# On ParametricS-Metric Spaces and Fixed-Point Type Theorems for Expansive Mappings

Authors: Nihal Taş; Nihal Yılmaz Özgür
Journal: Journal of Mathematics
 (2016)

---

## Abstract

We introduce the notion of a parametricS-metric space as generalization of a parametric metric space. Using some expansive mappings, we prove a fixed-point theorem on a parametric S-metric space. It is important to obtain new fixed-point theorems on a parametric S-metric space because there exist some parametric S-metrics which are not generated by any parametric metric. We expect that many mathematicians will study various fixed-point theorems using new expansive mappings (or contractive mappings) in a parametric S-metric space.


---

## Body

## 1. Introduction and Backgrounds

Contractive conditions have been started by studying Banach’s contraction principle. These conditions have been used in various fixed-point theorems for some generalized metric spaces. Then expansive conditions were introduced [1] and new fixed-point results were obtained using expansive mappings.Recently, the notion of anS-metric has been studied by some mathematicians. This notion was introduced by Sedghi et al. in 2012 [2] as follows.Definition 1 (see [2]).
LetX be a nonempty set and let S:X×X×X→0,∞ be a function. S is called an S-metric on X if,(S1)Sa,b,c=0 if and only if a=b=c,(S2)Sa,b,c≤Sa,a,x+Sb,b,x+Sc,c,x,
for eacha,b,c,x∈X. The pair X,S is called an S-metric space.Using the notion of anS-metric space, various meaningful fixed-point studies were obtained by some researchers (see [2–6] for more details).The relationship between a metric and anS-metric was studied and an example of an S-metric which is not generated by any metric was given in [3, 4].Later, the notion of a parametric metric space was introduced and some basic concepts such as a convergent sequence and a Cauchy sequence were defined in [7]. We recall the following definitions.Definition 2 (see [7]).
LetX be a nonempty set and let P:X×X×0,∞→0,∞ be a function. P is called a parametric metric on X if,(P1)Pa,b,t=0 if and only if a=b,(P2)Pa,b,t=Pb,a,t,(P3)Pa,b,t≤Pa,x,t+Px,b,t,
for eacha,b,x∈X and all t>0. The pair X,P is called a parametric metric space.Definition 3 (see [7]).
LetX,P be a parametric metric space and let an be a sequence in X:(1)
an converges to x if and only if there exists n0∈N such that (1)Pan,x,t<ε,for all n≥n0 and all t>0; that is, (2)limn→∞Pan,x,t=0.It is denoted by limn→∞⁡an=x.(2)
an is called a Cauchy sequence if, for all t>0, (3)limn,m→∞Pan,am,t=0.(3)
X,P is called complete if every Cauchy sequence is convergent.In the following definition, the concept of a parametricb-metric space as generalization of a parametric metric space was given.Definition 4 (see [8]).
LetX be a nonempty set, let s≥1 be a real number, and let P:X×X×0,∞→0,∞ be a function. P is called a parametric b-metric on X if,(Pb1)P(a,b,t)=0 if and only if a=b,(Pb2)P(a,b,t)=P(b,a,t),(Pb3)P(a,b,t)≤sP(a,x,t)+P(x,b,t),
for eacha,b,x∈X and all t>0. The pair (X,P) is called a parametric b-metric space.Notice that a parametricb-metric is sometimes called a parametric s-metric according to a real number s≥1 in the above definition (see [9]).Some fixed-point theorems have been still investigated using the notions of a parametric metric space and a parametricb-metric space for various contractive or expansive mappings (see [7–10] for more details). For example, Hussain et al. proved some fixed-point theorems on complete parametric metric spaces and triangular intuitionistic fuzzy metric spaces [7]. Also, Hussain et al. introduced the notion of parametric b-metric space and investigated some fixed-point results [8]. Jain et al. established some fixed-point, common fixed-point, and coincidence point theorems for expansive type mappings on parametric metric spaces and parametric b-metric spaces [10]. Rao et al. obtained two common fixed-point theorems on parametric s-metric spaces [9].The aim of this paper is to introduce the concept of a parametricS-metric and give some basic facts. We give two examples of a parametric S-metric which is not generated by any parametric metric. We prove some fixed-point results under various expansive mappings in a parametric S-metric space. Also, we verify our results with some examples.


