In this study, we generalize both b-metric spaces and 2-metric spaces into a new class of generalized metric spaces that we call b_{2}-metric spaces. Then, under various contractive circumstances in partially ordered spaces, we demonstrate a few fixed point theorems in b_{2}-metric space. Many Mathematician gave the concept of b_{2} -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 b_{2} -metric spaces and prove new results.
DOI | 10.11648/j.pamj.20231204.12 |
Published in | Pure and Applied Mathematics Journal (Volume 12, Issue 4, August 2023) |
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), 2024. Published by Science Publishing Group |
Fixed Point, b - Metric Space, 2-Metric Space, Partial Order Set, Generalized Contractive Mappings
[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., b_{2}-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 |
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
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
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
@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} }
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 -