# $\mathfrak{F}$-Bipolar metric spaces and fixed point theorems with applications

Volume 26, Issue 2, pp 184--195
Publication Date: November 05, 2021 Submission Date: August 03, 2021 Revision Date: September 06, 2021 Accteptance Date: September 25, 2021
• 227 Views

### Authors

S. Rawat - Department of Mathematics, H.N.B. Garhwal University, Uttarakhand-246174, India. R. C. Dimri - Department of Mathematics, H.N.B. Garhwal University, Uttarakhand-246174, India. A. Bartwal - Department of Mathematics, H.N.B. Garhwal University, Uttarakhand-246174, India.

### Abstract

In this paper, we propose a new generalization of metric spaces by the unification of two novel notions, namely $\mathfrak{F}$-metric spaces and bipolar metric spaces, under the name $\mathfrak{F}$-bipolar metric spaces. Further, in this newly generalized notion we provide a binary topology and prove some fixed point results. As applications of our result, we prove the existence and uniqueness of solution of integral equation and the existence of a unique solution in homotopy theory. We also give some non-trivial examples to vindicate our claims. Our fixed point results extend several results in the existing literature.

### Share and Cite

##### ISRP Style

S. Rawat, R. C. Dimri, A. Bartwal, $\mathfrak{F}$-Bipolar metric spaces and fixed point theorems with applications, Journal of Mathematics and Computer Science, 26 (2022), no. 2, 184--195

##### AMA Style

Rawat S., Dimri R. C., Bartwal A., $\mathfrak{F}$-Bipolar metric spaces and fixed point theorems with applications. J Math Comput SCI-JM. (2022); 26(2):184--195

##### Chicago/Turabian Style

Rawat, S., Dimri, R. C., Bartwal, A.. "$\mathfrak{F}$-Bipolar metric spaces and fixed point theorems with applications." Journal of Mathematics and Computer Science, 26, no. 2 (2022): 184--195

### Keywords

• $\mathfrak{F}$-Bipolar metric spaces
• fixed point
• completeness

•  54E50
•  54A20
•  54H25

### References

• [1] I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ul'yanovsk. Gos. Ped. Inst., 30 (1989), 26--37

• [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133--181

• [3] A. Bartwal, R. C. Dimri, G. Prasad, Some fixed point theorems in fuzzy bipolar metric spaces, J. Nonlinear Sci. Appl., 13 (2020), 196--204

• [4] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31--37

• [5] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5--11

• [6] M. Frechet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, 22 (1906), 1--72

• [7] M. Jleli, B. Samet, On a new generalization of Metric Spaces, J. Fixed Point Theory Appl., 20 (2018), 20 pages

• [8] S. N. Jothi, P. Thangavelu, Topology between two sets, J. Math. Sci. Comput. Appl., 1 (2011), 95--107

• [9] G. N. V. Kishore, R. P. Agarwal, B. S. Rao, R. V. N. S. Rao, Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications, Fixed Point Theory Appl., 2018 (2018), 13 pages

• [10] S. G. Matthews, Partial metric topology, The New York Academy of Sciences. Ann. N. Y. Acad. Sci., 1994 (1994), 183--197

• [11] A. Mutlu, U. Gürdal, Bipolar metric spaces and some fixed point theorems, J. Nonlinear Sci. Appl., 9 (2016), 5362--5373

• [12] A. Mutlu, K. Özkan, U. Gürdal, Coupled fixed point theorems on bipolar metric spaces, Eur. J. Pure Appl. Math., 10 (2017), 655--667

• [13] A. Mutlu, K. Özkan, U. Gürdal, Fixed Point Theorems For Multivalued Mappings On Bipolar Metric Spaces, Fixed Point Theory, 21 (2020), 271--280

• [14] A. Mutlu, K. Özkan, U. Gürdal, Locally and Weakly Contractive Principle in Bipolar Metric Spaces, TWMS J. Appl. Eng. Math., 10 (2020), 379--388

• [15] K. Özkan, U. Gürdal, The Fixed Point Theorem and Characterization of Bipolar Metric Completeness, Konuralp J. Math., 8 (2020), 137--143

• [16] K.Özkan, U. Gürdal, A. Mutlu, A Generalization of Amini-Harandi's Fixed Point Theorem with an Application to Nonlinear Mapping Theory, Fixed Point Theory, 21 (2020), 707--714

• [17] K.Özkan, U. Gürdal, A. Mutlu, Caristi's and Downing-Kirk's Fixed Point Theorems on Bipolar Metric Spaces, Fixed Point Theory, 22 (2021), 785--794

• [18] S. Shukla, Partial Rectangular Metric Spaces and Fixed Point Theorems, Sci. World J., 2014 (2014), 7 pages