ON THE GEOMETRY OF FIXED POINTS FOR SELF-MAPPINGS ON S-METRIC SPACES

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 x0mapping 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.


Introduction and Preliminaries
The notion of an S-metric space was introduced as a generalization of a metric space as follows: De…nition 1. [20] Let X be a nonempty set and S : X 3 ! [0; 1) be a function satisfying the following conditions for all x; y; z; a 2 X : (1) S(x; y; z) = 0 if and only if x = y = z, (2) S(x; y; z) S(x; x; a) + S(y; y; a) + S(z; z; a). Then S is called an S-metric on X and the pair (X; S) is called an S-metric space.
Many researchers have studied on S-metric spaces to obtain new …xed point results and some applications (see [7,9,10,15,21] and the references therein). Also, the relationship between a metric and an S-metric was investigated in various studies and some examples of an S-metric which is not generated by any metric were given (see [4,5,11] for more details).
Recently, the …xed circle problem (resp. …xed disc problem) raised by Özgür and Taş (see [12,18] and the references therein) has been studied as an geometric ON THE GEOM ETRY O F FIXED PO INTS 191 approach to the …xed-point theory on metric spaces and some generalized metric spaces (for example, S-metric spaces) (see [8,9,13,14,23,24]). Now we recall the notions of a circle and a disc on an S-metric space given in [13,20], respectively.
Let (X; S) be an S-metric space and T : X ! X be a self-mapping. A circle C S x0;r and a disc D S x0;r are de…ned as follows: If T x = x for all x 2 C S x0;r (resp. x 2 D S x0;r ), then the circle C S x0;r (resp. the disc D S x0;r ) is called as the …xed circle (resp. the …xed disc) of T . A recent solution to the …xed-circle problem was given using the notion of S-Zam…rescu type x 0 -mapping on S-metric spaces as follows: De…nition 2. [16] Let (X; S) be an S-metric space and T : X ! X be a selfmapping. Then T is called an S-Zam…rescu type x 0 -mapping if there exist x 0 2 X and a; b 2 [0; 1) such that for all x 2 X.
Let the number be de…ned as Theorem 3. [16] Let (X; S) be an S-metric space, T : X ! X be a self-mapping and be the real number de…ned in (1). If the following conditions hold: for each x 2 C S x0; , then C S x0; is a …xed circle of T , that is, C S x0; F ix(T ).
In this paper, we give new solutions to the …xed circle (resp. …xed disc) problem on S-metric spaces. In Section 2, we prove some …xed circle and …xed disc results using di¤erent 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.

Main Results
In this section, we inspire the methods given in [2,6,19,26] and use the number de…ned in (1) to obtain new …xed circle (resp. …xed disc) results on S-metric spaces. In [19], Pata proved a …xed point theorem to generalize the well-known Banach's contraction principle on a metric space. However, Berinde showed that the main result in [19] does not hold at least in two extremal cases for the used parameter ". The corrected version of this theorem was given with some necessary examples in [2]. In our results, we will not use the Picard iteration. Hence, our main results hold for all the parameters 2 [0; 1] and this situation will be supported by some illustrative examples given in the next section.
Let denotes the class of all increasing functions for all x 2 X and each 2 [0; 1], where kxk s = S(x; x; x 0 ).
Notice that kx 0 k s = S(x 0 ; x 0 ; x 0 ) = 0. Let us consider the inequality given in the notion of S-Pata type x 0 -mapping under the cases = 0 and = 1, respectively.
x; x 0 ) 2 and also for = 1, we get Theorem 5. Let (X; S) be an S-metric space, T : X ! X be an S-Pata type x 0 -mapping with x 0 2 X and be the real number de…ned in (1). Then C S x0; is a …xed circle of T , that is, C S x0;

