_{c}-Contractions and the Fixed-Circle Problem

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

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Fixed point theory has become the focus of many researchers lately (see [

What is (are) the necessary and sufficient condition(s) for a self-mapping

Various fixed-circle theorems have been obtained using different approaches on metric and some generalized metric spaces (see [

Motivated by the above studies, we present some new fixed-circle theorems using the ideas given in [

Let

At first, we recall some necessary definitions and a theorem related to fixed circle. A circle and a disc are defined on a metric space as follows, respectively:

Now we define new contractive conditions and give some fixed-circle results.

Assume that

Let

Now we show that

Assume that

Using Proposition 2, we rewrite the condition (

Using this inequality, we obtain the following fixed-circle result.

Let

In Definition 5, if we choose

From the similar arguments used in the proof of Proposition 2, the proof follows easily since

Using Proposition 3, we rewrite the condition (

Using this inequality, we obtain the following fixed-circle result.

It can be easily seen since

In Definition 5, if we choose

From the similar arguments used in the proof of Proposition 2, it can be easily proved.☐

By the similar arguments used in the proof of Theorem 3 and Definition 7, it can be easily checked.☐

Now we give two illustrative examples of our obtained results.

In the following example, we see that the converse statements of Theorems 2–5 are not always true.

In this section, we give some examples of discontinuous functions and obtain a discontinuity result related to fixed circle.

Consider the above examples, we give the following theorem.

From Theorem 2, we see that

We have presented new generalized fixed-circle results using new types of contractive conditions on metric spaces. The obtained results can be also considered as fixed-disc results. By means of some known techniques which are used to obtain some fixed-point results, we have generated useful fixed-circle theorems. As we have seen in the last section, our main results can be applied to other research areas.

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

This research received no external funding.

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

The authors declare no conflicts of interest.

_{c}-contractive and

_{c}-expanding mappings on metric spaces