# Fixed Point Theorems for Generalized Kannan-Type Mappings in a New Type of Fuzzy Metric Space

Authors: Mi Zhou; Xiao-lan Liu; Nicolae Adrian Secelean
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, first, we introduce a new type ofS∗−fuzzy metric space which is a generalization of fuzzy metric spaces. Second, we study the topological properties of S∗−fuzzy metric spaces. Finally, we extend Kannan-type mappings to generalized Kannan-type mappings under ϕ−gauge functions introduced by Fang in S∗−fuzzy metric spaces and prove the existence and uniqueness of fixed point for this kind of mappings. Furthermore, we also obtain the common fixed point theorems for weak compatibility along with E.A. property or CLRg property. Our results extend and improve very recent theorems in the related literature.


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## Body

## 1. Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy sets. Since then, one of the important problems is to obtain an adequate notion of fuzzy metric spaces. In 1975, Kramosil and Michálek [2] reformulated successfully the notion probabilistic metric space, introduced by Menger in 1942, in fuzzy context. After that, George and Veeramani [3] modified the concept of fuzzy metric spaces and defined a Hausdorff topology on this fuzzy space. Another approach for fuzzy metric spaces was proposed by Kaleva and Seikkala [4], by setting the distance between two points to be a nonnegative upper semicontinuous, normal, and convex fuzzy number. In the last several decades, there has been a tremendous development and growth in fuzzy mathematics. One of the branches is to obtain several kinds of generalized fuzzy metric proposed by using related generalized metric spaces: 2−metric spaces [5, 6], D−metric spaces [7], G−metric spaces [8], and D∗−metric spaces [9], respectively. Very recently, Sedghi et al. [10] introduced an S−metric space which is a generalization of D∗−metric spaces and G−metric spaces and justified their work by various examples and definitions related to the topology of S−metric spaces. Now, there arises a natural question: “how the fuzzy metric spaces can be generalized by using the concept of S−metric spaces?”In 1968, Kannan [11] introduced the Kannan-type mappings as follows.Definition 1.
(see [11]). Let X,d be a metric space and T be a mapping on X. We say that T is a Kannan-type mapping, if there exists 0≤k<1/2 such that(1)dTx,Ty≤kdx,Tx+dy,Ty,∀x,y∈X.
This kind of mapping is very important in metric fixed point theory. It is well known that Banach’s contraction mappings are continuous while Kannan-type mappings are not necessary continuous. This is a big difference between these two types of mappings. Again, it may also be noted that Banach’s contraction does not characterize metric completeness. In fact, Subrahmanyam [12] proved that every metric space X is complete if and only if every Kannan-type mapping has a fixed point. However, it was also pointed out that Kannan’s fixed point result is not an extension of Banach contraction principle. The above shows some of the reasons why the Kannan-type mappings and their generalizations have been considered as constituting an important class of mappings in fixed point theory.
On the other hand, an interesting class of problems in fixed point theory was addressed in recent times by use of gauge (control) functions. There are several gauge (control) functions which have been used to extend Sehgal’s contraction in probabilistic metric spaces. Some examples of such applications are in [13–18]. One of such gauge functions was introduced in the setup of complete Menger probabilistic metric spaces and fuzzy metric spaces by Fang [17], where the gauge function ϕ satisfies the condition: for each t>0, there exists r≥t such that limn⟶∞ϕnr=0, which is considered as weaker than the condition presented by Francisco et al. in [19]. It is perceived that the study of fixed points for contractions under some gauge (control) functions is an important category of problems in fixed point theory.
In this reason, in Section2, we introduce a new type of generalized fuzzy metric spaces called S∗−fuzzy metric spaces which is a generalization of Q−fuzzy metric spaces and M−fuzzy metric spaces. We also study the topological properties and prove some interesting results related its topology and convergent sequences. By using the gauge functions introduced by Fang in S∗−fuzzy metric spaces, we introduce in Section 3 the generalized Kannan-type mappings as an extension of the concept of Kannan-type mappings and prove the existence and uniqueness of fixed point for this kind of mapping. Furthermore, we also obtain the common fixed point theorems for weak compatibility along with E.A. or CLRg property.


## 2.S∗−Fuzzy Metric Spaces

We begin by recalling some basic definitions concerningS−metric spaces, fuzzy metric spaces, etc.Definition 2.
(see [10]). Let X be a nonempty set. An S−metric on X is a function S:X3⟶0,∞ that satisfies the following conditions for ∀x,y,z,a∈X:(S1) Sx,y,z=0⟺x=y=z.(S2) Sx,y,z≤Sx,x,a+Sy,y,a+Sz,z,a.
The pairX,S is called an S−metric space.Remark 1.
(see [10]). Every D∗−metric space is a G-metric space and every G-metric space is an S−metric space, but in general, the converse of these implications are not true (see [10] for more details).Definition 3.
(see [20]). A mapping ∗:Πi=1n0,1⟶0,1 is called an nth order t−norm if the following conditions are satisfied:(1)
∗0,0,…,0=0,∗a,1,1,…,1=a,∀a∈0,1.(2)
∗a1,a2,a3,…,an=∗a2,a1,a3,…,an=⋯=∗a2,a3,a4,…,an,a1.(3)
ai≥bi,i=1,2,3,…,n  implies  ∗a1,a2,a3,…,an≥∗b1,b2,b3,…,bn.(4)
∗∗a1,a2,a3,…,an,b2,b3,…,bn=∗a1,∗a2,a3,…,an,b2,b3,…,bn=∗a1,a2,∗a3,a4,…,an,b2,b3,b4,…,bn=∗a1,a2,a3,…,an−1,∗an,b2,b3,b4,…,bn.
Whenn=2, we have binary t−norm, which is well known as t−norm. Typical examples of t−norm are Lukasiewiczt−norm ∗La,b=maxa+b−1,0, product t−norm ∗Pa,b=a⋅b, and minimum t−norm ∗Ma,b=mina,b.Definition 4.
(see [21]). Let ∗ be a t−norm and let ∗n:0,1⟶0,1n∈ℕ be defined as follows:(2)∗1x=∗x,x,∗n+1x=∗∗nx,x,n∈ℕ,x∈0,1.
We say that thet−norm is of H−type, if ∗ is continuous and the sequence ∗nxn=1∞ is equicontinuous at x=1; i.e., for every ε∈0,1, there exists η∈0,1, such that if x∈1−η,1, then ∗nx>1−ε, ∀n∈ℕ. A trivial example of t−norm of H−type is ∗M.Definition 5.
