Hardy-Rogers Type Mappings for Fuzzy Metric Space

Mohit Kumar; Ritu Arora; Ajay Kumar

doi:doi:10.11648/j.pamj.20190806.11

| Peer-Reviewed |

Hardy-Rogers Type Mappings for Fuzzy Metric Space

Mohit Kumar, Ritu Arora, Ajay Kumar

Received: 23 July 2019 Accepted: 26 September 2019 Published: 24 December 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy set by Zadeh, where the concept of uncertainty has been introduced in the theory of sets in a non probabilistic manner. The several researchers were conducting the generalization of the concept of fuzzy sets. The present research paper focuses on the existence of fixed points in fuzzy metric space. Hardy-Rogers is to establish a fixed point theorem for three maps of a complete metric space. The contractive condition is generalized and the commuting condition of Jungck is replaced by the concept of weakly commuting. The three Hardy-Rogers type mappings are extended in fuzzy metric space and also extend to generalize non-expansive mapping define over a compact fuzzy metric space. The contractive condition is generalization of Hardy-Rogers and the commuting condition of Jungck is replace by the concept of weakly commuting. Our results deals with mappings satisfying a condition weaker than commutativity in complete fuzzy metric space and is the generalization in complete fuzzy metric space of Hardy-Rogers type mappings in complete metric space. We also provide some illustrative example to support our result. We apply also our main results to derive unique and common fixed point for contractive mappings.

DOI	10.11648/j.pamj.20190806.11
Published in	Pure and Applied Mathematics Journal (Volume 8, Issue 6, December 2019)
Page(s)	93-99
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

Weakly Commuting Mapping, Asymptotically Regular Mapping, Compact Fuzzy Metric Space, Fixed Point

References

[1]	George A. and Veeramani P., On some results in fuzzy metric spaces, Fuzzy Sets and System, 64 (1994), 395-399.
[2]	Rhoades B. E., Sessa S., Khan M. S. and Khan M. D., Some fixed point theorems for Hardy-Rogers type mappings, Internat. J. Math. &Math. Sci., 7 (1) (1984), 75-87.
[3]	Singh B. and Jain S., Semi-compatibility, compatibility and fixed point theorems in fuzzy metric space, Journal of the Chungcheong Mathematical Society, 18 (1) (2005).
[4]	Jungck G., Commuting mappings and fixed points, Amer. Math. Monthly, 83 (1976), 261-263.
[5]	Hardy G. E. and Rogers T. D., A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201-206.
[6]	Engl H. W., Weak convergence of asymptotically regular sequence for non expansive mapping and connections with certain cheishef-centers, Nonlinear Analysis TMA, 1 (5) (1977), 495-501.
[7]	Kramosil I. and Michalek J., Fuzzy metric and statistical metric spaces, Ky-bernetica, 11 (1975), 326-334.
[8]	Zadeh L. A., Fuzzy sets, Inform and Control, 89 (1965), 338-353.
[9]	Grabiec M., Fixed points in fuzzy metric spaces, Fuzzy Sets and System, 27 (1988), 385-389.
[10]	Mishra M. K. and Ojha D. B., An application of fixed point theorems in fuzzy metric spaces, IJAEST, 1 (2), 123-129.
[11]	Kannan R., Some results on fixed points IV, Fund. Math. LXXIV, (1972), 181-187.
[12]	Sessa S., On a weak commutativity condition in fixed point considerations, Publ. Inst. Math., 32 (46) (1982), 149-153.
[13]	Schweizer and Sklar, Statistical metric spaces, Pac. J. Math., 10 (1960), 385-389.
[14]	Mishra S. N., Mishra N. and Singh S. L., Common fixed point of maps in fuzzy metric space, Internat. J. Math. &Math. Sci., 17 (1994), 253-258.
[15]	Shen Y., Qiu D., and Chen W., On convergence of fixed points in fuzzy metric spaces, Abstract and Applied Analysis, 2013, Article ID 135202, 6 pages.
[16]	Soni R., Common fixed point theorems in fuzzy metric space with application, ArXiv: 1807.02387v1 [math. FA] (2018).

Cite This Article

Plain Text BibTeX RIS

APA Style

Mohit Kumar, Ritu Arora, Ajay Kumar. (2019). Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure and Applied Mathematics Journal, 8(6), 93-99. https://doi.org/10.11648/j.pamj.20190806.11

Copy | Download

ACS Style

Mohit Kumar; Ritu Arora; Ajay Kumar. Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure Appl. Math. J. 2019, 8(6), 93-99. doi: 10.11648/j.pamj.20190806.11

Copy | Download

AMA Style

Mohit Kumar, Ritu Arora, Ajay Kumar. Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure Appl Math J. 2019;8(6):93-99. doi: 10.11648/j.pamj.20190806.11