F ix(T ).
Proof. At …rst, we show that x 0 is a …xed point of T . On the contrary, assume that T x 0 6 = x 0 . Using the S-Pata type x 0 -mapping property, we obtain For = 0, by inequality (2), we …nd this is a contradiction. So, the assumption is false. This shows that T x 0 = x 0 and hence kT Let > 0 and x 2 C S x0; be any point such that T x 6 = x. Using the S-Pata type x 0 -mapping hypothesis, we obtain For = 0, by inequality (3), we get a contradiction with the de…nition of . Hence it should be T x = x. Consequently, T …xes the circle C S x0; and so C S x0; F ix(T ).
Corollary 6. Let (X; S) be an S-metric space, T : X ! X be an S-Pata type x 0 -mapping with x 0 2 X and be the real number de…ned in (1). Then T …xes whole of the disc D S x0; , that is, D S x0; F ix(T ).
Proof. By the similar arguments used in the proof of Theorem 5, the proof follows easily.
We de…ne another contractive condition to obtain a new …xed-circle result. then T is called an S-Pata Zam…rescu type x 0 -mapping with respect to 2 .
In the above de…nition, we consider the extremal cases = 0 and = 1, respectively. For = 0, we have and also for = 1, we get where L = (1) > 0. Now we investigate the relationship between the notions of an S-Zam…rescu type x 0 -mapping and an S-Pata Zam…rescu type x 0 -mapping.
Hence we get that an S-Zam…rescu type x 0 -mapping is a special case of an S-Pata Zam…rescu type x 0 -mapping with = , (x) = x 1 and = = 1. Now we prove the following …xed circle theorem.
Theorem 8. Let (X; S) be an S-metric space, T : X ! X be a self-mapping and be the real number de…ned in (1). If the following conditions hold: Proof. Using the condition (i), we can easily obtain that T x 0 = x 0 and hence kT x 0 k s = kx 0 k s = 0. Let = 0. Then we have C S x0; = fx 0 g. Clearly, C S x0; is a …xed circle of T , that is, C S x0; F ix(T ). Let > 0 and x 2 C S x0; be any point such that T x 6 = x. Using the conditions (i) and (ii), we obtain For = 0, using the inequality (4), we get a contradiction with the de…nition of . Consequently, it should be T x = x whence T …xes the circle C S x0; and so C S x0; F ix(T ).
Corollary 9. Let (X; S) be an S-metric space, T : X ! X be a self-mapping and be the real number de…ned in (1). If the following conditions hold: (i) T is an S-Pata Zam…rescu type x 0 -mapping with respect to 2 for x 0 2 X, then T …xes whole of the disc D S x0; , that is, D S x0; F ix(T ).
Proof. By the similar arguments used in the proof of Theorem 8, the proof follows easily.

Illustrative Examples and Some Remarks
In this section, we give some examples to show the validity of our results obtained in the previous section.
Example 11. Let X = R be the S-metric space with the S-metric de…ned by S(x; y; z) = jx zj + jx + z 2yj , for all x; y; z 2 R [11]. Let us de…ne the self-mapping T 1 : R ! R as for all x 2 R. Then T 1 is both an S-Pata type resp. the disc D S 0;1 = 1 2 ; 1 2 . Example 12. Let X = R be the S-metric space with the S-metric considered in Example 11. Let us de…ne the self-mapping T 2 : R ! R as for all x 2 R. Then T 2 is both an S-Pata type x 0 -mapping and an S-Pata Zam-…rescu type x 0 -mapping with x 0 = 0 (or x 0 = 3), = = = 1 and Also we obtain = 2. Consequently, T 2 …xes the circles C S 0;2 and C S 3;2 (resp. the discs D S 0;2 and D S 3;2 ).
Example 13. Let X = R be the S-metric space with the S-metric considered in Example 11. Let us de…ne the self-mapping T 3 : R ! R as for all x 2 R. Then T 3 is not an S-Pata type x 0 -mapping and an S-Pata Zam…rescu type x 0 -mapping with x 0 = 0. But T 3 …xes the circle C S 0;4 = f 2; 2g and the disc D S 0;4 = [ 2; 2]. Now, we give an example of a self-mapping that satis…es the conditions of Theorem 8 but does not satisfy the conditions of Theorem 5. The following remarks can be deduced from the obtained results and the given examples.

Remark 15. (i)
The point x 0 satisfying the conditions of an S-Pata type x 0mapping and an S-Pata Zam…rescu type x 0 -mapping is always a …xed point of the self-mapping T . Moreover, the choice of x 0 is independent from the number (see Example 11 and Example 12). Also the number of x 0 can be more than one (see Example 12).
(ii) The converse statements of Theorem 5, Corollary 6, Theorem 8 and Corollary 9 are not always true (see Example 13). That is, a self-mapping having a …xed circle (resp. …xed disc) need not to be an S-Pata type x 0 -mapping or an S-Pata Zam…rescu type x 0 -mapping with x 0 where the point x 0 is the center of the …xed circle (resp. …xed disc).

Conclusion
In this paper, we have presented some new solutions to the …xed circle problem on S-metric spaces. To do this, we have inspired by the Pata and Zam…rescu type methods. We have proved two main …xed circle theorems and some related results. Also, we have given necessary illustrative examples supporting our obtained results. On the other hand, there are many generalized metric spaces in the literature (for example, see [3,25] and the references therein). Hence, the …xed circle (resp. …xed disc) problem can be studied on these generalized metric spaces using similar approaches as a future work.
On the other hand, a related problem is the best proximity point problem since the best proximity point theorems investigate an optimal solution of the minimization problem fd (x; T x) : x 2 Ag for a mapping T : A ! B where A and B are two non-empty subsets of a metric space (see [1] and the references therein). In [6], the existence of best proximity point was investigated using the Pata type proximal mappings. In [17], the notion of a best proximity circle is introduced and some proximal contractions for a non-self-mapping are determined. In this context, a related future work is the investigation of the existence of a best proximity circle via the notions of p-proximal contraction and p-proximal contractive mapping de…ned in [22].