(see [3]). A 3−tuple X,M,∗ is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t−norm, and M is a fuzzy set on X2×0,∞ satisfying the following conditions for ∀x,y∈X and t,s>0:(FM1)Mx,y,t>0.(FM2)Mx,y,t=1⟺x=y.(FM3)Mx,y,t=My,x,t.(FM4)Mx,y,t∗My,z,s≤Mx,z,t+s.(FM5)Mx,y,⋅:0,∞⟶0,1 is a continuous function.
The pairM,∗ is called a fuzzy metric on X. Here, Mx,y,t is considered as the degree of nearness of x and y with respect to t.
Next, we define theS∗−fuzzy metric space by using the concept of S−metric.Definition 6.
A tripleX,S∗,∗ is called an S∗−fuzzy metric space (denoted S∗FMS) if X is an arbitrary nonempty set, ∗ is a continuous 3rdt−norm, and S∗ is a fuzzy set on X3×0,∞ satisfying the following conditions for ∀x,y,z,a∈X and r,s,t>0:S∗FM1S∗x,y,z,t>0.S∗FM2S∗x,y,z,t=1⟺x=y=z.S∗FM3S∗x,y,z,r+s+t≥∗S∗x,x,a,r,S∗y,y,a,s,S∗z,z,a,t.S∗FM4S∗x,y,z,⋅:0,∞⟶0,1 is a continuous function.Example 1.
LetX=ℝ be the real line and S be an S−metric on X defined by Sx,y,z=y+z−2x+y−z. Then, X,S is an S−metric space. Define ∗a,b,c=a⋅b⋅c, ∀a,b,c∈0,1. Let S∗ be the function on X3×0,∞ defined by S∗x,y,z,t=t/t+Sx,y,z, ∀x,y,z∈X and t>0.
It is easy to check thatX,S∗,∗ is an S∗−fuzzy metric space. Moreover, it is neither Q−fuzzy metric space nor M−fuzzy metric space because S∗ is not symmetric.Example 2.
LetX=ℝ and S be an S−metric defined by Sx,y,z=x−y+y−z, ∀x,y,z∈ℝ. Define ∗a,b,c=a⋅b⋅c, ∀a,b,c∈0,1, and let S∗ be the function on ℝ3×0,∞ defined by S∗x,y,z,t=exp−Sx,y,z/t, ∀x,y,z∈ℝ and t>0. It is also easy to verify that X,S∗,∗ is an S∗−fuzzy metric space.Example 3.
LetX,M,∗ be a fuzzy metric space with continuous t−norm ∗. If we denote S∗:X3×0,∞⟶0,1 by S∗x,y,z,t=Mx,y,t∗My,z,t∗Mz,x,t, ∀x,y,z∈X and t>0, then X,S∗,∗ is an S∗−fuzzy metric space.Proof.
(1)
It is easy to see that, for everyx,y,z∈X,t>0, S∗x,y,z,t>0 and S∗x,y,z,⋅ is continuous.(2)
S∗x,y,z,t=1⟺Mx,y,t=My,z,t=Mz,x,t=1⟺x=y=z, ∀x,y,z∈X, t>0.(3)
For allx,y,z∈X and t,r,s>0, we have(3)S∗x,y,z,t+r+s=Mx,y,t+r+s∗My,z,t+r+s∗Mz,x,t+r+s≥Mx,a,t+s2∗My,a,r+s2∗My,a,r+t2∗Mz,a,s+t2∗Mx,a,t+r2∗Mz,a,s+r2≥S∗x,x,a,t∗S∗y,y,a,r∗S∗z,z,a,s.
Therefore, the conditions of Definition6 are satisfied, and X,S∗,∗ is an S∗−fuzzy metric space.Example 4.
Suppose thatX1,S1∗,∗ and X2,S2∗,∗ are two S∗-fuzzy metric spaces. The fuzzy set D∗ is defined as(4)D∗x1,x2y1,y2,z1,z2,t=minS1∗x1,y1,z1,t,S2∗x2,y2,z2,t,∀x1,x2,y1,y2,z1,z2∈X1×X2andt>0 is an S∗−fuzzy metric on X1×X2.Proof.
It is not difficult to verify that the conditionsS∗FM1,S∗FM2, and S∗FM4 in Definition 6 hold.
It remains to prove that the conditionS∗FM3 holds. One has(5)D∗x1,x2,y1,y2,z1,z2,r+s+t=minS1∗x1,y1,z1,r+s+t,S2∗x2,y2,z2,r+s+t≥min∗S1∗x1,x1,a,r,S1∗y1,y1,a,s,S1∗z1,z1,a,t,∗S2∗x1,x1,a,r,S2∗y1,y1,a,s,S2∗z1,z1,a,t≥∗D∗x1,x2,x1,x2,a,a,r,D∗y1,y2,y1,y2,a,a,s,D∗z1,z2,z1,z2,a,a,t,∀x1,x2,y1,y2,z1,z2∈X1×X2andt>0, as required.
Consequently, the fuzzy setD∗ is an S∗−fuzzy metric on X1×X2.Proposition 1.
IfX,S∗,∗ is an S∗−fuzzy metric space, then S∗x,x,y,t=S∗y,y,x,t, ∀x,y∈X and t>0.Proof.
For eachε>0, from S∗FM3, we deduce(6)S∗x,x,y,2ε+t≥∗S∗x,x,x,ε,S∗x,x,x,ε,S∗y,y,x,t=S∗y,y,x,t,S∗y,y,x,2ε+t≥∗S∗y,y,y,ε,S∗y,y,y,ε,S∗x,x,y,t=S∗x,x,y,t.
By taking the limit in above equalities asε⟶0, we obtain S∗x,x,y,t=S∗y,y,x,t.Proposition 2.
S∗x,x,y,t is nondecreasing with respect to t, ∀x,y∈X.Proof.
Suppose thatS∗x,x,y,t>S∗x,x,y,s for some 0<t<s, with s/2>t.
Then,(7)∗S∗x,x,x,s2,S∗x,x,x,s2−t,S∗y,y,x,t≤S∗x,x,y,s<S∗x,x,y,t.
ByS∗FM2, we have(8)S∗x,x,x,s2=S∗x,x,x,s2−t=1.
Therefore, by Proposition1 and Definition 3, we have(9)S∗y,y,x,t=S∗x,x,y,t≤S∗x,x,y,s<S∗x,x,y,t,which is a contradiction. Hence, the conclusion holds.Proposition 3.
LetX,S∗,∗ be an S∗−fuzzy metric space. Then, for ∀x,y∈X, it follows that(1)
S∗x,y,y,t1≥S∗x,x,y,t2, ∀t1>t2>0.(2)
S∗x,y,x,t1≥S∗x,x,y,t2, ∀t1>t2>0.Proof.