## 2. ParametricS-Metric Spaces

In this section, we introduce the notion of “a parametricS-metric space” and give some basic properties of this space. Also, we investigate a relationship between a parametric metric and a parametric S-metric (resp., a parametric b-metric and a parametric S-metric).Definition 5.
LetX be a nonempty set and let PS:X×X×X×0,∞→0,∞ be a function. PS is called a parametric S-metric on X if,PS1PSa,b,c,t=0 if and only if a=b=c,PS2PSa,b,c,t≤PSa,a,x,t+PSb,b,x,t+PSc,c,x,t,
for eacha,b,c∈X and all t>0. The pair (X,PS) is called a parametric S-metric space.Now we give the following examples of parametricS-metric spaces.Example 6.
LetX=f∣f:0,∞→Rbe  a  function and let the function PS:X×X×X×0,∞→0,∞ be defined by (4)PSf,g,h,t=ft-ht+gt-ht,for each f,g,h∈X and all t>0. Then PS is a parametric S-metric and the pair X,PS is a parametric S-metric space.Example 7.
LetX=R and let the function PS:X×X×X×0,∞→0,∞ be defined by (5)PSa,b,c,t=gta-b+b-c+a-c,for each a,b,c∈R and all t>0, where g:(0,∞)→(0,∞) is a continuous function. Then PS is a parametric S-metric and the pair (R,PS) is a parametric S-metric space.Example 8.
LetX=R+∪{0} and let the function PS:X×X×X×(0,∞)→[0,∞) be defined by (6)PSa,b,c,t=0;ifa=b=c,gtmax⁡a,b,c;otherwise,for each a,b,c∈X and all t>0, where g:(0,∞)→(0,∞) is a continuous function. Then PS is a parametric S-metric and the pair (X,PS) is a parametric S-metric space.We prove the following lemma which can be considered as the symmetry condition in a parametricS-metric space.Lemma 9.
LetX,PS be a parametric S-metric space. Then we have (7)PSa,a,b,t=PSb,b,a,t,for each a,b∈X and all t>0.Proof.
Using the condition(PS2), we obtain(8)PSa,a,b,t≤2PSa,a,a,t+PSb,b,a,t=PSb,b,a,t,PSb,b,a,t≤2PSb,b,b,t+PSa,a,b,t=PSa,a,b,t.From inequalities (8), we have(9)PSa,a,b,t=PSb,b,a,t.Now we give the relationship between a parametric metric and a parametricS-metric in the following lemma.Lemma 10.
LetX,P be a parametric metric space and let the function PSP:X×X×X×0,∞→0,∞ be defined by (10)PSPa,b,c,t=Pa,c,t+Pb,c,t,for each a,b,c∈X and all t>0. Then PSP is a parametric S-metric and the pair X,PSP is a parametric S-metric space.Proof.
It can be easily seen from Definitions2 and 5.We call the parametric metricPSP as the parametric S-metric generated by P. Notice that there exist parametric S-metrics PS satisfying PS≠PSP for all parametric metrics. We give some examples.Example 11.
LetX=R and let the function PS:X×X×X×0,∞→0,∞ be defined by (11)PSa,b,c,t=ta-c+a+c-2b,for each a,b,c∈R and all t>0. Then PS is a parametric S-metric and the pair (R,PS) is a parametric S-metric space. We have PS≠PSP; that is, PS is not generated by any parametric metric P.Example 12.
