| Peer-Reviewed

Some Fixed Point Theorems on b2 - Metric Spaces

Received: 25 August 2023     Accepted: 14 September 2023     Published: 27 September 2023
Views:       Downloads:
Abstract

In this study, we generalize both b-metric spaces and 2-metric spaces into a new class of generalized metric spaces that we call b2-metric spaces. Then, under various contractive circumstances in partially ordered spaces, we demonstrate a few fixed point theorems in b2-metric space. Many Mathematician gave the concept of b2 -metric spaces as a generalization of 2-metric space. The purpose of this research article to established some results of 2-metric space proved by the Arun Garg et al. in b2 -metric spaces and prove new results.

Published in Pure and Applied Mathematics Journal (Volume 12, Issue 4)
DOI 10.11648/j.pamj.20231204.12
Page(s) 72-78
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), 2023. Published by Science Publishing Group

Keywords

Fixed Point, b - Metric Space, 2-Metric Space, Partial Order Set, Generalized Contractive Mappings

References
[1] Czerwik, S: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5-11 (1993).
[2] Czerwik, S: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Modena 46, 263-276 (1998).
[3] Gähler, VS: 2-metrische Räume und ihre topologische Struktur. Math. Nachr. 26, 115-118 (1963).
[4] Hussain, N, Parvaneh, V, Roshan, JR, Kadelburg, Z: Fixed points of cyclic weakly (ψ, ϕ, L, A, B)- contractive mappings in ordered b-metric spaces with applications. Fixed Point Theory Appl. 2013, Article ID 256 (2013).
[5] Dung, NV, Le Hang, VT: Fixed point theorems for weak C-contractions in partially ordered 2-metric spaces. Fixed Point Theory Appl. 2013, Article ID 161 (2013).
[6] Naidu, SVR, Prasad, JR: Fixed point theorems in 2-metric spaces. Indian J. Pure Appl. Math. 17 (8), 974-993 (1986).
[7] Aliouche, A, Simpson, C: Fixed points and lines in 2-metric spaces. Adv. Math. 229, 668- 690 (2012).
[8] Deshpande, B, Chouhan, S: Common fixed point theorems for hybrid pairs of mappings with some weaker conditions in 2-metric spaces. Fasc. Math. 46, 37-55 (2011).
[9] Freese, RW, Cho, YJ, Kim, SS: Strictly 2-convex linear 2-normed spaces. J. Korean Math. Soc. 29 (2), 391-400 (1992).
[10] Iseki, K: Fixed point theorems in 2-metric spaces. Math. Semin. Notes 3, 133-136 (1975).
[11] Iseki, K: Mathematics on 2-normed spaces. Bull. Korean Math. Soc. 13 (2), 127-135 (1976).
[12] Lahiri, BK, Das, P, Dey, LK: Cantor’s theorem in 2-metric spaces and its applications tofixed point problems. Taiwan. J. Math. 15, 337-352 (2011).
[13] Lai, SN, Singh, AK: An analogue of Banach’s contraction principle in 2-metric spaces. Bull. Aust. Math. Soc. 18, 137-143 (1978).
[14] Popa, V, Imdad, M, Ali, J: Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces. Bull. Malays. Math. Soc. 33, 105-120 (2010).
[15] Ahmed, MA: A common fixed point theorem for expansive mappings in 2-metric spaces and its application. Chaos Solitons Fractals 42 (5), 2914-2920 (2009).
[16] Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973) 17.
[17] Ðukic, D, Kadelburg, Z, Radenovic, S: Fixed points of Geraghty-type mappings in various generalized metric spaces. Abstr. Appl. Anal. 2011, Article ID 561245 (2011).
[18] Berinde, V: On the approximation of fixed points of weak contractive mappings. Carpath. J. Math. 19, 7-22 (2003).
[19] Berinde, V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9, 43-53 (2004).
[20] Berinde, V: General contractive fixed point theorems for ´ Ciric-type almost contraction in metric spaces. Carpath. J. Math. 24, 10-19 (2008).
[21] Berinde, V: Some remarks on a fixed point theorem for ´ Ciric-type almost contractions. Carpath. J. Math. 25, 157-162 (2009).
[22] Babu, GVR, Sandhya, ML, Kameswari, MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 24, 8-12 (2008).
[23] Roshan, JR, Parvaneh, V, Sedghi, S, Shobkolaei, N, Shatanawi, W: Common fixed points of almost generalized (ψ, ϕ)s -contractive mappings in ordered b-metric spaces. Fixed Point Theory Appl. 2013, Article ID 159 (2013).
[24] Ciric, L, Abbas, M, Saadati, R, Hussain, N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 217, 5784-5789 (2011).
[25] Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc.30, 1-9 (1984).
[26] Fathollahi, S, Hussain, N, Khan, LA: Fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces. Fixed Point Theory Appl. 2014, Article ID 6 (2014).
[27] Mustafa et al., b2-Metric spaces and some fixed point theorems. Fixed Point Theory and Applications2014 2014: 144.
[28] A. Garg, Z. K. Ansari and R. Shrivastava, Some common fixed point theorems in 2-Metric Space, South Asian J Math, 2011, 1 (3), 106-110.
[29] Fréchet, M. M. Sur quelques points du calcul fonctionnel. Rend. Circ. Matem. Palermo 22, 1–72 (1906). https://doi.org/10.1007/BF03018603
Cite This Article
  • APA Style