The conclusion easily follows from Definition6 and Proposition 2.Theorem 1.
LetX,S∗,∗ be an S∗−fuzzy metric space. For every x∈X,t>0, the open ball Bx,r,t with center x and radius 0<r<1 is defined by(10)Bx,r,t=y∈X:S∗y,y,x,t>1−r.
Then,TS∗:=T⊂X:x∈T⟺∃t>0,r∈0,1such thatBx,r,t⊆T is a topology on X.Proof.
It is obvious that∅ and X belong to TS∗.
LetTii∈I⊆TS∗ and T=∪i∈ITi. We will show that T∈TS∗.
Choosex∈T. There exists i0∈I such that x∈Ti0. As Ti0∈TS∗, there exist t>0 and r∈0,1 such that Bx,r,t⊆Ti0. Thus, Bx,r,t⊆∪i∈ITi=T.
Now, letTii=1n⊆TS∗ and T=∩Tin. We will show that T∈TS∗.
Ifx∈T, then x∈Ti, ∀i∈1,2,…,n. Thus, one can find ti>0 and ri∈0,1 such that Bx,ri,ti⊆Ti, ∀i∈1,2,…,n. Putting r=minr1,r2,…,rnandt=mint1,t2,…,tn, one obtains Bx,r,t⊆Bx,ri,ti, ∀i. for all i in 1,2,…,n, Indeed, for y∈Bx,r,t, we have S∗y,y,x,t>1−r≥1−ri, ∀i. for all i in 1,2,…,n, Since, t≤ti, we have S∗y,y,x,t≤S∗y,y,x,ti, ∀i∈1,2,…,n. Thus, S∗y,y,x,ti≥1−ri, ∀i∈1,2,…,n. Hence, y∈Bx,ri,ti, ∀i∈1,2,…,n. Therefore, Bx,r,t⊆Ti, ∀i∈1,2,…,n, so Bx,r,t⊆∩Tin=T.Theorem 2.
EveryS∗−fuzzy metric space is Hausdorff.Proof.
LetX,S∗,∗ be an S∗−fuzzy metric space, and let xandy be two distinct points of X. Then, 0<S∗x,x,y,t<1.
For somet>0, we denote r=S∗x,x,y,t. For every r0, r<r0<1, we can find r1 such that r1∗r1∗r1≥r0. Now, we consider the open balls B1x,1−r1,t/3 and B2y,1−r1,t/3. We claim that(11)B1x,1−r1,t3∩B2y,1−r1,t3=∅.
Indeed, if there existsz∈B1x,1−r1,t/3∩B2y,1−r1,t/3, then(12)r=S∗x,x,y,t≥∗S∗x,x,z,t3,S∗x,x,z,t3,S∗y,y,z,t3>∗r1,r1,r1≥r0>r,which is a contradiction. Therefore, X,S∗,∗ is Hausdorff.Definition 7.
LetX,S∗,∗ be an S∗−fuzzy metric space and A⊆X. We say that A is FS∗− bounded if, for every x,y∈A and t>0, there exists r∈0,1 such that S∗x,x,y,t≥1−r. Furthermore, a sequence xn in X is said to be FS∗−bounded if, for ∀t>0 and all n∈ℕ, there exists r∈0,1 such that xn∈B¯x,r,t, where B¯x,r,t is a closed ball with center x and radius 0<r<1 defined by B¯x,r,t=y∈X:S∗y,y,x,t≥1−r.Definition 8.
A sequencexn in X,S∗,∗ is convergent tox∈X, if S∗xn,xn,x,t⟶1 or S∗x,x,xn,t⟶1 as n⟶∞ for each t>0. That is, for each ε>0 and t>0, there exists n0∈ℕ such that, for ∀n≥n0, S∗xn,xn,x,t>1−ε or S∗x,x,xn,t>1−ε. Furthermore, a subset Y of an S∗−fuzzy metric space X,S∗,∗ is said to be closed if xn⊆Y and xn⟶x imply x∈Y.Definition 9.
A sequencexn in an S∗−fuzzy metric space X,S∗,∗ is said to be Cauchy if, for each ε>0 and t>0, there exists n0∈ℕ such that S∗xn,xn,xm,t>1−ε or S∗xm,xm,xn,t>1−ε, ∀m,n>n0. If every Cauchy sequence in X is convergent in X, then X is called a completeS∗−fuzzy metric space.Remark 2.
(1)
LetX,S∗,∗ be an S∗−fuzzy metric space. A sequence xn is convergent to x if and only if xn is convergent to x in the topology TS∗. That is,(13)xn⟶TS∗x⟺∀r∈0,1,t>0,∃n0∈ℕ:xn∈Bx,r,t,∀n>n0,⟺∀r∈0,1,t>0,∃n0∈ℕ:S∗xn,xn,x,t>1−r,∀n>n0,⟺limn→∞S∗xn,xn,x,t=1,∀t>0.(2)
It is easy to prove that theS∗−fuzzy metric space X,S∗,∗ is complete if and only if the S−metric space is complete where S∗x,y,z,t=t/t+Sx,y,z, ∀x,y,z∈X and t>0.Lemma 1.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type. Let xn be a sequence in X with xn⟶x and xn⟶y as n⟶∞. Then, x=y.Proof.
Assume thatxn converges to x and y. Then, S∗x,x,xn,r⟶n1, for each r>0, and S∗y,y,xn,t−2r⟶n1, for each t−2r>0:(14)S∗x,x,y,t≥∗S∗x,x,xn,r,S∗x,x,xn,r,S∗y,y,xn,t−2r⟶n1.
Therefore,x=y.Lemma 2.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type and xn be a convergent sequence. Then,(1)
xn is FS∗− bounded and its limit is unique.(2)
xn is a Cauchy sequence.(3)
Every subsequence ofxn converges to the same limit.(4)
Every Cauchy sequence isFS∗− bounded.(5)
If every Cauchy sequencexn in X has a subsequence xn such that xnk⟶x, then xn⟶x.Proof.
