C om mun.Fac.Sci.U niv.A nk.Ser. A 1 M ath. Stat. Volum e 69, N umb er 2, Pages 1184–1192 (2020) D O I: 10.31801/cfsuasm as.616325
ISSN 1303–5991 E-ISSN 2618–6470
Received by the editors: S eptem ber 06, 2019; Accepted: Ju ly 09, 2020
ON THE GEOMETRY OF FIXED POINTS FOR
SELF-MAPPINGS ON S-METRIC SPACES
Nihal TA¸S and Nihal ÖZGÜR
Bal¬kesir University, Department of Mathematics, 10145, Bal¬kesir, TURKEY
Abstract. In this paper, we focus on some geometric properties related to the set F ix(T ), the set of all …xed 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 x0 -mapping and an S-Pata Zam…rescu type x0-mapping. Using these notions, we propose new solutions to the …xed circle (resp. …xed disc) problem. Also, we give some illustrative examples of our main results.
1. 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 : X3 ! [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 re-sults 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 stud-ies 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 Ta¸s (see [12, 18] and the references therein) has been studied as an geometric
2020 Mathematics Subject Classi…cation. Primary 54H25; Secondary 47H09, 47H10.
Keywords and phrases. S-metric space, …xed circle, S-Pata type x0-mapping, S-Pata Zam-…rescu type x0-mapping.
nihaltas@balikesir.edu.tr; nyozgur@yahoo.com-Corresponding author 0000-0002-4535-4019; 0000-0002-8152-1830.
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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 CS
x0;r and a disc D
S
x0;r are de…ned as follows:
CxS0;r = fx 2 X : S(x; x; x0) = rg and DxS0;r= fx 2 X : S(x; x; x0) rg , where r 2 [0; 1). If T x = x for all x 2 CS x0;r (resp. x 2 D S
x0;r), then the circle C
S
x0;r (resp. the
disc DxS0;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 x0-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 self-mapping. Then T is called an S-Zam…rescu type x0-mapping if there exist x02 X
and a; b 2 [0; 1) such that S(T x; T x; x) > 0 =)
S(T x; T x; x) max aS(x; x; x0);
b
2[S(T x0; T x0; x) + S(T x; T x; x0)] , for all x 2 X.
Let the number be de…ned as
= inf fS(T x; T x; x) : T x 6= x; x 2 Xg . (1) 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:
(i) T is an S-Zam…rescu type x0-mapping with x02 X,
(ii) S(T x; T x; x0) for each x 2 CxS0; ,
then CxS0; is a …xed circle of T , that is, CxS0; 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.
2. 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 : [0; 1] ! [0; 1) with (0) = 0.
De…nition 4. Let (X; S) be an S-metric space, T : X ! X be a self-mapping, 0, 1 and 2 [0; ] be any constants. Then T is called an S-Pata type x0-mapping if there exist x02 X and 2 such that
S(T x; T x; x) > 0 =) S(T x; T x; x) 1 2 kxks+ ( ) [1 + kxks+ kT xks] ,
for all x 2 X and each 2 [0; 1], where kxks= S(x; x; x0).
Notice that kx0ks= S(x0; x0; x0) = 0. Let us consider the inequality given in the
notion of S-Pata type x0-mapping under the cases = 0 and = 1, respectively.
For = 0, we have
S(T x; T x; x) > 0 =) S(T x; T x; x) 12kxks=S(x; x; x 0)
2 and also for = 1, we get
S(T x; T x; x) > 0 =) S(T x; T x; x) (1) [1 + kxks+ kT xks]
= L [1 + kxks+ kT xks]
= L [1 + S(x; x; x0) + S(T x; T x; x0)] ,
where L = (1) > 0.
Theorem 5. Let (X; S) be an S-metric space, T : X ! X be an S-Pata type x0-mapping with x0 2 X and be the real number de…ned in (1). Then CxS0; is a
…xed circle of T , that is, CS
x0; F ix(T ).
Proof. At …rst, we show that x0is a …xed point of T . On the contrary, assume that
T x06= x0. Using the S-Pata type x0-mapping property, we obtain
S(T x0; T x0; x0)
1
2 kx0ks+ ( ) [1 + kx0ks+ kT x0ks] . (2) For = 0, by inequality (2), we …nd
S(T x0; T x0; x0) 0,
this is a contradiction. So, the assumption is false. This shows that T x0= x0 and
hence kT x0ks= kx0ks= 0.
