Common Fixed Point Theorems For F-Contractions In $G$-Metric Spaces Using Compatible Mappings

Abstract

Using a mapping $F:\mathbb{R_{+}}\rightarrow \mathbb{R}$, Wardowski [1] introduce a new type of contraction called $F$-contraction and prove a new fixed point theorem concerning $F$-contraction. In the present article, we prove some fixed point theorems with helping compatible maps for type $1$ and type $2$ $F$-contraction in complete $G$-metric spaces.

Keywords

• Fixed point,$G$-metric space,$F$-contraction,Compatible mapping

References

1. [1] D. Wardowski, Fixed Point of a new type of contactive mappings in complete metric spaces, Fixed Point Theory and Applications, 94, 6 pp., (2012), DOI:10.1186/1687-1812-2012-94.
2. [2] S. Banach, Surles operations dansles ensembles abstracits et leur application aux equations integrales, Fund Math., 3, (1922), 133-181.
3. [3] Z. Mustafa and B. Sims, A new approach to a generalized metric spaces, J. Nonlinear Convex Anal., 7, (2), (2006), 289-297.
4. [4] M. Abbas, A. R. Khan and T. Nazir, Coupled common fixed point results in two generalized metric spaces, Applied Mathematics and Computation, Vol:217 No.13 pp. (2011), 6328-6336.
5. [5] M. Abbas, T. Nazir and D. Doric, Common fixed point of mappings satisfying (E.A) property in generalized metric spaces, Applied Mathematics and Computation, Vol:218, No.14 pp., (2012), 7665-7670.
6. [6] Z. Mustafa, W. Shatanawi and M. Bataineh, Existence of fixed point results in G-metric spaces, Int. J. Math. and Math. Sci., page 10, (2009), DOI:10.1155/2009/283028.
7. [7] H.S. Ding and E. Karapınar, A note on some coupled fixed point theorems on G-metric space, Journal of Inequalities and Applications, Vol:2012, article 170, (2012), DOI: 10.1186/1029-242X-2012-170.
8. [8] G. Jungck, Commuting mappings and fixed point, Amer. Math. Monthly, Vol:83, No.4 (1976), 261-263.

9. [9] G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9, (1986), 771-779.
10. [10] M. Kumar, Compatible Maps in G-Metric spaces, Int. Journal of Math. Anaysis, Vol:6, No.29 (2012), 1415-1421.
11. [11] M. Cosentino, and P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat Vol:28, No.4 pp., (2014), 715-722.
12. [12] J. Ahmad, A. Rawashdeh and A. Azam, New fixed point theorems for generalized F-contractions in complete metric space, Fixed Point Theory and Applications, 80, (2015), DOI: 10.1186/s13663-015-0333-2.
13. [13] R. Batra and S. Vashistha, Fixed points of an F-contraction on metric spaces with a graph, Int. J. Comput. Math., Vol:91, (12), (2014), 2483-2490.
14. [14] S. Chong, J. K. Kim, L. Wong, and J. Tang, Existence of fixed points for generalized F-contractions and generalized F-suzuki contractions in metric spaces, Global Journal of pure and applied Mathematics, ISSN 0973-1768, Vol:12, No.6 (2016), pp. 4867-4882.
15. [15] G. Mınak, A. Helvacı, and I. Altun, C´ iric´ type generalized F-contractions on complete metric spaces and fixed point results, Filomat, 28, (6), (2014), 1143-1151.
16. [16] H. Piri and P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 210, (2014), DOI:10.1186/1687-1812-2014,210.
17. [17] D. Wardowski and N. Van Dung, Fixed points of F -weak contractions on complete metric spaces, Demonstr. Math., Vol:XLVII No.1, (2014), 146-155.
18. [18] R. Batra, S. Vashistha and R. Kumar, A coincidence point theorem for F-contractions on metric spaces equipped with an altered distance, J. Math.Comput. Sci., 4, (5), (2014), 826-833.