# Fixed Point Theorems for New Type Contractive Mappings

Authors: Müzeyyen Sangurlu Sezen
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we introduce some new concepts of contractions calledγ-contractions and γ-weak contractions. We prove some fixed point theorems for mappings providing γ-contractions and γ-weak contractions unlike known results in the literature. Also, we present a few examples to illustrate the validity of the results obtained in the paper.


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## Body

## 1. Introduction and Preliminaries

The fixed point theory is very important concept in mathematics. In 1922, Banach created a famous result called Banach contraction principle in the concept of the fixed point theory [1]. Later, most of the authors intensively introduced many works regarding the fixed point theory in various of spaces. The concept of a fuzzy metric space was introduced in different ways by some authors (see [2, 3]). Gregori and Sapena [4] introduced the notion of fuzzy contractive mappings and gave some fixed point theorems for complete fuzzy metric spaces in the sense of George and Veeramani, and also for Kramosil and Michalek’s fuzzy metric spaces which are complete in Grabiec’s sense. Mihet [5] developed the class of fuzzy contractive mappings of Gregori and Sapena, considered these mappings in non-Archimedean fuzzy metric spaces in the sense of Kramosil and Michalek, and obtained a fixed point theorem for fuzzy contractive mappings. At the same time, there are lots of different types of fixed point theorems presented by many authors by expanding the Banach’s result in the literature (see [5–12]).In this work, using a mappingγ:[0,1)→R we introduce some new types of contractions called γ-contractions and γ-weak contractions. Later, we prove some fixed point theorems for mappings providing γ-contractions and γ-weak contractions. Some examples are supplied in order to support the usability of our results. Our main results are substantially different and useful compared to some known results in the existing literature.Definition 1 ([13]).
A binary operation∗:[0,1]×[0,1]→[0,1] is called a continuous triangular norm (in short, continuous t-norm) if it satisfies the following conditions: (TN-1)
∗ is commutative and associative,(TN-2)
∗ is continuous,(TN-3)
∗(a,1)=a for every a∈[0,1],(TN-4)
∗(a,b)≤∗(c,d) whenever a≤c, b≤d and a,b,c,d∈[0,1].An arbitraryt-norm ∗ can be extended (by associativity) in a unique way to a nary operator taking for (x1,x2,…,xn)∈[0,1]n, n∈N; the value ∗(x1,x2,…,xn) is defined, in [14], by ∗I˙=10xi=1, ∗I˙=1nxi=∗(∗I˙=1n-1xi,xn)=∗(x1,x2,…,xn).Definition 2 ([15]).
A fuzzy metric space is an ordered triple(X,M,∗) such that X is a nonempty set, ∗ is a continuous t-norm, and M is a fuzzy set on X2×(0,∞), satisfying the following conditions, for all x,y,z∈X, s,t>0:(FM-1)
M(x,y,t)>0,(FM-2)
M(x,y,t)=1 iff x=y,(FM-3)
M(x,y,t)=M(y,x,t),(FM-4)
M(x,z,t+s)≥M(x,y,t)∗M(y,z,s),(FM-5)
M(x,y,·):(0,∞)→[0,1] is continuous.
If, in the above definition, the triangular inequality (FM-4) is replaced by
(NA)M(x,z,max⁡{t,s})≥M(x,y,t)∗M(y,z,s) for all x,y,z∈X, s,t>0, or equivalently,(1)Mx,z,t≥Mx,y,t∗My,z,tand then the triple (X,M,∗) is called a non-Archimedean fuzzy metric space [16].Definition 3.
Let(X,M,∗) be a fuzzy metric space. Then see the following. (i)
A sequence{xn} in X is said to converge to x in X, denoted by xn→x, if and only if limn→∞⁡M(xn,x,t)=1 for all t>0; i.e., for each r∈(0,1) and t>0, there exists n0∈N such that M(xn,x,t)>1-r for all n≥n0 [3, 17].(ii)
A sequence{xn} is a M-Cauchy sequence if and only if for all ε∈(0,1) and t>0, there exists n0∈N such that M(xn,xm,t)≥1-ε for all m>n≥n0 [15, 17]. A sequence {xn} is a G-Cauchy sequence if and only if limn→∞⁡M(xn,xn+p,t)=1 for any p>0 and t>0 [4, 14, 18].(iii)
The fuzzy metric space(X,M,∗) is called M-complete (G-complete) if every M-Cauchy (G-Cauchy) sequence is convergent.