LetX=f∣f:0,∞→Rbeafunction and let the function PS:X×X×X×0,∞→0,∞ be defined by (12)PSf,g,h,t=eft-eht+eft+eht-2egt,for each f,g,h∈X and all t>0. Then PS is a parametric S-metric and the pair X,PS is a parametric S-metric space. We have PS≠PSP; that is, PS is not generated by any parametric metric P.In the following lemma, we see the relationship between a parametricb-metric and a parametric S-metric.Lemma 13.
LetX,PS be a parametric S-metric space and let the function P:X×X×0,∞→0,∞ be defined by (13)Pa,b,t=PSa,a,b,t,for each a,b∈X and all t>0. Then P is a parametric b-metric and the pair X,P is a parametric b-metric space.Proof.
Using condition(PS1), we see that conditions (Pb1) and (Pb2) are satisfied. Now we show that condition (Pb3) is satisfied. Using condition (PS2) and Lemma 9, we have (14)Pa,x,t=PSa,a,x,t≤2PSa,a,b,t+PSx,x,b,t=2Pa,b,t+Pb,x,t,Pa,x,t=PSx,x,a,t≤2PSx,x,b,t+PSa,a,b,t=Pa,b,t+2Pb,x,t,which implies that (15)Pa,x,t≤32Pa,b,t+Pb,x,t.Then P is a parametric b-metric with s=3/2.Remark 14.
Notice that the minimum value ofs is 3/2. So it should be s≠1; that is, P does not define a parametric metric in Lemma 13.Definition 15.
LetX,PS be a parametric S-metric space and let an be a sequence in X:(1)
an converges to x if and only if there exists n0∈N such that (16)PSan,an,x,t<ε,for all n≥n0 and all t>0; that is, (17)limn→∞PSan,an,x,t=0.It is denoted by limn→∞⁡an=x.(2)
an is called a Cauchy sequence if, for all t>0, (18)limn,m→∞PSan,an,am,t=0.(3)
X,PS is called complete if every Cauchy sequence is convergent.Lemma 16.
LetX,PS be a parametric S-metric space. If an converges to x, then x is unique.Proof.
Letlimn→∞⁡an=x and let limn→∞⁡an=y with x≠y. Then there exists n1,n2∈N such that (19)PSan,an,x,t<ε4,PSan,an,y,t<ε2,for each ε>0, all t>0, and n≥n1,n2. If we take n0=maxn1,n2, then, using condition PS2 and Lemma 9, we get (20)PSx,x,y,t≤2PSx,x,an,t+PSy,y,an,t<ε2+ε2=ε,for each n≥n0. Therefore PSx,x,y,t=0 and x=y.Lemma 17.
LetX,PS be a parametric S-metric space. If an converges to x, then an is Cauchy.Proof.
By the similar arguments used in the proof of Lemma16, we can easily see that an is a Cauchy sequence.As a consequence of Lemma10 and Definition 15, we obtain the following corollary.Corollary 18.
LetX,P be a parametric metric space and let X,PSP be a parametric S-metric space, where PSP is generated by parametric metric P. Then we have the following:(1)
an→x in X,P if and only if an→x in X,PSP.(2)
an is Cauchy in X,P if and only if an is Cauchy in X,PSP.(3)
X,P is complete if and only if (X,PSP) is complete.Definition 19.
LetX,PS be a parametric S-metric space and let T:X→X be a self-mapping of X. T is said to be a continuous mapping at x in X if (21)limn→∞PSTan,Tan,Tx,t=0,for any sequence an in X and all t>0 such that (22)limn→∞PSan,an,x,t=0.