Let = 0. Then we have CS
x0; = fx0g. Clearly, C
S
x0; is a …xed circle of T , that
Let > 0 and x 2 CS
x0; be any point such that T x 6= x. Using the S-Pata type
x0-mapping hypothesis, we obtain
S(T x; T x; x) 1 2 kxks+ ( ) [1 + kxks+ kT xks] . (3)
For = 0, by inequality (3), we get
S(T x; T x; x) 12kxks=
S(x; x; x0)
2 =2,
a contradiction with the de…nition of . Hence it should be T x = x. Consequently, T …xes the circle CS
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 x0-mapping with x0 2 X and be the real number de…ned in (1). Then T …xes
whole of the disc DS
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. De…nition 7. Let (X; S) be an S-metric space, T : X ! X be a self-mapping,
0, 1 and 2 [0; ] be any constants. If there exist x02 X and 2 such
that
S(T x; T x; x) > 0 =) S(T x; T x; x) 1 2 MS(x; x0)
+ ( ) [1 + kxks+ kT xks+ kT x0ks] ,
for all x 2 X and each 2 [0; 1], where kxks= S(x; x; x0) and
MS(x; y)
= max S(x; x; y);S(T x; T x; x) + S(T y; T y; y)
2 ;
S(T y; T y; x) + S(T x; T x; y)
2 ,
then T is called an S-Pata Zam…rescu type x0-mapping with respect to 2 .
In the above de…nition, we consider the extremal cases = 0 and = 1, respec-tively. For = 0, we have
S(T x; T x; x) > 0 =) S(T x; T x; x) 12MS(x; x0)
and also for = 1, we get
S(T x; T x; x) > 0 =) S(T x; T x; x) (1) [1 + kxks+ kT xks+ kT x0ks]
= L [1 + kxks+ kT xks+ kT x0ks] ,
where L = (1) > 0.
Now we investigate the relationship between the notions of an S-Zam…rescu type x0-mapping and an S-Pata Zam…rescu type x0-mapping.
Let = maxfa; bg in De…nition 2 and let us consider Bernoulli’s inequality 1 + pt (1 + t)p, p 1, t 2 [ 1; 1). Then we get S(T x; T x; x) > 0 =) S(T x; T x; x) max aS(x; x; x0);b2[S(T x0; T x0; x) + S(T x; T x; x0)] maxnS(x; x; x0);S(T x0;T x0;x)+2S(T x;T x;x0) o maxnS(x; x; x0);S(T x;T x;x)+S(T x2 0;T x0;x0);S(T x0;T x0;x)+2S(T x;T x;x0) o 1 2 max n S(x; x; x0);S(T x;T x;x)+S(T x2 0;T x0;x0);S(T x 0;T x0;x)+S(T x;T x;x0) 2 o + + 21 h 1 + max n kxks; kxks+kT xks+kT x0ks 2 oi 1 2 MS(x; x0) + 1 + 1 [1 + kxks+ kT xks+ kT x0ks] 1 2 MS(x; x0) + 1 [1 + kxks+ kx0ks+ kT xks+ kT x0ks] 1 2 MS(x; x0) + 1 [1 + kxks+ kx0ks+ kT xks+ kT x0ks] .
Hence we get that an S-Zam…rescu type x0-mapping is a special case of an S-Pata
Zam…rescu type x0-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:
(i) T is an S-Pata Zam…rescu type x0-mapping with respect to 2 for x02 X,
(ii) S(T x; T x; x0) for each x 2 CxS0; ,
then CxS0; is a …xed circle of T , that is, CxS0; F ix(T ).
Proof. Using the condition (i), we can easily obtain that T x0 = x0 and hence
kT x0ks = kx0ks= 0. Let = 0. Then we have CxS0; = fx0g. Clearly, C
S x0; is a
…xed circle of T , that is, CS
x0; F ix(T ).
Let > 0 and x 2 CS
x0; be any point such that T x 6= x. Using the conditions
(i) and (ii), we obtain
S(T x; T x; x) 1 2 MS(x; x0) + ( ) [1 + kxks+ kT xks+ kT x0ks] 1 2 max ; S(T x; T x; x) 2 + ( ) [1 + kxks+ kT xks] . (4)
For = 0, using the inequality (4), we get
S(T x; T x; x) 12max ;S(T x; T x; x)
2 ,
a contradiction with the de…nition of . Consequently, it should be T x = x whence T …xes the circle CxS0; and so CxS0; 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 x0-mapping with respect to 2 for x02 X,
(ii) S(T x; T x; x0) for each x 2 DxS0;
then T …xes whole of the disc DSx0; , that is, DSx0; F ix(T ).