## 2. Main Results

Definition 4.
Letγ:[0,1)→R be a strictly increasing, continuous mapping and for each sequence {an}n∈N of positive numbers limn→∞⁡an=1 if and only if limn→∞⁡γ(an)=+∞. Let Γ be the family of all γ functions.
A mappingT:X→X is said to be a γ-contraction if there exists a δ∈(0,1) such that(2)MTx,Ty,t<1⇒γMTx,Ty,t≥γMx,y,t+δfor all x,y∈X and γ∈Γ.Example 5.
Letγ∈Γ. The different types of the mapping γ are as follows:(γ1)
1/(1-x),(γ2)
1/(1-x)+x,(γ3)
1/(1-x2),(γ4)
1/1-x.Remark 6.
From(γ1) and (2) it is easy to conclude that every γ- contraction T is a contractive mapping, that is,(3)MTx,Ty,t>Mx,y,tfor all x,y∈X, such that Tx≠Ty. Thus every γ-contraction is a continuous mapping.Theorem 7.
Let(X,M,∗) be a non-Archimedean fuzzy metric space and let T:X→X be a γ-contraction. Then T has a unique fixed point in X.Proof.
Letx0∈X be arbitrary and fixed. Define sequence {xn} by (4)Txn=xn+1foralln∈N.If xn=xn+1 then xn+1 is a fixed point of T; then the proof is finished. Suppose that xn≠xn+1 for all n∈N. Therefore by (2), we get (5)γMTxn-1,Txn,t≥γMxn-1,xn,t+δ.
Repeating this process, we get(6)γMTxn-1,Txn,t≥γMxn-1,xn,t+δ=γMTxn-2,Txn-1,t+δ≥γMxn-2,xn-1,t+2δ⋯≥γMx0,x1,t+nδ.
Lettingn→∞, from (6) we get(7)limn→∞γMTxn-1,Txn,t=+∞.Then, we have(8)limn→∞MTxn-1,Txn,t=1.Now, we want to show that {xn} is a Cauchy sequence. Suppose to the contrary, we assume that {xn} is not a Cauchy sequence. Then there are ε∈(0,1) and t0>0 such that for all k∈N there exist n(k),m(k)∈N with n(k)>m(k)>k and(9)Mxnk,xmk,t0≤1-ε.Assume that m(k) is the least integer exceeding n(k) satisfying inequality (9). Then, we have(10)Mxmk-1,xnk,t0>1-εand so, for all k∈N, we get(11)1-ε≥Mxnk,xmk,t0≥Mxmk-1,xmk,t0∗Mxmk-1,xnk,t0≥Mxmk-1,xmk,t0∗1-ε.
By takingk→∞ in (11) and using (8), we obtain(12)limk→∞Mxnk,xmk,t0=1-ε.From (FM-4), we get(13)Mxmk+1,xnk+1,t0≥Mxmk+1,xmk,t0∗Mxmk,xnk,t0∗Mxnk1,xnk+1,t0.
Taking the limit ask→∞ in (13), we obtain(14)limk→∞Mxnk+1,xmk+1,t0=1-ε.By applying inequality (2) with x=xm(k) and y=xn(k)(15)γMxnk+1,xmk+1,t≥γMxnk,xmk,t+δ.Taking the limit as k→∞ in (15), applying (2), from (12), (14), and continuity of γ, we obtain (16)γ1-ε≥γ1-ε+δwhich is a contradiction. Thus {xn} is a Cauchy sequence in X. From the completeness of (X,M,∗) there exists z∈X such that(17)limn→∞xn=z.
Finally, the continuity ofT yields M(Tz,z,t)=limn→∞⁡M(Txn,xn,t)=limn→∞⁡M(xn+1,xn,t)=1. Now, we show that T has a unique fixed point. Suppose that z1 and z2 are two fixed points of T. Indeed, if for z1,z2∈X, Tz1=z1≠z2=Tz2, then we get(18)γMz1,z2,t≥γMz1,z2,t+δwhich is a contradiction. Thus, T has a unique fixed point. Hence, the proof is completed.Example 8.