## 3. Some Fixed-Point Results

In this section, we give some fixed-point results for expansive mappings in a complete parametricS-metric space.Definition 20.
Let(X,PS) be a parametric S-metric space and let T be a self-mapping of X.
( S P 1 ) There exist real numbers kii∈1,2,3 satisfying ki≥0i∈2,3 and k1>1 such that (23)PSTa,Ta,Tb,t≥k1PSa,a,b,t+k2PSTa,Ta,a,t+k3PSTb,Tb,b,t,for each a,b∈X and all t>0.Theorem 21.
LetX,PS be a complete parametric S-metric space and let T be a surjective self-mapping of X. If T satisfies condition (SP1), then T has a unique fixed point in X.Proof.
Using the hypothesis, it can be easily seen thatT is injective. Indeed, if we take Ta=Tb, then, using condition (SP1), we get (24)0=PSTa,Ta,Ta,t≥k1PSa,a,b,t+k2PSTa,Ta,a,t+k3PSTa,Ta,b,t,for all t>0 and so PS(a,a,b,t)=0; that is, we have a=b since k1>1.
Let us denote the inverse mapping ofT by F. Let a0∈X and define the sequence an as follows: (25)a1=Fa0,a2=Fa1=F2a0,…,an+1=Fan=Fn+1a0,….Suppose that an≠an+1 for all n. Using condition (SP1) and Lemma 9, we have (26)PSan-1,an-1,an,t=PSTT-1an-1,TT-1an-1,TT-1an,t≥k1PST-1an-1,T-1an-1,T-1an,t+k2PSTT-1an-1,TT-1an-1,T-1an-1,t+k3PSTT-1an,TT-1an,T-1an,t=k1PSFan-1,Fan-1,Fan,t+k2PSan-1,an-1,Fan-1,t+k3PSan,an,Fan,t=k1PSan,an,an+1+k2PSan-1,an-1,an,t+k3PSan,an,an+1,t=k1+k3PSan,an,an+1,t+k2PSan-1,an-1,an,t,which implies that(27)1-k2PSan-1,an-1,an,t≥k1+k3PSan,an,an+1,t.Clearly, we have k1+k3≠0. Hence, we obtain(28)PSan,an,an+1,t≤1-k2k1+k3PSan-1,an-1,an,t.If we put k=1-k2/k1+k3, then we get k<1, since k1+k2+k3>1. Repeating this process in condition (28), we find(29)PSan,an,an+1,t≤knPSa0,a0,a1,t,for all t>0.
Letm,n∈N with m>n≥1. Using inequality (29) and condition (PS2), we have (30)PSan,an,am,t≤2kn1-kPSa0,a0,a1,t.If we take limit for n,m→∞, we obtain (31)limn,m→∞PSan,an,am,t=0.Therefore {an} is Cauchy. Then there exists y∈X such that (32)limn→∞an=y,since (X,PS) is a complete parametric S-metric space. Using the surjectivity hypothesis, there exists a point x∈X such that Tx=y. From condition (SP1), we have (33)PSan,an,y,t=PSTan+1,Tan+1,Tx,t≥k1PSan+1,an+1,x,t+k2PSan,an,an+1,t+k3PSy,y,x,t.If we take limit for n→∞, we obtain (34)0≥k1+k3PSy,y,x,t,which implies that y=x and Ty=y.
Now we show the uniqueness ofy. Let z be another fixed point of T with y≠z. Using condition (SP1) and Lemma 9, we get (35)PSy,y,z,t=PSTy,Ty,Tz,t≥k1PSy,y,z,t+k2PSy,y,y,t+k3PSz,z,z,t=k1PSy,y,z,t,which implies that y=z, since k1>1. Consequently, T has a unique fixed point y.We give some examples which satisfy the conditions of Theorem21.Example 22.
LetX=R+∪0 be the complete S-metric space with the S-metric defined in Example 8. Let us define the self-mapping T:R+∪0→R+∪0 as (36)Tx=βx,for all x∈R with β>1, and the function g:0,∞→0,∞ as (37)gt=t2,for all t∈0,∞. Then T satisfies the conditions of Theorem 21 with k1=β and k2=k3=0. Then T has a unique fixed point x=0 in X.Example 23.
LetX=R+∪0 be the complete S-metric space with the S-metric defined in Example 8. Let us define the self-mapping T:R+∪0→R+∪0 as (38)Tx=x+lnx+1,for all x∈R with β>1, and the function g:0,∞→0,∞ as (39)gt=t3+t2+t+1,for all t∈(0,∞). Then T satisfies the conditions of Theorem 21 with k1=min⁡ln(x+1)/x:x≠0∈X and k2=k3=0. Then T has a unique fixed point x=0 in X.If we takek2=k3 in condition (SP1), then we obtain the following corollary.Corollary 24.
Let(X,PS) be a complete parametric S-metric space and let T be a surjective self-mapping of X. If there exist real numbers kii∈1,2 satisfying k1>1 and k2≥0 such that (40)PSTa,Ta,Tb,t≥k1PSa,a,b,t+k2maxPSTa,Ta,a,t,PSTb,Tb,b,t,for each a,b∈X and all t>0, then T has a unique fixed point in X.If we takek1=k and k2=k3=0 and k1=k and k2=0 in Theorem 21 and Corollary 24, respectively, then we obtain the following corollaries.Corollary 25.
LetX,PS be a complete parametric S-metric space and let T be a surjective self-mapping of X. If there exists a real number k>1 such that (41)PSTa,Ta,Tb,t≥kPSa,a,b,t,for each a,b∈X and all t>0, then T has a unique fixed point in X.Corollary 26.
LetX,PS be a complete parametric S-metric space and let T be a surjective self-mapping of X. If there exist a positive integer m and a real number k>1 such that (42)PSTma,Tma,Tmb,t≥kPSa,a,b,t,for each a,b∈X and all t>0, then T has a unique fixed point in X.Proof.
From Corollary25, by a similar way used in the proof of Theorem 21, it can be easily seen that Tm has a unique fixed point a in X. Also we have (43)Ta=TTma=Tm+1a=TmTaand so we obtain that Ta is a fixed point for Tm. We get Ta=a, since a is the unique fixed point.



---

*Source: 4746732-2016-11-07.xml*