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 Fc-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 Fc-contraction to discontinuous self-mappings which have fixed circles.

fixed circle Ćirić type Fc-contraction Hardy–Rogers type Fc-contraction Reich type Fc-contraction Chatterjea type Fc-contraction Classification primary 54H25 secondary 47H10
1. Introduction

Fixed point theory has become the focus of many researchers lately (see [1,2,3,4]). One of the main important results of fixed point theory is when we show that a self mapping on a metric space under some specific conditions has a unique fixed point. In some cases when we do not have uniqueness of the fixed point, such a map fixes a circle which we call a fixed circle, the fixed-circle problem arises naturally in practice. There exist a lot of examples of self-mappings that map a circle onto itself and fixes all the points of the circle, whereas the circle is not fixed by the self-mapping. For example, let C,d be the usual metric space and C0,1 be the unit circle. Let us consider the self-mappings T1:CC and T2:CC defined by T1z=1z¯ifz00ifz=0 and T2z=1zifz00ifz=0, for all zC where z¯ is the complex conjugate of the complex number z. Then, we have Ti(C0,1)=C0,1 (i=1,2), but C0,1 is the fixed circle of T1 while it is not the fixed circle of T2 (especially T2 fixes only two points of the unit circle). Thus, a natural question arises as follows:

What is (are) the necessary and sufficient condition(s) for a self-mapping T that make a given circle as the fixed circle of T? Therefore, it is important to investigate new fixed-circle results.

Various fixed-circle theorems have been obtained using different approaches on metric and some generalized metric spaces (see [5,6,7,8,9] for more details). For example, in , fixed-circle results were proved using the Caristi’s inequality on metric spaces. In , it was given a fixed-circle theorem for a self-mapping that maps a given circle onto itself. In , it was extended known fixed-circle results in many directions and introduced a new notion called as an Fc-contraction. In addition, some generalized fixed-circle theorems were investigated on an S-metric space (see [6,7]).

Motivated by the above studies, we present some new fixed-circle theorems using the ideas given in [10,11]. In , it was proved some fixed-point results using an F-contraction of the Hardy-Rogers-type and in , it was obtained a fixed-point theorem using a Ćirić type generalized F-contraction. We generate some fixed-circle results from these types of contractions using Wardowski’s technique. For some fixed-point results obtained by this technique, one can consult the references [10,11,12,13]. In Section 2, we define the notions of a Ćirić type Fc-contraction, Hardy-Rogers type Fc-contraction, Reich type Fc-contraction and Chatterjea type Fc-contraction. Using these concepts, we prove some results related to the fixed-circle problem. In Section 3, we present an application of our obtained results to a discontinuous self-mapping that has a fixed circle.

2. New Fixed-Circle Results via Some Classical Techniques

Let (X,d) be a metric space and T:XX be a self-mapping in the whole paper. Now we investigate some new fixed-circle theorems using the ideas of some classical fixed-point theorems.

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:Cu0,r=uX:d(u,u0)=r and Du0,r=uX:d(u,u0)r.

(). Let Cu0,r be a circle on X. If Tu=u for every uCu0,r then the circle Cu0,r is said to be a fixed circle of T.

(). Let F be the family of all functions F:(0,)R such that

(F1) F is strictly increasing,

(F2) For each sequence αn in 0, the following holds lim n α n = 0 if and only if lim n F ( α n ) = ,

(F3) There exists k(0,1) such that limα0+αkF(α)=0.

(). If there exist t>0, FF and u0X such that for all uX the following holds: d ( u , T u ) > 0 t + F ( d ( u , T u ) ) F ( d ( u 0 , u ) ) , then T is said to be an Fc-contraction on X.

(). Let T be an Fc-contractive self-mapping with u0X and r = min d ( u , T u ) : u T u .

Then Cu0,r is a fixed circle of T. Especially, T fixes every circle Cu0,ρ where ρ<r.