(1)
First, we show that the convergent sequencexn is FS∗−bounded. Suppose that xn⟶x. Then, for each ε∈0,1 and t>0, there exists n0∈ℕ such that S∗xn,xn,x,t/3>1−ε, ∀n≥n0. Choose x0∈X and s∈0,1 such that S∗x0,x0,x,t/3>1−s and let k∈0,1 be such that(15)minS∗xn,xn,x,t3:n=1,2,3,…,n0=1−k.Then, we can find a numberr∈0,1 such that(16)min∗1−ε,1−ε,1−s,∗1−k,1−k,1−s=1−r.So, for∀n∈ℕ, we have(17)S∗xn,xn,x,t≥∗S∗xn,xn,x,t3,S∗xn,xn,x,t3,S∗x0,x0,x,t3≥1−r.Thus, for∀n∈ℕ, xn⊆B¯x,r,t, that is, xn is FS∗−bounded (the bar means the closure).Now, we show that the limit ofxn is unique. We suppose that the convergent sequence xn has two distinct limits x and y. For any ε∈0,1 and t>0, we can find a number r∈0,1 such that ∗1−r,1−r,1−r>1−ε. Set ε=1−S∗x,x,y,t. From our assumption, there exist n1andn2∈ℕ such that S∗xn,xn,x,t/3>1−r, ∀n≥n1, and S∗xn,xn,y,t/3>1−r, ∀n≥n2. Taking n0=maxn1,n2, we have for n≥n0,(18)S∗x,x,y,t≥∗S∗xn,xn,x,t3,S∗xn,xn,x,t3,S∗xn,xn,y,t3≥∗1−r,1−r,1−r>1−ε=S∗x,x,y,t,which impliesx=y.(2)
For eachr,t>0 with t−2r>0, one can find n0∈ℕ such that, for p∈ℕ,(19)S∗xn,xn,x,r⟶n1,forr>0,S∗xn+p,xn+p,x,t−2r⟶n1,forr,t−2r>0,S∗xn,xn,xn+p,t≥∗S∗xn,xn,x,r,S∗xn,xn,x,r,S∗xn+p,xn+p,x,t−2r=∗1,1,1⟶n1.Therefore,xn is a Cauchy sequence.(3)
Letxnk be a subsequence of xn. If xn⟶x, then for each ε∈0,1 and t>0, there exists n0∈ℕ such that S∗xn,xn,x,t>1−ε, ∀n≥n0. If k≥n0, then n0≤k≤nk and so S∗xnk,xnk,x,t>1−ε which implies that xnk converges to x.(4)
Letxn be a Cauchy sequence in X. Then, for each ε∈0,1 and t>0, there exists n0∈ℕ such that S∗xn,xn,xm,t>1−ε, ∀m,n≥n0. So, for n≥n0, we have S∗xn,xn,xn0,t>1−ε. Let r=1−minS∗xn,xn,xn0,t:n=1,2,…,n0−1 and choose s=maxε,r. Then, xn⊆B¯x,r,t, that is, xn is FS∗−bounded.(5)
Lett>0 and ε∈0,1. We can find a number r∈0,1 such that ∗1−r,1−r,1−r≥1−ε. Since xn is a Cauchy sequence in X, there exists n0∈ℕ such that S∗xn,xn,xm,t/3>1−r, ∀m,n≥n0. From xnk⟶kx, there exists a positive integer ik such that ik>n0 and S∗xik,xik,x,t/3>1−r. For ∀n≥n0, we have(20)S∗xn,xn,x,t≥∗S∗xn,xn,xik,t3,S∗xn,xn,xik,t3,S∗xik,xik,x,t3>∗1−r,1−r,1−r>1−ε,soxn⟶x.This completes the proof.Lemma 3.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type. Let xn and yn be two sequences in X with xn⟶nx, yn⟶ny, and S∗x,x,y,tn⟶nS∗x,x,y,t. Then, S∗xn,xn,yn,tn⟶nS∗x,x,y,t.Proof.
Sincexn⟶x, yn⟶y, and S∗x,x,y,tn⟶S∗x,x,y,t as n⟶∞, there exists n0∈ℕ such that t−tn<δ for n≥n0 and δ<t/2.
By the nondecreasing property ofS∗x,x,y,t with respect to t, we have(21)S∗xn,xn,yn,tn≥S∗xn,xn,yn,t−δ≥∗S∗xn,xn,x,δ3,S∗xn,xn,x,δ3,S∗yn,yn,x,t−5δ3≥∗S∗xn,xn,x,δ3,S∗xn,xn,x,δ3,∗S∗yn,yn,y,δ6,S∗yn,yn,y,δ6,S∗y,y,x,t−2δ,S∗x,x,y,t+2δ≥S∗x,x,y,tn+δ≥∗S∗x,x,xn,δ3,S∗x,x,xn,δ3,S∗y,y,xn,tn+δ3≥∗S∗x,x,xn,δ3,S∗x,x,xn,δ3,∗S∗y,y,yn,δ6,S∗y,y,yn,δ6,S∗xn,xn,yn,tn.
Sinceδ is arbitrary chosen, using Definition 8 and the continuity of S∗x,x,y,t with respect to t, for large enough n, we have(22)S∗x,x,y,t≥S∗xn,xn,yn,tn≥S∗y,y,x,t=S∗x,x,y,t.
Therefore,limn⟶∞S∗xn,xn,yn,tn=S∗x,x,y,t.Remark 3.
From Lemma3, we can conclude that S∗ is a continuous function on X3×0,∞ in every S∗−fuzzy metric space X,S∗,∗.Lemma 4.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type. If there exists k∈0,1 such that S∗x,x,y,kt≥S∗x,x,y,t, ∀x,y∈X,t>0, and limt⟶∞S∗x,x,y,t=1, then x=y.Proof.
If there existsk∈0,1 such that S∗x,x,y,kt≥S∗x,x,y,t, ∀x,y∈Xandt>0. Then, we have(23)S∗x,x,y,t≥S∗x,x,y,tk≥S∗x,x,y,tk2≥⋯≥S∗x,x,y,tkn,for all positive integer n. Taking the limit as n⟶∞, we have(24)S∗x,x,y,t=1.
Hence, we havex=y.Lemma 5.
LetX,S∗,∗ be an S∗−fuzzy metric. For each r∈0,1, define a function Sr∗ by Sr∗x,y,z=inftt≥0:S∗x,y,z,t>1−r, ∀x,y,z∈X. Then,(1)
Sr∗x,y,z<t⟺S∗x,y,z,t>1−r.(2)
Sr∗x,y,z=0,∀r∈0,1⟺x=y=z.(3)
Sr∗x,x,y=Sr∗y,y,x,∀x,y∈X.(4)
If∗=∗M, then, for every r∈0,1, Sr∗x,y,z≤Sr∗x,x,a+Sr∗y,y,a+Sr∗z,z,a∀x,y,z,a∈X.Proof.
It is not difficult to see that (1)–(3) hold.
Now, we prove (4).
Let anyr∈0,1 and ε>0. For ∀x,y,z,a∈X, we have(25)S∗x,x,a,Sr∗x,x,a+ε3>1−r,S∗y,y,a,Sr∗y,y,a+ε3>1−r,S∗z,z,a,Sr∗z,z,a+ε3>1−r.
