Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces

Gauri Shanker Sao; Swati Verma

doi:doi:10.11648/j.dmath.20200502.11

| Peer-Reviewed

Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces

Gauri Shanker Sao, Swati Verma

Published in International Journal of Discrete Mathematics (Volume 5, Issue 2)

Received: 18 August 2020 Accepted: 12 October 2020 Published: 30 October 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.

Published in	International Journal of Discrete Mathematics (Volume 5, Issue 2)
DOI	10.11648/j.dmath.20200502.11
Page(s)	10-14
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

Keywords

Fuzzy Metric Space, Completeness, Continuity, Contraction Function, Fixed Point, Lebesgue Property, G-complete

References

[1]	Alberto Branciari: A fixed point theorem for mappings satisfying a general con-tractive condition of integral type, International Journal of Mathematics and Mathematical sciences 29 (9) (2002) 531-536.
[2]	Amiller George and Prasanthini Veeramani: On some results in fuzzy metric spaces, fuzzy set and systems 64 (3) (1994) 395-399.
[3]	Mariusz Grabiec: Fixed point in fuzzy metric spaces, fuzzy sets and systems 27 (1988) 385-389.
[4]	Valentin Gregori, Juan Jose Minana, Almanzor Sapena: Completable fuzzy metric spaces, Top. appl. 225 (2017) 103-111.
[5]	Valentin Gregori, Juan Jose Minana, Almanzor Sapena: On Banach contraction principles in fuzzymetric spaces xed point theory 19 (2018) 235-248.
[6]	Valentin Gregory, Salvador Romaguera, Almanzor Sapena: Some questions in fuzzy metric spaces, fuzzysets and system 204 (2012) 71-85.
[7]	Shandong Heipern: Fuzzy mappings and _fixed point theorems, J. Math Anal Appl83 (1981) 566-569.
[8]	Nawab Hussain, Marwan Kutbi and P. Salini: Fixed point in_α-complete metric space with application Hindawi pub cor (2014) 11.
[9]	Ivan Kramosil and Jeremy Michalek: Fuzzy metrics and statistical metric spaces ky-bernetika 11 (5) (1975) 336-344.
[10]	Jose Rodriguez-Lopez, Salvador Romaguera: The Hausdorff Fuzzy metric space on compact sets, Fuzzy sets and Systems 147 (2) (2004) 273-183.
[11]	Almanzor Sapena: A contribution to the study of fuzzy metric spaces, App. Gen. Topology 2 (2001) 63-76.
[12]	Reed Vasuki, Prasanthin. Veeramani: Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy sets and system 135 (2003) 415-417.
[13]	Prasanthin Veeramani: Best approximation in fuzzy metric spaces, J. Fuzzy math. 9 (2001) 75-80.
[14]	Binod Chandra Tripathi, Szemeredi. Paul and Nand Ram Das: A fixed point theorem in a generalized fuzzy metric space Bol. Soc. Paran, Mat, 32 (2) (2014) 221-227.
[15]	Lotfi Aliasker Zadeh: Fuzzy sets, Information and control 8 (3) (1965) 338-353.

Cite This Article

Plain Text BibTeX RIS

APA Style

Gauri Shanker Sao, Swati Verma. (2020). Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. International Journal of Discrete Mathematics, 5(2), 10-14. https://doi.org/10.11648/j.dmath.20200502.11

Copy | Download

ACS Style

Gauri Shanker Sao; Swati Verma. Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. Int. J. Discrete Math. 2020, 5(2), 10-14. doi: 10.11648/j.dmath.20200502.11

Copy | Download

AMA Style

Gauri Shanker Sao, Swati Verma. Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. Int J Discrete Math. 2020;5(2):10-14. doi: 10.11648/j.dmath.20200502.11

Copy | Download

@article{10.11648/j.dmath.20200502.11,
  author = {Gauri Shanker Sao and Swati Verma},
  title = {Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces},
  journal = {International Journal of Discrete Mathematics},
  volume = {5},
  number = {2},
  pages = {10-14},
  doi = {10.11648/j.dmath.20200502.11},
  url = {https://doi.org/10.11648/j.dmath.20200502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20200502.11},
  abstract = {In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces
AU  - Gauri Shanker Sao
AU  - Swati Verma
Y1  - 2020/10/30
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20200502.11
DO  - 10.11648/j.dmath.20200502.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 10
EP  - 14
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20200502.11
AB  - In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.
VL  - 5
IS  - 2
ER  -

Copy | Download

Author Information

Gauri Shanker Sao

Department of Mathematics, Government Edpuganti Raghavendra Rao Post Graduate Science College, Atal Bihari Vajpayee Vishwavidyalaya University, Bilaspur, India
Swati Verma

Department of Mathematics, Omprakash Prakash Jindal University, Raigarh, India

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Gauri Shanker Sao, Swati Verma. (2020). Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. International Journal of Discrete Mathematics, 5(2), 10-14. https://doi.org/10.11648/j.dmath.20200502.11

Copy | Download

ACS Style

Gauri Shanker Sao; Swati Verma. Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. Int. J. Discrete Math. 2020, 5(2), 10-14. doi: 10.11648/j.dmath.20200502.11

Copy | Download

AMA Style

Gauri Shanker Sao, Swati Verma. Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces. Int J Discrete Math. 2020;5(2):10-14. doi: 10.11648/j.dmath.20200502.11

Copy | Download

@article{10.11648/j.dmath.20200502.11,
  author = {Gauri Shanker Sao and Swati Verma},
  title = {Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces},
  journal = {International Journal of Discrete Mathematics},
  volume = {5},
  number = {2},
  pages = {10-14},
  doi = {10.11648/j.dmath.20200502.11},
  url = {https://doi.org/10.11648/j.dmath.20200502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20200502.11},
  abstract = {In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces
AU  - Gauri Shanker Sao
AU  - Swati Verma
Y1  - 2020/10/30
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20200502.11
DO  - 10.11648/j.dmath.20200502.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 10
EP  - 14
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20200502.11
AB  - In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.
VL  - 5
IS  - 2
ER  -

Copy | Download