Araştırma Makalesi
BibTex RIS Kaynak Göster

A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces

Yıl 2023, Sayı: 43, 73 - 82, 30.06.2023

Abhishikta Das Tarapada Bag

https://doi.org/10.53570/jnt.1277026

Öz

This paper contains the equivalence between tvs-G cone metric and G-metric using a scalarization function $\zeta_p$, defined over a locally convex Hausdorff topological vector space. This function ensures that most studies on the existence and uniqueness of fixed-point theorems on G-metric space and tvs-G cone metric spaces are equivalent. We prove the equivalence between the vector-valued version and scalar-valued version of the fixed-point theorems of those spaces. Moreover, we present that if a real Banach space is considered instead of a locally convex Hausdorff space, then the theorems of this article extend some results of G-cone metric spaces and ensure the correspondence between any G-cone metric space and the G-metric space.

Anahtar Kelimeler

G-metric space G-cone metric space tvs-G cone metric space

Kaynakça

  • S Gahler, \emph{2-Metrische Raume Und Ihre Topologische Struktur}, Mathematische Nachrichten 26 (1-4) (1963) 115--148.
  • S. Czerwik, \emph{Contraction Mappings in b-Metric Spaces}, Acta Mathematica et Informatica Universitatis Ostraviensis 1 (1) (1993) 5--11.
  • S. Czerwik, \emph{Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces}, Atti del Seminario Matematico e Fisico dell'Universita di Modena 46 (1998) 263--276.
  • W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
  • B. C. Dhage, \emph{Generalized Metric Spaces Mappings with Fixed Point}, Bulletin of Calcutta Mathematical Society 84 (1992) 329--336.
  • Z. Mustafa, B. Sims, \emph{A New Approach to Generalized Metric Spaces}, Journal of Nonlinear Convex Analysis 7 (2) (2006) 289--297.
  • S. Sedghi, N. Shobe, A. Aliouche, \emph{A Generalization of Fixed Point Theorems in S-Metric Spaces}, Matematoqki Vesnik 64 (3) (2012) 258--266.
  • H. L. Guang, Z. Xian, \emph{Cone Metric Space and Fixed Point Theorems of Contractive Mapping}, Journal of Mathematical Analysis and Applications 322 (2) (2007) 1468--1476.
  • A. Das, T. Bag, \emph{A Study on Parametric S-metric Spaces}, Communications in Mathematics and Applications 13 (3) (2022) 921--933.
  • A. Das, T. Bag, \emph{A Generalization to Parametric Metric Spaces}, International Journal of Nonlinear Analysis and Application 14 (1) (2023), 229–244.
  • G. Rano, T. Bag, S.K. Samanta, \emph{Fuzzy Metric Space and Generating Space of Quasi-Metric Family}, Annals of Fuzzy Mathematics and Informatics 11 (2016) 183--195.
  • I. Beg, M. Abbas, T. Nazir, \emph{Generalized Cone Metric Spaces}, The Journal of Nonlinear Sciences and Its Applications 3 (1) (2010) 21--31.
  • G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer Berlin, Heidelberg, 2005.
  • W. S. Du, \emph{On Some Nonlinear Problems Induced by an Abstract Maximal Element Principle}, Journal of Mathematical Analysis and Applications 347 (2) (2008) 391--399.
  • C. Gerth, P. Weidner, \emph{Nonconvex Separation Theorems and Some Applications in Vector Optimization}, Journal of Optimization Theory and Applications 67 (2) (1990) 297--320.
  • A. Göpfert, C. Tammer, C. Z. Linescu, \emph{On the Vectorial Ekeland's Variational Principle and Minimal Points in Product Spaces}, Nonlinear Analysis: Theory, Methods and Applications 39 (7) (2000) 909--922.
  • A. Göpfert, C. Tammer, H. Riahi, C. Z. Linescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.
  • W. Du, \emph{A Note on Cone Metric Fixed Point Theory and Its Equivalence}, Nonlinear Analysis: Theory, Methods and Applications 72 (5) (2010) 2259--2261.
  • I. J. Lin, C. M. Chen, C. H. Chen, T. Y. Cheng, \emph{A Note on tvs-G-Cone Metric Fixed Point Theory}, Journal of Applied Mathematics 2012 (2012) Article ID 407071 10 pages.
  • Z. Mustafa, H. Obiedat, F. Awawdeh, \emph{Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces}, Fixed Point Theory and Applications 2008 (2008) Article Number 189870 12 pages.