Hence, fromS∗FMS3 and ∗=∗M, one obtains(26)S∗x,y,z,Sr∗x,x,a+Sr∗y,y,a+Sr∗z,z,a+ε≥minS∗x,x,a,Sr∗x,x,a+ε3,S∗y,y,a,Sr∗y,y,a+ε3,S∗z,z,a,Sr∗z,z,a+ε3>min1−r,1−r,1−r=1−r,which implies(27)Sr∗x,y,z≤Sr∗x,x,a+Sr∗y,y,a+Sr∗z,z,a+ε.
Lettingε⟶0, we have(28)Sr∗x,y,z≤Sr∗x,x,a+Sr∗y,y,a+Sr∗z,z,a,∀x,y,z,a∈X.Definition 10.
LetX,S∗,∗ be an S∗−fuzzy metric space and A be a nonempty subset of X. We define S∗−fuzzy diameter DAS∗ as(29)DAS∗=supt>0infS∗x,x,y,t:x,y∈A.Theorem 3.
TheS∗−fuzzy diameter DAS∗ has the following properties:(1)
DAS∗=1⇔A is a singleton.(2)
IfA⊆B, then DAS∗≥DBS∗.(3)
For anyx,y∈A, S∗x,x,y,t≥DAS∗, ∀t>0.(4)
IfA=x,y, then DAS∗=S∗x,x,y,t, ∀t>0.Proof.
The conclusions can be easily obtained from Definition10.


## 3. Fixed Point Theorems for Kannan-Type Mappings underϕ−Gauge Functions

Definition 11.
(see [17]). Let Φw denote the class of all functions ϕ:ℝ+⟶ℝ+ satisfying the following condition: for each t>0, there exists r≥t such that limn⟶∞ϕnr=0.Lemma 6.
(see [17]). If ϕ∈Φw, then for ∀t>0, there exists r≥t such that ϕr<t.Definition 12.
A mapψ:0,1×0,1⟶0,1 is said to be a Ψ−function, if(1)
ψ is monotonically increasing and continuous.(2)
ψx,x>x,∀0<x<1.(3)
ψ1,1=1,ψ0,0=0.Definition 13.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type. A mapping f:X⟶X is said to be a ϕ−contraction if there exists a function ϕ∈Φw such that(30)S∗fx,fy,fz,ϕt≥S∗x,y,z,t,∀x,y,z∈X and t>0.Lemma 7.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type and a self-mapping f on X be a ϕ−contraction. Let xn+1=fxn be the iterative sequence generated by x0∈X, ∀n∈ℕ. If there exists a function ϕ∈Φw such that(1)
ϕt>0,∀t>0.(2)
S∗xn,xn,xn+1,ϕt≥S∗xn−1,xn−1,xn,t, ∀n∈ℕ and t>0.(3)
limt⟶∞S∗x,x,y,t=1, ∀x,y∈X.
Thenxn is a Cauchy sequence in X.Proof.
From Assumption 1, it follows thatϕnt>0, ∀t>0.
From Assumption 2, we obtain, by induction,(31)S∗xn,xn,xn+1,ϕnt≥S∗x0,x0,x1,t,∀n∈ℕ and t>0.
Now, we will prove that, for∀t>0,(32)limn⟶∞S∗xn,xn,xn+1,t=1.
SinceS∗x0,x0,x1,t⟶1 as t⟶∞, ∀ε∈0,1, there exists t1>0 such that S∗x0,x1,x1,t>1−ε. Since ϕ∈Φw, there exists t2≥t1 such that limn⟶∞ϕnt2=0. Thus, for ∀t>0, there exists n0≥1 such that ϕnt2<t, ∀n≥n0. Using the inequalities t>ϕnt2, t2≥t1, (31), and the monotonicity property of S∗x,y,z,⋅ with respect to t, we have(33)S∗xn,xn,xn+1,t≥S∗xn,xn,xn+1,ϕnt2≥S∗x0,x0,x1,t2≥S∗x0,x0,x1,t1>1−ε,∀t>0. Thus, (32) holds.
Sinceϕ∈Φw, by Lemma 6, we deduce that, for any t>0, there exists r≥t such that ϕr<t.
Letn≥1 be given. Now, we show that for ∀k≥1,(34)S∗xn,xn,xn+k,t≥∗kS∗xn,xn,xn+1,t−ϕr2.
From (34), we obtain(35)S∗xn,xn,xn+1,t≥∗S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2≥∗1S∗xn,xn,xn+1,t−ϕr2.
Thus, (34) holds for k=1. Assume that (34) holds for some k≥1. Since ∗ is monotone, from S∗FM3 in Definition 6 and, then, by conditions (31) and (34), it follows that(36)S∗xn,xn,xn+k+1,t=S∗xn,xn,xn+k+1,t−ϕr2+t−ϕr2+ϕr≥∗S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2,S∗xn+k+1,xn+k+1,xn+1,ϕr≥∗S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2,S∗xn+k,xn+k,xn,r≥∗S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2,S∗xn+k,xn+k,xn,t≥∗S∗xn,xn,xn+1,t−ϕr2,S∗xn,xn,xn+1,t−ϕr2,∗kS∗xn,xn,xn+1,t−ϕr2=∗k+1S∗xn,xn,xn+1,t−ϕr2,which completes the claim.
Next, we will show thatxn is a Cauchy sequence in X, that is, limn,m,l⟶∞S∗xn,xm,xl,t=1, ∀t>0.
Fixt>0 and 0<ε<1. Since ∗nt is equicontinuous at t=1 and ∗n1=1, there exists δ>0 such that ∗ns>1−ε for s∈1−δ,1 and n≥1.
We first prove thatlimn,m⟶∞S∗xn,xn,xm,t∈1−δ,1, ∀n≥n0. Since t−ϕr/2>0, from (32), there exists n0≥1 such that S∗xn,xn,xn+1,t−ϕr/2>1−δ, ∀n≥n0.
Hence, by (34), we have S∗xn,xn,xn+k,t>1−ε, ∀k≥1.
Thus, we have shown that, for∀t>0,(37)limn,m⟶∞S∗xn,xn,xm,t=1.
ByS∗FM3 with a=xm in Definition 6, we have that, for ∀t>0,(38)S∗xn,xm,xl,t≥∗S∗xn,xn,xm,t3,S∗xm,xm,xm,t3,S∗xl,xl,xm,t3=∗S∗xn,xn,xm,t3,1,S∗xl,xl,xm,t3.
From (37), it follows that, for ∀t>0,(39)limn,m⟶∞S∗xn,xn,xm,t3=1,liml,m⟶∞S∗xl,xl,xm,t3=1.