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

If there exist t>0, FF and u0X such that for all uX the following holds: d ( u , T u ) > 0 t + F ( d ( u , T u ) ) F ( m ( u , u 0 ) ) , where m ( u , v ) = max d ( u , v ) , d ( u , T u ) , d ( v , T v ) , 1 2 d ( u , T v ) + d ( v , T u ) , then T is said to be a Ćirić type Fc-contraction on X.

If T is a Ćirić type Fc-contraction with u0X then we have Tu0=u0.

Assume that Tu0u0. From the definition of a Ćirić type Fc-contraction, we get d(u0,Tu0)>0t+F(d(u0,Tu0))F(m(u0,u0))=Fmaxd(u0,u0),d(u0,Tu0),d(u0,Tu0),12d(u0,Tu0)+d(u0,Tu0)=F(d(u0,Tu0)), a contradiction because of t>0. Then we have Tu0=u0.☐

Let T be a Ćirić type Fc-contraction with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then Cu0,r is a fixed circle of T. Especially, T fixes every circle Cu0,ρ with ρ<r.

Let uCu0,r. Since d(u0,Tu)=r, the self-mapping T maps Cu0,r into (or onto) itself. If Tuu, by the definition of r, we have d(u,Tu)r. So using the Ćirić type Fc-contractive property, Proposition 1 and the fact that F is increasing, we get F(r)F(d(u,Tu))F(m(u,u0))t<F(m(u,u0))=Fmaxd(u,u0),d(u,Tu),d(u0,Tu0),12d(u,Tu0)+d(u0,Tu)=Fmaxr,d(u,Tu),0,r=F(d(u,Tu)), a contradiction. Therefore, d(u,Tu)=0 and so Tu=u. Consequently, Cu0,r is a fixed circle of T.

Now we show that T also fixes any circle Cu0,ρ with ρ<r. Let uCu0,ρ and assume that d(u,Tu)>0. By the Ćirić type Fc-contractive property, we have F(d(u,Tu))F(m(u,u0))t<F(m(u,u0))=F(d(u,Tu)), a contradiction. Thus we obtain d(u,Tu)=0 and Tu=u. So, Cu0,ρ is a fixed circle of T.☐

Let T be a Ćirić type Fc-contractive self-mapping with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then T fixes the disc Du0,r.

If there exist t>0, FF and u0X such that for all uX the following holds: d ( u , T u ) > 0 t + F ( d ( u , T u ) ) F α d ( u , u 0 ) + β d ( u , T u ) + γ d ( u 0 , T u 0 ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) , where α + β + γ + δ + η = 1 , α , β , γ , δ , η 0 and α 0 , then T is said to be a Hardy-Rogers type Fc-contraction on X.

If T is a Hardy-Rogers type Fc-contraction with u0X then we have Tu0=u0.

Assume that Tu0u0. From the definition of a Hardy-Rogers type Fc-contraction, we get d(u0,Tu0)>0t+F(d(u0,Tu0))Fαd(u0,u0)+βd(u0,Tu0)+γd(u0,Tu0)+δd(u0,Tu0)+ηd(u0,Tu0)=Fβ+γ+δ+ηd(u0,Tu0)<F(d(u0,Tu0)), a contradiction because of t>0. Then we have Tu0=u0.☐

Using Proposition 2, we rewrite the condition (3) as follows:d(u,Tu)>0t+F(d(u,Tu))Fαd(u,u0)+βd(u,Tu)+δd(u,Tu0)+ηd(u0,Tu), where α+β+δ+η1,α,β,δ,η0andα0.

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

Let T be a Hardy-Rogers type Fc-contraction with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then Cu0,r is a fixed circle of T. Especially, T fixes every circle Cu0,ρ with ρ<r.