Yıl 2023, Sayı: 43, 73 - 82, 30.06.2023

Abhishikta Das Tarapada Bag

https://doi.org/10.53570/jnt.1277026

Öz

Kaynakça

  • S Gahler, \emph{2-Metrische Raume Und Ihre Topologische Struktur}, Mathematische Nachrichten 26 (1-4) (1963) 115--148.
  • S. Czerwik, \emph{Contraction Mappings in b-Metric Spaces}, Acta Mathematica et Informatica Universitatis Ostraviensis 1 (1) (1993) 5--11.
  • S. Czerwik, \emph{Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces}, Atti del Seminario Matematico e Fisico dell'Universita di Modena 46 (1998) 263--276.
  • W. Kirk, N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
  • B. C. Dhage, \emph{Generalized Metric Spaces Mappings with Fixed Point}, Bulletin of Calcutta Mathematical Society 84 (1992) 329--336.
  • Z. Mustafa, B. Sims, \emph{A New Approach to Generalized Metric Spaces}, Journal of Nonlinear Convex Analysis 7 (2) (2006) 289--297.
  • S. Sedghi, N. Shobe, A. Aliouche, \emph{A Generalization of Fixed Point Theorems in S-Metric Spaces}, Matematoqki Vesnik 64 (3) (2012) 258--266.
  • H. L. Guang, Z. Xian, \emph{Cone Metric Space and Fixed Point Theorems of Contractive Mapping}, Journal of Mathematical Analysis and Applications 322 (2) (2007) 1468--1476.
  • A. Das, T. Bag, \emph{A Study on Parametric S-metric Spaces}, Communications in Mathematics and Applications 13 (3) (2022) 921--933.
  • A. Das, T. Bag, \emph{A Generalization to Parametric Metric Spaces}, International Journal of Nonlinear Analysis and Application 14 (1) (2023), 229–244.
  • G. Rano, T. Bag, S.K. Samanta, \emph{Fuzzy Metric Space and Generating Space of Quasi-Metric Family}, Annals of Fuzzy Mathematics and Informatics 11 (2016) 183--195.
  • I. Beg, M. Abbas, T. Nazir, \emph{Generalized Cone Metric Spaces}, The Journal of Nonlinear Sciences and Its Applications 3 (1) (2010) 21--31.
  • G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer Berlin, Heidelberg, 2005.
  • W. S. Du, \emph{On Some Nonlinear Problems Induced by an Abstract Maximal Element Principle}, Journal of Mathematical Analysis and Applications 347 (2) (2008) 391--399.
  • C. Gerth, P. Weidner, \emph{Nonconvex Separation Theorems and Some Applications in Vector Optimization}, Journal of Optimization Theory and Applications 67 (2) (1990) 297--320.
  • A. Göpfert, C. Tammer, C. Z. Linescu, \emph{On the Vectorial Ekeland's Variational Principle and Minimal Points in Product Spaces}, Nonlinear Analysis: Theory, Methods and Applications 39 (7) (2000) 909--922.
  • A. Göpfert, C. Tammer, H. Riahi, C. Z. Linescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.
  • W. Du, \emph{A Note on Cone Metric Fixed Point Theory and Its Equivalence}, Nonlinear Analysis: Theory, Methods and Applications 72 (5) (2010) 2259--2261.
  • I. J. Lin, C. M. Chen, C. H. Chen, T. Y. Cheng, \emph{A Note on tvs-G-Cone Metric Fixed Point Theory}, Journal of Applied Mathematics 2012 (2012) Article ID 407071 10 pages.
  • Z. Mustafa, H. Obiedat, F. Awawdeh, \emph{Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces}, Fixed Point Theory and Applications 2008 (2008) Article Number 189870 12 pages.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Abhishikta Das 0000-0002-2860-424X

Tarapada Bag 0000-0002-8834-7097

Yayımlanma Tarihi 30 Haziran 2023
Gönderilme Tarihi 4 Nisan 2023
Yayımlandığı Sayı Yıl 2023 Sayı: 43

Kaynak Göster

APA Das, A., & Bag, T. (2023). A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. Journal of New Theory(43), 73-82. https://doi.org/10.53570/jnt.1277026
AMA Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. Haziran 2023;(43):73-82. doi:10.53570/jnt.1277026
Chicago Das, Abhishikta, ve Tarapada Bag. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory, sy. 43 (Haziran 2023): 73-82. https://doi.org/10.53570/jnt.1277026.
EndNote Das A, Bag T (01 Haziran 2023) A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. Journal of New Theory 43 73–82.
IEEE A. Das ve T. Bag, “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”, JNT, sy. 43, ss. 73–82, Haziran 2023, doi: 10.53570/jnt.1277026.
ISNAD Das, Abhishikta - Bag, Tarapada. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory 43 (Haziran 2023), 73-82. https://doi.org/10.53570/jnt.1277026.
JAMA Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. 2023;:73–82.
MLA Das, Abhishikta ve Tarapada Bag. “A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces”. Journal of New Theory, sy. 43, 2023, ss. 73-82, doi:10.53570/jnt.1277026.
Vancouver Das A, Bag T. A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces. JNT. 2023(43):73-82.
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