On the geometry of fixed points of self-mappings on S-metric spaces

Year 2020, Volume: 69 Issue: 2, 1184 - 1192, 31.12.2020

Nihal Özgür , Nihal Taş

https://doi.org/10.31801/cfsuasmas.616325

Cited By: 5

Abstract

In this paper, we focus on some geometric properties related to the set Fix(T), the set of all fixed points of a mapping T:X→X, on an S-metric space (X,S). For this purpose, we present the notions of an S-Pata type x₀-mapping and an S-Pata Zamfirescu type x₀-mapping. Using these notions, we propose new solutions to the fixed circle (resp. fixed disc) problem. Also, we give some illustrative examples of our main results. In this paper, we give new solutions to the fixed circle (resp. fixed disc) problem on S-metric spaces. In Section 2, we prove some fixed circle and fixed disc results using different approaches. In Section 3, we give some illustrative examples of our obtained results and deduce some important remarks. In Section 4, we summarize our study and recommend some future works.

Keywords

$S$-metric space, fixed circle, $S$-Pata type $x_{0}$-mapping, $S$-Pata Zamfirescu type $x_{0}$-mapping

Supporting Institution

Balikesir University

Project Number

BAP 2018 /021

Thanks

This work is financially supported by Balikesir University under the Grant no. BAP 2018 /021.

References

• Altun, İ., Aslantaş, M., Şahin, H., Best proximity point results for p-proximal contractions, Acta Math. Hungar., (2020). https://doi.org/10.1007/s10474-020-01036-3. Berinde, V., Comments on some fixed point theorems in metric spaces, Creat. Math. Inform., 27 (1) (2018), 15-20.
• Dosenovic, T., Radenovic, S. and Sedghi, S., Generalized metric spaces: survey, TWMS J. Pure Appl. Math., 9 (1) (2018), 3-17.
• Gupta, A., Cyclic contraction on S-metric space, Int. J. Anal. Appl., 3 (2) (2013), 119-130.
• Hieu, N. T., Ly, N. T., Dung, N. V., A generalization of Ciric quasi-contractions for maps on S-metric spaces, Thai J. Math., 13 (2) (2015), 369-380.
• Jacob, G. K., Khan, M. S., Park, C., Yun, S., On generalized Pata type contractions, Mathematics, 6 (2018), 25.
• Mlaiki, N., α-ψ-contractive mapping on S-metric space, Math. Sci. Lett., 4 (1) (2015), 9-12.
• Mlaiki, N., Çelik, U., Taş, N., Özgür, N. Y., Mukheimer, A., Wardowski type contractions and the fixed-circle problem on S-metric spaces, J. Math., (2018), Article ID 9127486.
• Mlaiki, N., Özgür, N. Y., Taş, N., New fixed-point theorems on an S-metric space via simulation functions, Mathematics, 7(7) (2019), 583.
• Özgür, N. Y. , Taş, N., Some fixed point theorems on S-metric spaces, Mat. Vesnik, 69 (1) (2017), 39-52.
• Özgür, N. Y., Taş, N., Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25), Math. Sci. (Springer), 11 (1) (2017), 7-16.
• Özgür, N. Y., Taş, N., Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc., 42 (4) (2019), 1433-1449.
• Özgür, N. Y., Taş, N., Fixed-circle problem on S-metric spaces with a geometric viewpoint, Facta Universitatis. Series: Mathematics and Informatics, 34 (3) (2019), 459-472.
• Özgür, N. Y., Taş, N., Çelik, U., New fixed-circle results on S-metric spaces, Bull. Math. Anal. Appl., 9 (2) (2017), 10-23.
• Özgür, N. Y., Taş, N., The Picard theorem on S-metric space, Acta Math. Sci., 38B (4) (2018), 1245-1258.
• Özgür, N. Y., Taş, N., A new solution to the Rhoades' open problem with an application, arXiv preprint. arXiv:1910.12304
• Özgür, N. Y., Taş, N., Pata Zamfirescu type fixed-disc results with a proximal application, arXiv preprint. arXiv:1910.12302
• Özgür, N. Y., Fixed-disc results via simulation functions, Turkish J. Math., 43 (6) (2019), 2794-2805.
• Pata, V., A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), 299-305.
• Sedghi, S., Shobe, N., Aliouche, A., A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik, 64 (3) (2012), 258-266.
• Sedghi, S., Dung, N. V., Fixed point theorems on S-metric spaces, Mat. Vesnik, 66 (1) (2014), 113-124.
• Şahin, H., Aslantas, M., Altun, İ., Feng-Liu type approach to best proximity point results for multivalued mappings. J. Fixed Point Theory Appl. 22 (2020), 11.
• Taş, N., Suzuki-Berinde type fixed-point and fixed-circle results on S-metric spaces, J. Linear Topol. Algebra, 7 (3) (2018), 233-244.
• Taş, N., Various types of fixed-point theorems on S-metric spaces, J. BAUN Inst. Sci. Technol., 20 (2) (2018), 211-223.
• Taş, N., Özgür, N. Y., New generalized fixed point results on S_{b}-metric spaces, arXiv preprint. arXiv:1703.01868
• Zamfirescu, T., Fixed point theorems in metric spaces, Arch. Math., 23 (1972), 292-298.