Let uCu0,r. Using the Hardy-Rogers type Fc-contractive property, Proposition 2 and the fact that F is increasing, we get F(r)F(d(u,Tu))Fαd(u,u0)+βd(u,Tu)+δd(u,Tu0)+ηd(u0,Tu)t<F(αr+βd(u,Tu)+δr+ηr)F((α+β+δ+η)d(u,Tu))F(d(u,Tu)), a contradiction. Therefore, d(u,Tu)=0 and so Tu=u. Consequently, Cu0,r is a fixed circle of T. By the similar arguments used in the proof of Theorem 2, T also fixes any circle Cu0,ρ with ρ<r.☐

Let T be a Hardy-Rogers type Fc-contractive self-mapping with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then T fixes the disc Du0,r.

If we consider α=1 and β=γ=δ=η=0 in Definition 5, then we get the notion of an Fc-contractive mapping.

In Definition 5, if we choose δ=η=0, then we obtain the following definition.

If there exist t>0, FF and u0X such that for all uX the following holds: d ( u , T u ) > 0 t + F ( d ( u , T u ) ) F α d ( u , u 0 ) + β d ( u , T u ) + γ d ( u 0 , T u 0 ) , where α + β + γ < 1 and α , β , γ 0 , then T is said to be a Reich type Fc-contraction on X.

If a self-mapping T on X is a Reich type Fc-contraction with u0X then we have Tu0=u0.

From the similar arguments used in the proof of Proposition 2, the proof follows easily since β+γ<1.☐

Using Proposition 3, we rewrite the condition (4) as follows:d(u,Tu)>0t+F(d(u,Tu))Fαd(u,u0)+βd(u,Tu), where α+β<1andα,β0.

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

Let T be a Reich type Fc-contraction with u0X and r be defined as in (1). Then Cu0,r is a fixed circle of T. Especially, T fixes every circle Cu0,ρ with ρ<r.

It can be easily seen since F(r)F(d(u,Tu))F((α+β)d(u,Tu))<F(d(u,Tu)).

Let T be a Reich type Fc-contractive self-mapping with u0X and r be defined as in (1). Then T fixes the disc Du0,r.

In Definition 5, if we choose α=β=γ=0 and δ=η, then we obtain the following definition.

If there exist t>0, FF and u0X such that for all uX the following holds: d ( u , T u ) > 0 t + F ( d ( u , T u ) ) F η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) , where η 0 , 1 2 , then T is said to be a Chatterjea type Fc-contraction on X.

If a self-mapping T on X is a Chatterjea type Fc-contraction with u0X then we have Tu0=u0.

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

Let T be a Chatterjea type Fc-contraction with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then Cu0,r is a fixed circle of T. Especially, T fixes every circle Cu0,ρ with ρ<r.

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

Let T be a Chatterjea type Fc-contractive self-mapping with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then T fixes the disc Du0,r.

Now we give two illustrative examples of our obtained results.

Let X=1,2,e31,e3,e3+1 be the metric space with the usual metric. Let us define the self-mapping T:XX as T u = 2 if u = 1 u otherwise , for all uX.

The Ćirić type Fc-contractive self-mapping T: The self-mapping T is a Ćirić type Fc-contractive self-mapping with F=lnu, t=ln(e31) and u0=e3. Indeed, we get d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0 for u=1 and m ( u , u 0 ) = m ( 1 , e 3 ) = max d ( 1 , e 3 ) , d ( 1 , 2 ) , 1 2 d ( 1 , e 3 ) + d ( e 3 , 2 ) = max e 3 1 , 1 , e 3 3 2 = e 3 1 .

Then, we have t + F ( d ( u , T u ) ) = ln ( e 3 1 ) + ln ( d ( 1 , 2 ) ) = ln ( e 3 1 ) ln ( d ( m ( u , u 0 ) ) ) = ln ( e 3 1 ) .

The Hardy-Rogers type Fc-contractive self-mapping T: The self-mapping T is a Hardy-Rogers type Fc-contractive self-mapping with F=lnu, t=ln(e3)ln3, α=β=13, δ=η=0 and u0=e3. Indeed, we get d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0 for u=1 and α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) = 1 3 d ( 1 , e 3 ) + d ( 1 , 2 ) = 1 3 e 3 1 + 1 = e 3 3 .