Thus, by using the continuity of∗, we conclude that(40)limn,m⟶∞S∗xn,xm,xm,t≥∗limn,m⟶∞S∗xn,xm,xm,t3,1,liml,m⟶∞S∗xl,xl,xm,t3=∗1,1,1=1.
Therefore, we have proved that, for∀t>0,(41)limn,m,l⟶∞S∗xn,xm,xl,t=1.
This shows thatxn is a Cauchy sequence in X.Lemma 8.
Letxn be an S∗−fuzzy metric space X,S∗,∗, where ∗ is a continuous 3rdt−norm of H−type. If there exists a function ϕ∈Φw such that S∗x,x,y,ϕt≥S∗x,x,y,t, ∀t>0, and limt⟶∞S∗x,x,y,t=1, then x=y.Proof.
SinceS∗ is monotone, it is obvious that ϕt>0, ∀t>0. By induction, we have ϕnt>0, with ∀t>0andn≥1 and also have, from the assumption, that(42)S∗x,x,y,ϕnt≥S∗x,x,y,t,∀t>0,n≥1.
To prove thatx=y, we need to verify that S∗x,x,y,t=1 for ∀t>0. On the contrary, suppose that there exists t0>0 such that S∗x,x,y,t0<1. Since limt⟶∞S∗x,x,y,t=1, there exists t1>t0 such that(43)S∗x,x,y,t>S∗x,x,y,t0,∀t>t1.
Sinceϕ∈Φw, there exists t2≥t1 such that limn⟶∞ϕnt2=0. Therefore, we can choose large enough n0≥1 such that ϕn0t2<t0. By the monotonicity of S∗, using (42) and (43), we have(44)S∗x,x,y,t0≥S∗x,x,y,ϕn0t2≥S∗x,x,y,t2>S∗x,x,y,t0,which is a contradiction. Therefore, S∗x,x,y,t=1, ∀t>0. Consequently, x=y.Theorem 4.
LetX,S∗,∗ be a complete S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type. If the mapping f:X⟶X is a ϕ−contraction, then f has a unique fixed point x∈X and, for ∀x0∈X, limnfnx0=x.Proof.
Letx0 be an arbitrary point in X and let us define the sequence xn by xn+1=fnx0, ∀n≥0.
Sincef is a ϕ−contraction, by (30), we have that, for ∀t>0,(45)S∗xn,xn,xn+1,ϕt=S∗fxn−1,fxn−1,fxn,ϕt≥S∗xn−1,xn−1,xn,t.
By Lemma7, we conclude that xn is a Cauchy sequence in X. Since X is complete, there exists x∈X such that limn⟶∞xn=x, that is, for ∀t>0, limn⟶∞S∗xn,xn,x,t=1.
Now, we will prove thatx is a fixed point of f.
Sinceϕ∈Φw, by Lemma 6, we have that for ∀t>0, there exists r≥t such that ϕr<t. Putting a=fxn in S∗FM3 of Definition 6, we have(46)S∗x,x,fx,t≥∗S∗x,x,fxn,t−ϕr2,S∗x,x,fxn,t−ϕr2,S∗fx,fx,fxn,ϕr≥∗S∗x,x,xn+1,t−ϕr2,S∗x,x,xn+1,t−ϕr2,S∗x,x,xn,r.
Sincelimn⟶∞xn=limn⟶∞xn+1=x and ∗ is continuous, taking limit as n⟶∞ in above inequality, one has(47)S∗fx,fx,x,t≥∗1,1,1=1.
So,fx=x, that is, x is a fixed point of f.
Next, suppose thaty with x≠y is another fixed point of f. Then, for ∀t>0, we have(48)S∗x,x,y,ϕt=S∗fx,fx,fy,ϕt≥S∗x,x,y,t,which implies, according to Lemma 8, that x=y. Therefore, f has a unique fixed point.Example 5.
LetX=0,∞ and ∗a,b,c=a⋅b⋅c, ∀a,b,c∈X. Define the functions S∗:X3×0,∞⟶0,∞ by S∗x,y,z,t=t/t+Sx,y,z, ∀x,y,z∈X, and Sx,y,z=x−z+y−z. Then, X,S∗,∗ is an S∗−fuzzy metric space.
Letf:X⟶X be a mapping defined by fx=x/2 and ϕ:ℝ+⟶ℝ+ be defined by(49)ϕt=t2,0≤t<1;−t3+43,1≤t≤2;t−1,2<t<∞.
Notice thatϕt∈Φw and ϕt≥t/2, ∀t≥0.
Now, we verify thatf satisfies the ϕ−contraction condition:(50)S∗fx,fy,fz,ϕt=ϕtϕt+fx−fz+fy−fz=ϕtϕt+x/2−z/2+y/2−z/2≥t/2t/2+x/2−z/2+y/2−z/2=tt+x−z+y−z=S∗x,y,z,t.
This shows thatf is a ϕ−contraction; then, f has a fixed point in X. Indeed, 0 is the fixed point of f.Definition 14.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type and assume that ψ is a Ψ−function. A mapping f:X⟶X is called generalized Kannan-type mapping under a gauge function ϕ if, for ∀x,y∈X,(51)S∗fx,fx,fy,ϕt≥ψS∗x,x,fx,t1a,S∗y,y,fy,t2b,where t,t1,t2>0 with t=t1+t2, a,b>0 with 0<a+b<1, ϕ∈Φw.Lemma 9.
LetX,S∗,∗ be a complete S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type and f be a generalized Kannan-type mapping under a gauge function ϕ on X. Assume that limn⟶∞S∗x,x,y,t=1, ∀x,y∈X,t>0. If xn=fxn−1 is the iterative sequence generating by x0∈X, then limn⟶∞S∗xn+1,xn+1,xn,ϕt=1, ∀t>0.Proof.
Letx0∈X, xn=fxn−1, n∈ℕ, and t, t1, t2, a, and b be positive real numbers with 0<a+b<1. From (51), for t=t1+t2, we have(52)S∗xn+1,xn+1,xn,ϕt=S∗fxn,fxn,fxn−1,ϕt≥ψS∗xn,xn,xn+1,t1a,S∗xn−1,xn−1,xn,t2b.
For∀t>0, putting t1=at/a+b,t2=bt/a+b and c=a+b in the above inequality, we have(53)S∗xn+1,xn+1,xn,ϕt≥ψS∗xn,xn,xn+1,tc,S∗xn−1,xn−1,xn,tc,n∈ℕ.
Now, we claim that(54)S∗xn+1,xn+1,xn,tc≥S∗xn,xn,xn−1,tc,t>0,n∈ℕ.
Suppose, to the contrary, there existst>0 such that(55)S∗xn+1,xn+1,xn,tc<S∗xn,xn,xn−1,tc,n∈ℕ.