Proof. By the similar arguments used in the proof of Theorem 8, the proof follows easily.
Remark 10. If a self-mapping T satis…es the conditions of Theorem 8, then we have kT x0ks= kx0ks= 0. Therefore, Theorem 8 coincides with Theorem 5 in the
case where MS(x; x0) = kxks for all x 2 X. On the other hand, if T satis…es the
conditions of Theorem 5 then clearly, T satis…es the conditions of Theorem 8 since MS(x; x0) kxks.
3. 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 T1: R ! R as
T1x =
x ; x 2 [ 2; 2]
x + 12 ; x 2 ( 1; 2) [ (2; 1) ,
for all x 2 R. Then T1 is both an S-Pata type x0-mapping and an S-Pata
Zam-…rescu type x0-mapping with x0= 0, = = = 1 and
(x) = 01 ; x = 0
2 ; x 2 (0; 1]
.
Also we have = 1. Consequently, by Theorem 5 and Theorem 8 (resp. Corollary 6 and Corollary 9), T1…xes the circle C0;1S = 12;
1
2 resp. the disc D0;1S = 12; 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 T2: R ! R as
T2x =
x ; x 2 [ 4; 1)
x + 1 ; x 2 ( 1; 4) ,
for all x 2 R. Then T2 is both an S-Pata type x0-mapping and an S-Pata
Zam-…rescu type x0-mapping with x0= 0 (or x0= 3), = = = 1 and
(x) = 01 ; x = 0
2 ; x 2 (0; 1]
.
Also we obtain = 2. Consequently, T2 …xes the circles C0;2S and C3;2S (resp. the
discs DS
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 T3: R ! R as
T3x =
x ; x 2 [ 2; 2]
0 ; x 2 ( 1; 2) [ (2; 1) ,
for all x 2 R. Then T3is not an S-Pata type x0-mapping and an S-Pata Zam…rescu
type x0-mapping with x0 = 0. But T3 …xes the circle C0;4S = f 2; 2g and the disc
DS
0;4 = [ 2; 2].
Now, we give an example of a self-mapping that satis…es the conditions of The-orem 8 but does not satisfy the conditions of TheThe-orem 5.
Example 14. Let X = R be the S-metric space with the usual S-metric de…ned by S(x; y; z) = jx zj + jy zj ,
for all x; y; z 2 X [21]. Now, we de…ne the self-mapping T4: X ! X by
T4x = 5 3x ; jxj = 1 x ; jxj 6= 1 . We have = inf fS(T4x; T4x; x) : jxj = 1g = inf f2 jT4x xj : jxj = 1g = inf 2 5 3x x : jxj = 1 = inf 4 3jxj : jxj = 1 = 4 3.
Then, it is easy to verify that T4 is not an S-Pata type x0-mapping for the point
x0= 0 independent from the choice of the parameters , , and the function .
But, if we choose = = = 1 then T4 is an S-Pata Zam…rescu type x0-mapping
for the point x0= 0 with the function
(x) = 01 ; x = 0
4 ; x 2 (0; 1]
.
Clearly, T4 …xes the circle
C0;S4 3 = x : 2 jx 0j = 4 3 = x : jxj = 2 3 = 2 3; 2 3 and the disc DS
0;4
3 = x : jxj
2 3 .
The following remarks can be deduced from the obtained results and the given examples.
Remark 15. (i) The point x0 satisfying the conditions of an S-Pata type x0
-mapping and an S-Pata Zam…rescu type x0-mapping is always a …xed point of the
self-mapping T . Moreover, the choice of x0 is independent from the number (see
Example 11 and Example 12). Also the number of x0 can be more than one (see
Example 12).
(ii) The converse statements of Theorem 5, Corollary 6, Theorem 8 and Corol-lary 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 x0-mapping or an S-Pata
Zam…rescu type x0-mapping with x0 where the point x0 is the center of the …xed
circle (resp. …xed disc).
4. 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 minimiza-tion 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 re-lated 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].
Acknowledgement. Both authors are supported by the Scienti…c Research Projects Unit of Balikesir University under the project number BAP 2018 / 021.
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