Copy | Download

@article{10.11648/j.pamj.20190806.11,
  author = {Mohit Kumar and Ritu Arora and Ajay Kumar},
  title = {Hardy-Rogers Type Mappings for Fuzzy Metric Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {8},
  number = {6},
  pages = {93-99},
  doi = {10.11648/j.pamj.20190806.11},
  url = {https://doi.org/10.11648/j.pamj.20190806.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190806.11},
  abstract = {The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy set by Zadeh, where the concept of uncertainty has been introduced in the theory of sets in a non probabilistic manner. The several researchers were conducting the generalization of the concept of fuzzy sets. The present research paper focuses on the existence of fixed points in fuzzy metric space. Hardy-Rogers is to establish a fixed point theorem for three maps of a complete metric space. The contractive condition is generalized and the commuting condition of Jungck is replaced by the concept of weakly commuting. The three Hardy-Rogers type mappings are extended in fuzzy metric space and also extend to generalize non-expansive mapping define over a compact fuzzy metric space. The contractive condition is generalization of Hardy-Rogers and the commuting condition of Jungck is replace by the concept of weakly commuting. Our results deals with mappings satisfying a condition weaker than commutativity in complete fuzzy metric space and is the generalization in complete fuzzy metric space of Hardy-Rogers type mappings in complete metric space. We also provide some illustrative example to support our result. We apply also our main results to derive unique and common fixed point for contractive mappings.},
 year = {2019}
}

Copy | Download

TY - JOUR
T1 - Hardy-Rogers Type Mappings for Fuzzy Metric Space
AU - Mohit Kumar
AU - Ritu Arora
AU - Ajay Kumar
Y1 - 2019/12/24
PY - 2019
N1 - https://doi.org/10.11648/j.pamj.20190806.11
DO - 10.11648/j.pamj.20190806.11
T2 - Pure and Applied Mathematics Journal
JF - Pure and Applied Mathematics Journal
JO - Pure and Applied Mathematics Journal
SP - 93
EP - 99
PB - Science Publishing Group
SN - 2326-9812
UR - https://doi.org/10.11648/j.pamj.20190806.11
AB - The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy set by Zadeh, where the concept of uncertainty has been introduced in the theory of sets in a non probabilistic manner. The several researchers were conducting the generalization of the concept of fuzzy sets. The present research paper focuses on the existence of fixed points in fuzzy metric space. Hardy-Rogers is to establish a fixed point theorem for three maps of a complete metric space. The contractive condition is generalized and the commuting condition of Jungck is replaced by the concept of weakly commuting. The three Hardy-Rogers type mappings are extended in fuzzy metric space and also extend to generalize non-expansive mapping define over a compact fuzzy metric space. The contractive condition is generalization of Hardy-Rogers and the commuting condition of Jungck is replace by the concept of weakly commuting. Our results deals with mappings satisfying a condition weaker than commutativity in complete fuzzy metric space and is the generalization in complete fuzzy metric space of Hardy-Rogers type mappings in complete metric space. We also provide some illustrative example to support our result. We apply also our main results to derive unique and common fixed point for contractive mappings.
VL - 8
IS - 6
ER -

Copy | Download

Author Information

Mohit Kumar

Department of Mathematics, Uttaranchal College of Applied and Life Sciences, Uttaranchal University, Dehradun, India
Ritu Arora

Department of Mathematics, Kanya Gurukul Campus Gurukula Kangri Vishwavidyalaya, Haridwar, India
Ajay Kumar

Department of Mathematics, Shaheed Smt. Hansa Dhanai Govt, Degree College Agrora, Tehri Garhwal, India

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Mohit Kumar, Ritu Arora, Ajay Kumar. (2019). Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure and Applied Mathematics Journal, 8(6), 93-99. https://doi.org/10.11648/j.pamj.20190806.11

Copy | Download

ACS Style

Mohit Kumar; Ritu Arora; Ajay Kumar. Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure Appl. Math. J. 2019, 8(6), 93-99. doi: 10.11648/j.pamj.20190806.11

Copy | Download

AMA Style

Mohit Kumar, Ritu Arora, Ajay Kumar. Hardy-Rogers Type Mappings for Fuzzy Metric Space. Pure Appl Math J. 2019;8(6):93-99. doi: 10.11648/j.pamj.20190806.11

Copy | Download

@article{10.11648/j.pamj.20190806.11,
  author = {Mohit Kumar and Ritu Arora and Ajay Kumar},
  title = {Hardy-Rogers Type Mappings for Fuzzy Metric Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {8},
  number = {6},
  pages = {93-99},
  doi = {10.11648/j.pamj.20190806.11},
  url = {https://doi.org/10.11648/j.pamj.20190806.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190806.11},
  abstract = {The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy set by Zadeh, where the concept of uncertainty has been introduced in the theory of sets in a non probabilistic manner. The several researchers were conducting the generalization of the concept of fuzzy sets. The present research paper focuses on the existence of fixed points in fuzzy metric space. Hardy-Rogers is to establish a fixed point theorem for three maps of a complete metric space. The contractive condition is generalized and the commuting condition of Jungck is replaced by the concept of weakly commuting. The three Hardy-Rogers type mappings are extended in fuzzy metric space and also extend to generalize non-expansive mapping define over a compact fuzzy metric space. The contractive condition is generalization of Hardy-Rogers and the commuting condition of Jungck is replace by the concept of weakly commuting. Our results deals with mappings satisfying a condition weaker than commutativity in complete fuzzy metric space and is the generalization in complete fuzzy metric space of Hardy-Rogers type mappings in complete metric space. We also provide some illustrative example to support our result. We apply also our main results to derive unique and common fixed point for contractive mappings.},
 year = {2019}
}

Copy | Download