By property ofψ−function and (53), we have(56)S∗xn+1,xn+1,xn,ϕt≥ψS∗xn+1,xn+1,xn,tc,S∗xn,xn,xn−1,tc≥ψS∗xn+1,xn+1,xn,tc,S∗xn+1,xn+1,xn,tc>S∗xn+1,xn+1,xn,tc,which is a contradiction. So (53) and (54) imply the following inequality:(57)S∗xn+1,xn+1,xn,ϕt≥S∗xn,xn,xn−1,tc,t>0,n∈ℕ.
If we apply induction to the above inequality, we see that(58)S∗xn+1,xn+1,xn,ϕt≥S∗x1,x1,x0,tcn,t>0,n∈ℕ.
Our additional assumption onS∗−fuzzy metric spaces implies that limn⟶∞S∗x1,x1,x0,t/cn=1.
Thus, by taking the limit asn⟶∞, we have that, for all t>0,(59)limn⟶∞S∗xn+1,xn+1,xn,ϕt=1,t>0.Theorem 5.
LetX,S∗,∗ be an S∗−fuzzy metric space, where ∗ is a continuous 3rdt−norm of H−type and assume that limt⟶∞S∗x,y,z,t=1, ∀x,y,z∈X, t>0. Let f:X⟶X be a generalized Kannan-type mapping under a gauge function ϕ on X. Then, f has a unique fixed point x∈X and, for every x0∈X, limnfnx0=x.Proof.
Choosex0∈X and let xn=fxn−1 be the iterative sequence generated by x0. We will show that xn is a Cauchy sequence. Indeed, if not so, by definition, there exists 0<ε<1 for which we can find t>0 and subsequences xmk and xnk of xn with nk>mk>k for all positive k such that(60)S∗xmk,xmk,xnk,ϕt≤1−ε,t>0.
So, for∀t,t1,t2>0 with t=t1+t2 and a,b>0 with 0<a+b<1, we have(61)1−ε≥S∗xmk,xmk,xnk,ϕt=S∗fxmk−1,fxmk−1,fxnk−1,ϕt≥ψS∗xmk−1,xmk−1,fxmk−1,t1a,S∗xnk−1,xnk−1,fxnk−1,t2b=ψS∗xmk−1,xmk−1,xmk,t1a,S∗xnk−1,xnk−1,xnk,t2b.
Therefore,(62)1−ε≥ψS∗xmk−1,xmk−1,xmk,t1a,S∗xnk−1,xnk−1,xnk,t2b.
By Lemma9, for ∀t>0, we get(63)limn⟶∞S∗xn+1,xn+1,xn,ϕt=1.
So we can choosek large enough such that(64)S∗xmk−1,xmk−1,xmk,t1a>1−ε,S∗xnk−1,xnk−1,xnk,t2b>1−ε.
Therefore, from (62) and (64) and the definition of Ψ−function, it follows that(65)1−ε≥ψ1−ε,1−ε>1−ε,which is a contradiction. So xn is a Cauchy sequence. The completeness of S∗−fuzzy metric space X,S∗,∗ implies that limn⟶∞xn=x for some x∈X. Now, we claim that fx=x. If possible, let S∗x,x,fx,ϕt<1, for some t>0.
Since0<b<1, we have chosen η1,η2,t1,t2>0 such that(66)ϕt=η1+η2+ϕt1+t2,t2b>ϕt.
Then, we have(67)S∗x,x,fx,ϕt≥∗S∗x,x,xn,η1,S∗x,x,xn,η2,S∗fx,fx,xn,ϕt1+t2=∗S∗x,x,xn,η1,S∗x,x,xn,η2,S∗xn,xn,fx,ϕt1+t2≥∗S∗x,x,xn,η1,S∗x,x,xn,η2,ψS∗xn,xn,xn+1,t1a,S∗xn−1,xn−1,xn,t2b≥∗S∗x,x,xn,η1,S∗x,x,xn,η2,ψS∗xn,xn,xn+1,t1a,S∗xn−1,xn−1,xn,ϕt.
By Lemma9 and the convergence of xn to x, there exists a positive integer N1 such that, for ∀n>N1,(68)S∗x,x,xn,η1>S∗x,x,fx,ϕt,S∗x,x,xn,η2>S∗x,x,fx,ϕt,S∗xn,xn,xn+1,t1a>S∗x,x,fx,ϕt.
Thus, it follows that(69)S∗x,x,fx,ϕt>S∗x,x,fx,ϕt,which is a contradiction. Hence, S∗x,x,fx,t=1, ∀t>0; therefore, x is a fixed point of f.
In order to prove the uniqueness, suppose thatx and y are two fixed points of f. Therefore, with all of the above assumptions on a,b,t1,t2, for ∀t>0, we have(70)S∗x,x,y,ϕt=S∗fx,fx,fy,ϕt≥ψS∗x,x,fx,t1a,S∗y,y,fy,t2b=ψS∗x,x,x,t1a,S∗y,y,fy,t2b=ψ1,1=1.
Thus,x=y, completing the proof.
Next, we prove the common fixed point theorems for weakly compatible mappings along withE.A. property and CLRg property.Definition 15.
Letf and g be two self-mappings of an S∗−fuzzy metric space X,S∗,∗. The mappings are said to be weakly compatible, if they commute at their coincidence points; that is, fx=gx implies fgx=gfx.Definition 16.
Letf and g be two self-mappings of an S∗−fuzzy metric space X,S∗,∗. We say that f and g satisfy the E.A. property if there exists a sequence xn such that limn⟶∞fxn=limn⟶∞gxn=x, for some x∈X.Remark 4.
Note that the weakly compatible andE.A. property are independent to each other as is follows from the following examples.Example 6.
LetX,S∗,∗ be an S∗−fuzzy metric space, where X=0,2 and S∗ is defined by S∗x,y,z,t=t/t+x−z+y−z, ∀x,y,z∈X and t>0. ∗ is defined by ∗a,b,c=a⋅b⋅c, ∀a,b,c∈0,1.
Definef,g:X⟶X by(71)fx=2−x,x∈0,1,0,x∈1,2,gx1,x∈0,1,32,x∈1,2.
Then, for the sequencexn=1−1/n,n≥2, we have(72)limn⟶∞f1−1n=limn⟶∞1+1n=limn⟶∞g1−1n=1∈0,2.
Thus,f and g satisfy E.A. property.
Furthermore,f and g are weakly compatible, since x=1 is their coincidence point and f1=g1=1=fg1=gf1.Example 7.
LetX,S∗,∗ be an S∗−fuzzy metric space, where X=ℝ+,∗a,b,c=a⋅b⋅c,∀a,b,c∈0,1, and S∗x,y,z,t=t/t+x−z+y−z+x−y, ∀x,y,z∈X and t>0.