Then, we have t + F ( d ( u , T u ) ) = ln ( e 3 ) ln 3 + ln ( d ( 1 , 2 ) ) = ln ( e 3 ) ln 3 ln ( d ( α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) ) ) = ln ( e 3 ) ln 3 .

The Reich type Fc-contractive self-mapping T: The self-mapping T is a Reich type Fc-contractive self-mapping with F=lnu, t=ln(e3)ln4, α=β=14 and u0=e3. Indeed, we get d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0 for u=1 and α d ( u , u 0 ) + β d ( u , T u ) = 1 4 d ( 1 , e 3 ) + d ( 1 , 2 ) = 1 4 e 3 1 + 1 = e 3 4 .

Then, we have t + F ( d ( u , T u ) ) = ln ( e 3 ) ln 4 + ln ( d ( 1 , 2 ) ) = ln ( e 3 ) ln 4 ln ( d ( α d ( u , u 0 ) + β d ( u , T u ) ) ) = ln ( e 3 ) ln 4 .

The Chatterjea type Fc-contractive self-mapping T: The self-mapping T is a Chatterjea type Fc-contractive self-mapping with F=lnu, t=ln23e31, η=13 and u0=e3. Indeed, we get d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0 for u=1 and η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) = 1 3 d ( 1 , e 3 ) + d ( e 3 , 2 ) = 1 3 e 3 1 + e 3 2 = 2 e 3 3 1 .

Then, we have t + F ( d ( u , T u ) ) = ln 2 3 e 3 1 + ln ( d ( 1 , 2 ) ) = ln 2 3 e 3 1 ln ( η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) ) = ln 2 3 e 3 1 .

Also, we obtain r = min d ( u , T u ) : u T u = d ( 1 , 2 ) = 1 .

Consequently, T fixes the circle Ce3,1=e31,e3+1 and the disc De3,1=e31,e3,e3+1.

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

Let x0X be any point and the self-mapping T:XX be defined as T u = u if u D u 0 , μ u 0 if u D u 0 , μ , for all uX with μ>0. Then T is not a Ćirić type Fc-contractive self-mapping (resp. Hardy-Rogers type Fc-contractive self-mapping, Reich type Fc-contractive self-mapping and Chatterjea type Fc-contractive self-mapping). But T fixes every circle Cx0,ρ where ρμ.

3. An Application to Discontinuity Problem

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

Let X=1,2,e31,e3,e3+1 be the metric space with the usual metric. Let us define the self-mapping T:XX as T u = 2 if u < e 3 1 u if u e 3 1 , for all uX. As in Example 1, it is easily verified that the self-mapping T is a Ćirić type Fc-contractive self-mapping and Ce3,1=e31,e3+1 is a fixed circle of T. We note that the self-mapping T is continuous at the point e3+1 while the self-mapping T is discontinuous at the point e31.

Let X=1,2,e31,e3,e3+1 be the metric space with the usual metric. Let us define the self-mapping T:XX as T u = 2 if u < e 3 1 e 3 1 if e 3 1 u < e 3 u if e 3 u e 3 + 1 u 1 if u > e 3 + 1 , for all uX. As in Example 1, it is easily checked that the self-mapping T is a Ćirić type Fc-contractive self-mapping and Ce3,1=e31,e3+1 is a fixed circle of T. We note that the self-mapping T is discontinuous at the center e3 and on the circle Ce3,1.

Consider the above examples, we give the following theorem.

Let T be a Ćirić type Fc-contraction with u0X and r be defined as in (1). If d(u0,Tu)=r for all uCu0,r then Cu0,r is a fixed circle of T. Also T is discontinuous at uCu0,r if and only if limvum(u,v)0.

From Theorem 2, we see that Cu0,r is a fixed circle of T. Used the idea given in Theorem 2.1 on page 1240 in , we see that T is discontinuous at uCu0,r if and only if limvum(u,v)0.☐

4. Conclusions

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.

Author Contributions

Funding

This research received no external funding.

Acknowledgments

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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