In this study, we give some existence and uniqueness theorems for fixed circles of self-mappings on a metric space with some illustrative examples. Recently, real-valued neural networks with discontinuous activation functions have been a great importance in practice. Hence we give some new results for discontinuity at fixed circle on a metric space.

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of $F_c$ -contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type $F_c$ -contraction to discontinuous self-mappings which have fixed circles.

Recently, S-metric spaces are introduced as a generalization of metric spaces. In this paper, we consider the relationships between of an S-metric space and a metric space, and give an example of an S-metric which does not generate a metric. Then, we introduce new contractive mappings on S-metric spaces and investigate relationships among them by counterexamples. In addition, we obtain new fixed point theorems on S-metric spaces.