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