Niknam
    Y1  - 2014/01/30
    PY  - 2014
    N1  - https://doi.org/10.11648/j.pamj.20130206.14
    DO  - 10.11648/j.pamj.20130206.14
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 191
    EP  - 196
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20130206.14
    AB  - In this paper a generalization of a modular metric which is also a generalization of cone metric, is introduced and some of its topological properties are studied. Next, a fixed point theorem in this space is proved and finally by an example, it is proved that the fixed point result of the paper "Ch. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mapping in modular metric spaces, Fixed Point Theory Appl. (2011)" is not true.
    VL  - 2
    IS  - 6
    ER  - 

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Author Information
  • Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran

  • Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran

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