LetX=[0,1), a∗b=min{a,b},(19)Mx,y,t=11+max⁡x,y,x≠y,1,x=y,for all t>0. Let γ:[0,1)→R such that γ(x)=1/(1-x) for all x∈[0,1) and define T:X→X by T(x)=2x2/5 for all x∈X. Clearly, (X,M,∗) is a non-Archimedean fuzzy metric space.
Case 1. We assume that x<y for all x,y∈(0,1). Since x2<x and y2<y, then max⁡{x,y}>max⁡{Tx,Ty}. So, there exists δ∈(0,1) such that (20)1max⁡Tx,Ty+1≥1max⁡x,y+1+δ.Therefore(21)1+max⁡Tx,Tymax⁡Tx,Ty≥1+max⁡x,ymax⁡x,y+δ,and so(22)1+max⁡Tx,Ty1+max⁡Tx,Ty-1≥1+max⁡x,y1+max⁡x,y-1+δ,and, then, we get(23)11-1/1+max⁡Tx,Ty≥11+1/max⁡x,y+δ,which implies(24)11-MTx,Ty,t≥11+Mx,y,t+δ.That is,(25)γMTx,Ty,t≥γMx,y,t+δ.Case 2. Let x=0 and y∈(0,1). Since x2=0, y2<y, then max⁡{x,y}=y>y2=max⁡{Tx,Ty}. Hence, we have (26)MTx,Ty,t=11+max⁡Tx,Ty>11+max⁡x,y=Mx,y,t. So, there exists δ∈(0,1) such that (27)11-MTx,Ty,t≥11-Mx,y,t+δ.That is,(28)γMTx,Ty,t≥γMx,y,t+δ. Therefore, T is a γ- contraction. Then all the conditions of Theorem 7 hold and T has a unique fixed point x=0.Definition 9.
Let(X,M,∗) be a non-Archimedean fuzzy metric space. A mapping T:X→X is said to be a γ-weak contraction if there exists δ∈(0,1) such that (29)MTx,Ty,t<1⇒γMTx,Ty,t≥γmin⁡Mx,y,t,Mx,Tx,t,My,Ty,t+δfor all x,y∈X and γ∈Γ.Remark 10.
Everyγ-contraction is a γ-weak contraction. But the converse is not true.Example 11.
LetX=A∪B where A={1/10,1/2,1,2,3}, B=[4,6]. a∗b=min⁡{a,b} and M(x,y,t)=min⁡{x,y}/max⁡{x,y} for all t>0. Clearly, (X,M,∗) is a complete non-Archimedean fuzzy metric space. Let γ:[0,1)→R such that γ(x)=1/1-x for all x∈[0,1) and define T:X→X by (30)Tx=110,x∈A,12,x∈B.Since T is not continuous, T is not γ- contraction by Remark 6.
Now, we show thatT is a γ-weak contraction for all x∈A and y∈B.
Case 1. Let x=1 and y∈B, (31)MTx,Ty,t=15>12y=min⁡xy,110x,12y=min⁡Mx,y,t,Mx,Tx,t,My,Ty,t).Then we have (32)γ11-1/5>γ11-1/2y.So, there exists δ∈(0,1) such that (33)γMTx,Ty,t≥γmin⁡Mx,y,t,Mx,Tx,t,My,Ty,t+δ.Case 2. Let x∈{2,3} and y∈B, (34)MTx,Ty,t=15>110x=min⁡xy,110x,12y=min⁡Mx,y,t,Mx,Tx,t,My,Ty,t.Then we have (35)γ11-1/5>γ11-1/10x.So, there exists δ∈(0,1) such that (36)γMTx,Ty,t≥γmin⁡Mx,y,t,Mx,Tx,t,My,Ty,t+δ.Case 3. Let x∈{1/10,1/2} and y∈B, (37)MTx,Ty,t=15>xy=min⁡xy,110x,12y=min⁡Mx,y,t,Mx,Tx,t,My,Ty,t.Then we have (38)γ11-1/5>γ11-x/y. So, there exists δ∈(0,1) such that (39)γMTx,Ty,t≥γmin⁡Mx,y,t,Mx,Tx,t,My,Ty,t+δ. Therefore, T is a γ-weak contraction.Theorem 12.
Let(X,M,∗) be a non-Archimedean fuzzy metric space and let T:X→X be a γ-weak contraction. Then T has a unique fixed point in X.Proof.
Letx0∈X be arbitrary and fixed. Define sequence {xn} by (40)Txn=xn+1foralln∈N.If xn=xn+1 then xn+1 is a fixed point of T; then the proof is finished. Suppose that xn≠xn+1 for all n∈N. Therefore by (29), we get(41)γMTxn-1,Txn,t≥γmin⁡Mxn-1,xn,t,Mxn-1,Txn-1,t),Mxn,Txn,t+δ=γmin⁡Mxn-1,xn,t,Mxn-1,xn,t),Mxn,xn+1,t+δ=γmin⁡Mxn-1,xn,t,Mxn,xn+1,t+δ.
