Existence of fixed points in quasi metric spaces

Year 2020, Volume: 69 Issue: 1, 266 - 275, 30.06.2020

Sumit Chandok , Saurabh Manro

https://doi.org/10.31801/cfsuasmas.532381

Cited By: 3

Abstract

In this paper, we obtain some new fixed point theorems for two pairs of weakly compatible mappings in the setting of quasi metric spaces using $C$- class functions. Several interesting corollaries are also deduced. The results obtained extend various well known results of the literature in the setting of quasi metric space. We also construct an example to demonstrate

the usability of the proved results.

Keywords

$C$-class function, quasi-metric space, property E.A, common (E.A) property, weak compatibility

References

• Aamri, M. and El Moutawakil, D., Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181--188.
• Agarwal, R.P., Karapinar, E., O'Regan, D. and Roldan-Lopez-de-Hierro, A. F., Fixed Point Theory in Metric Type Spaces, Springer, 2015.
• Abbas, M. and Rhoades, B.E., Common fixed point results for noncommuting mapping without continuity in generalized metric spaces, Appl. Math. Comput., 215 (2009) 262--269.
• Abbas, M. Nazir, T. and Vetro, P., Common fixed point results for three maps in G-metric spaces, Filomat, 25 (2011) 1--17. Aydi, H., Shatanawi, W. and Vetro, C., On generalized weakly G-contraction mapping in G-metric spaces, Comput. Math. Appl., 62 (2011) 4222--4229.
• Chandok, S. and Ansari, A.H., Some results on generalized nonlinear contractive mappings, Comm. Opt. Theory, 2017 (2017), Article Id 27.
• Chandok, S., Mustafa, Z. and Postolache, M., Coupled common fixed point theorems for mixed g-monotone mappings in partially ordered G-metric spaces, Univ. Politehnica Sci. Bull. Ser. A: Appl. Math. Phys., 75(4)(2013), 13-26.
• Chandok, S., Sintunavarat, W. and Kumam, P., Some coupled common fixed points for a pair of mappings in partially ordered G-metric spaces, Math. Sci. (2013), 7-24.
• Ciric, L., Some Recent Results in Metrical Fixed Point Theory, University of Belgrade, Belgrad, 2003.
• Dhage, B.C., Generalized metric space and mapping with fixed point, Bull. Calcutta Math. Soc., 84 (1992) 329--336.
• Di Bari, C. and Vetro, P., Common fixed points in generalized metric spaces, Appl. Math. Comput., 218 (2012) 7322--7325.
• Gahler, S., 2-Metrische raume and ihre topologische struktur, Math. Nachr., 26 (1963) 115--148.
• Gopal, D., Imdad, M. and Vetro, C., Common fixed point theorems for mappings satisfying common property (E.A.) in symmetric spaces, Filomat, 25 (2011) 59--78.
• Huang, H., Deng, G. and Radenovic, S., Fixed point theorems for C-class functions in b-metric spaces and strong applications, J. Nonlinear Sci. Appl., 10 (2017), 5853-5868.
• Jungck, G., Commuting mappings and fixed points, Amer. Math. Monthly, 16, (1976) 261--263.
• Khan, M.S., Swaleh, M. and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Australian Math. Soc., 30(1), (1984), 1-9.
• Kirk, W. and Shahzad, N., Fixed Point Theory in Distance Spaces, Springer International Publishing Switzerland 2014.
• Kunzi, H.P.A., Nonsymmetric topology, in: Proc. Colloquium on topology, 1993, Szekszard, Hungary, Colloq. Math. Soc. Janos Bolyai Math. Studies, 4 (1995), 303-338.
• Lui, W., Wu, J. and Li, Z., Common fixed points of single-valued and multivalued maps, Int. J. Math. Math. Sci., 19(2005), 3045--3055.
• Manro, S., Bhatia, S. S. and Kumar, S., Expansion mapping theorems in G-metric spaces, Internat. J. Contem. Math. Sci., 5 (2010), 2529--2535.
• Mustafa, Z. and Sims, B., A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289--297.
• Mustafa, Z. and Sims, B., Some remarks concerning D-metric spaces, Proc. Int. Conf. on Fixed Point Theory and Applications, Valencia (Spain), July (2003), 189--198.
• Saadati, R., Vaezpour, S. M., Vetro, P. and Rhoades, B. E., Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling, 52 (2010) 797--801.
• Shatanawi, W., Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 181650, 9 pages doi:10.1155/2010/181650.
• Todorcevic, Vesna, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer Nature Switzerland AG 2019.
• Todorcevic, Vesna, Subharmonic behavior and quasiconformal mappings, Anal. Math. Phys. https://doi.org/10.1007/s13324-019-00308-8