    Bheem Singh Patel, Zaheer Kareem Ansari, Dharmendra Kumar, Arun Garg. (2023). Some Fixed Point Theorems on b2 - Metric Spaces. Pure and Applied Mathematics Journal, 12(4), 72-78. https://doi.org/10.11648/j.pamj.20231204.12

    Copy | Download

    ACS Style

    Bheem Singh Patel; Zaheer Kareem Ansari; Dharmendra Kumar; Arun Garg. Some Fixed Point Theorems on b2 - Metric Spaces. Pure Appl. Math. J. 2023, 12(4), 72-78. doi: 10.11648/j.pamj.20231204.12

    Copy | Download

    AMA Style

    Bheem Singh Patel, Zaheer Kareem Ansari, Dharmendra Kumar, Arun Garg. Some Fixed Point Theorems on b2 - Metric Spaces. Pure Appl Math J. 2023;12(4):72-78. doi: 10.11648/j.pamj.20231204.12

    Copy | Download

  • @article{10.11648/j.pamj.20231204.12,
      author = {Bheem Singh Patel and Zaheer Kareem Ansari and Dharmendra Kumar and Arun Garg},
      title = {Some Fixed Point Theorems on b2 - Metric Spaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {12},
      number = {4},
      pages = {72-78},
      doi = {10.11648/j.pamj.20231204.12},
      url = {https://doi.org/10.11648/j.pamj.20231204.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20231204.12},
      abstract = {In this study, we generalize both b-metric spaces and 2-metric spaces into a new class of generalized metric spaces that we call b2-metric spaces. Then, under various contractive circumstances in partially ordered spaces, we demonstrate a few fixed point theorems in b2-metric space. Many Mathematician gave the concept of b2 -metric spaces as a generalization of 2-metric space. The purpose of this research article to established some results of 2-metric space proved by the Arun Garg et al. in b2 -metric spaces and prove new results.},
     year = {2023}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Some Fixed Point Theorems on b2 - Metric Spaces
    AU  - Bheem Singh Patel
    AU  - Zaheer Kareem Ansari
    AU  - Dharmendra Kumar
    AU  - Arun Garg
    Y1  - 2023/09/27
    PY  - 2023
    N1  - https://doi.org/10.11648/j.pamj.20231204.12
    DO  - 10.11648/j.pamj.20231204.12
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 72
    EP  - 78
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20231204.12
    AB  - In this study, we generalize both b-metric spaces and 2-metric spaces into a new class of generalized metric spaces that we call b2-metric spaces. Then, under various contractive circumstances in partially ordered spaces, we demonstrate a few fixed point theorems in b2-metric space. Many Mathematician gave the concept of b2 -metric spaces as a generalization of 2-metric space. The purpose of this research article to established some results of 2-metric space proved by the Arun Garg et al. in b2 -metric spaces and prove new results.
    VL  - 12
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Madhyanchal Professional University, Bhopal, India

  • Department of Mathematics, JSS Academy of Technical Education, Noida, India

  • Department of Mathematics Satyawati College (Evening), University of Delhi, Delhi, India

  • Department of Mathematics, Madhyanchal Professional University, Bhopal, India

  • Sections