If there existsn∈N such that (42)min⁡Mxn-1,xn,t,Mxn,xn+1,t=Mxn,xn+1,t, from (41) becomes (43)γMTxn-1,Txn,t=Mxn,xn+1,t≥γMxn,xn+1,t+δ>Mxn,xn+1,t which is a contradiction, therefore,(44)min⁡Mxn-1,xn,t,Mxn,xn+1,t=Mxn-1,xn,tfor all n∈N. That is, from property of γ, (41), and (44), we get(45)Mxn,xn+1,t>Mxn-1,xn,t.Thus, from (41), we have (46)γMxn,xn+1,t≥γMxn-1,xn,t+δfor all n∈N. It implies that (47)γMxn,xn+1,t≥γMxn-1,xn,t+nδ.By taking n→∞ in (47), we get(48)limn→∞γMxn,xn+1,t=limn→∞γMTxn-1,Txn,t=+∞.Then, we have(49)limn→∞MTxn-1,Txn,t=1.The proof that {xn} is a Cauchy sequence can be shown as in Theorem 7. From the completeness of (X,M,∗) there exists z∈X such that(50)limn→∞xn=z. Now, we show that z is a fixed point of T. Since γ is continuous, there are two cases.
Case 1. For each n∈N, there exists in∈N such that xin+1=Tz and in>in-1, where i0=1. Then, we get(51)z=limn→∞xin+1=limn→∞Tz=Tz.This proves that z is a fixed point of T.
Case 2. There exists n0∈N such that xn+1≠Tz for all n≥n0. That is, M(Txn,Tz,t)<1 for all n≥n0. It follows from (29), property of γ,(52)γMxn+1,Tz,t=γMTxn,Tz,t≥γmin⁡Mxn,z,t,Mxn,Txn,t,Mz,Tz,t+δ=γmin⁡Mxn,z,t,Mxn,xn+1,t,Mz,Tz,t+δ.If M(z,Tz,t)<1, then we have(53)limn→∞Mxn,z,t=1,and there exists n1∈N such that for all n≥n1, we get(54)min⁡Mxn,z,t,Mxn,xn+1,t,Mz,Tz,t=Mz,Tz,t.From (52), we have(55)γMxn+1,Tz,t≥γMz,Tz,t+δfor all n≥max{n0,n1}. Since γ is continuous, taking the limit as n→∞ in (55), we obtain(56)γMz,Tz,t≥γMz,Tz,t+δwhich is a contradiction. Therefore, Mz,Tz,t=1; that is, z is a fixed point of T.
Now, we prove that the fixed point ofT is unique. Let z1,z2 be two fixed points of T. Suppose that z1≠z2; then we have Tz1≠Tz2. It follows from (29) that we have(57)γMz1,z2,t=γMTz1,Tz2,t≥γminMz1,z2,t,Mz1,Tz1,t,Mz2,Tz2,t+δ=γminMz1,z2,t,Mz1,z1,t,Mz2,z2,t+δ=γMz1,z2,t+δwhich is a contradiction. Then, M(z1,z2,t)=1, that is, z1=z2. Therefore, the fixed point of T is unique.Example 13.
Let(X,M,∗) be the non-Archimedean fuzzy metric space and let T be considered in Example 11. Let γ:[0,1)→R such that γ(x)=1/(1-x2) for all x∈[0,1). So, T is a γ-weak contraction. Therefore, Theorem 12 can be applicable to T and the unique fixed point of T is 1/10.Conclusion 14.
In this paper, we introduce new contraction types in non-Archimedean fuzzy metric spaces and presented new fixed point results. Our results can be expanded and solutions to new problems can be produced in this way. Also, a new more general contraction can be achieved or common fixed point theorems for a class of mappings can be obtained usingγ-contractions and also our result can be extended to other spaces.



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