New Fixed-Figure Results on Metric Spaces

Abstract

Geometric properties of the non-unique fixed points of a self-mapping F are investigated on a metric space in the framework of the fixed-figure problem. Mainly, we introduce new types of self-mappings of which fixed point set contains a certain geometric figure (e.g. an Apollonius circle, a Cassini curve, a circle, an ellipse or a hyperbola). This geometric figure is called a fixed figure (a fixed Apollonius circle, a fixed Cassini curve and so on) of the corresponding self-mapping. Then, using the classical techniques of fixed point theory and appropriate auxiliary numbers, we give new fixed-figure results. These kind geometric results are important in terms of applications in the cases of non-unique fixed points. © 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.