Definef,g:X⟶X by(73)fx=0,0<x≤1,1,x>1,x=0,gx=xthe integer part ofx∀x∈X.
Consider the sequencexn=1+1/n, n≥2. Then, we have(74)limn⟶∞fxn=1=limn⟶∞gxn.
At the same time, for the sequenceyn=1−1/n, n≥2, one has(75)limn⟶∞fyn=0=limn⟶∞gyn.
Thus,f and g satisfy the E.A. property. However, f and g are not weakly compatible since each u∈0,1 and v∈0,1 are coincidence points of f and g while they do not commute. Moreover, they commute at x=0,1,2,… but none of them are coincidence points of f and g. Hence, E.A. property does not imply weak compatibility.Definition 17.
Letf and g are two self-mappings of an S∗−fuzzy metric space X,S∗,∗. We say that f and g satisfy the common limit in the range of gCLRg property, if there exists a sequence xn such that limn⟶∞fxn=limn⟶∞gxn=gx, for some x∈X.Definition 18.
Letf and g be two self-mappings of an S∗−fuzzy metric space X,S∗,∗. The pair of mappings f,g is said to be generalized Kannan-type mappings under a gauge function ϕ if, for ∀x,y∈X,(76)S∗fx,fx,fy,ϕt≥ψS∗gx,gx,fx,t1a,S∗fy,fy,gy,t2b,where t,t1,t2,a,b>0 with t=t1+t2, 0<a+b<1, ϕ∈Φw.
In what follows, we prove a result concerning weakly compatible mappings along withE.A. property.Theorem 6.
Letf and g be two self-mappings of an S∗−fuzzy metric space X,S∗,∗ with a continuous 3rdt−norm of H−type. Suppose that the following conditions are satisfied:(1)
f and g satisfy E.A. property.(2)
The pairf,g is generalized Kannan-type mappings under a gauge function ϕ.(3)
gX is a closed subspace of X.(4)
f and g are weakly compatible on X, provided 0<S∗x,y,z,t<1, ∀t>0.
Then,f and g have a unique common fixed point.Proof.
Sincef and g satisfy E.A. property, there exists a sequence xn in X such that(77)limn⟶∞fxn=limn⟶∞gxn=u∈X.
SincegX is a closed subspace of X, every convergent sequence of points of gX has a limit in gX. Therefore, there is a∈X such that(78)u=limn⟶∞fxn=limn⟶∞gxn=ga∈gX.
We now show thatfa=u=ga.
If possible, let0<S∗ga,ga,fa,ϕt<1, for some t>0. Since 0<b<1, we can choose η1,η2,t1,t2>0 such that(79)ϕt=η1+η2+ϕt1+t2,t2b>ϕt.
Then, we get(80)S∗ga,ga,fa,ϕt≥∗S∗ga,ga,fxn,η1,S∗ga,ga,fxn,η2,S∗fa,fa,fxn,ϕt1+t2=∗S∗ga,ga,fxn,η1,S∗ga,ga,fxn,η2,S∗fxn,fxn,fa,ϕt1+t2≥∗S∗ga,ga,fxn,η1,S∗ga,ga,fxn,η2,ψS∗gxn,gxn,fxn,t1a,S∗fa,fa,ga,t2b≥∗S∗ga,ga,fxn,η1,S∗ga,ga,fxn,η2,ψS∗gxn,gxn,fxn,t1a,S∗fa,fa,ga,ϕt.
By (78), there exists a positive integer N2 such that, for ∀n>N2,(81)minS∗ga,ga,fxn,η1,S∗ga,ga,fxn,η2,S∗gxn,gxn,fxn,t1a>S∗ga,ga,fa,t.
Then, we have(82)S∗ga,ga,fa,ϕt>S∗ga,ga,fa,ϕt,which is a contradiction. Hence, S∗ga,ga,fa,ϕt=1, ∀t>0, which implies fa=u=ga.
Sincef and g are weakly compatible, it follows that gfa=fga; i.e., gu=fu.
Now, we will prove thatu is a common fixed point of f and g.
From condition (2), we deduce(83)S∗fu,fu,fxn,ϕt≥ψS∗gu,gu,fxn,t1a,S∗fxn,fxn,gxn,t2b,where t=t1+t2. Taking the limit as n⟶∞, we have(84)S∗fu,fu,u,ϕt≥ψS∗gu,gu,fu,t1a,S∗fu,fu,gu,t2b=ψ1,1=1.
Hence,fu=u=gu; that is, u is a common fixed point of f and g.
In order to prove the uniqueness, letuandv be two common fixed points of f and g. Therefore, for ∀t1,t2,t,a,b>0 with t=t1+t2, 0<a+b<1, one has(85)S∗u,u,v,ϕt=S∗fu,fu,fv,ϕt≥ψS∗gu,gu,fu,t1a,S∗fv,fv,gv,t2b=ψS∗u,u,u,t1a,S∗v,v,v,t2b=ψ1,1=1.
Hence,u=v; that is, u is a unique common fixed point of f and g.
Finally, we present a theorem for weakly compatible mappings along withCLRg property as follows.Theorem 7.
Letf and g be two self-mappings of an S∗−fuzzy metric space X,S∗,∗ with a continuous 3rdt−norm of H−type. Suppose that the following conditions hold:(1)
f and g satisfy CLRg property.(2)
The pairf,g is generalized Kannan-type mappings under a gauge function ϕ.(3)
f and g are weakly compatible on X, provided 0<S∗x,y,z,t<1, ∀t>0.
Then,f and g have a unique common fixed point.Proof.
Sincef and g satisfy the CLRg property, there exists a sequence xn in X such that(86)limn⟶∞fxn=limn⟶∞gxn=gx∈X.
We will show thatfx=gx.
If possible, let0<S∗ga,ga,fa,ϕt<1, for some t>0. Since 0<b<1, we can choose η1,η2,t1,t2>0 such that(87)ϕt=η1+η2+ϕt1+t2,t2b>ϕt.
Then, we have(88)S∗fx,fx,gx,ϕt≥∗S∗fx,fx,fxn,η1,S∗fx,fx,fxn,η2,S∗gx,gx,fxn,ϕt1+t2=∗S∗fx,fx,fxn,η1,S∗fx,fx,fxn,η2,S∗fxn,fxn,fx,ϕt1+t2≥∗S∗fx,fx,fxn,η1,S∗fx,fx,fxn,η2,ψS∗gxn,gxn,fxn,t1a,S∗fx,fx,gx,t2b≥∗S∗fx,fx,fxn,η1,S∗fx,fx,fxn,η2,ψS∗gxn,gxn,fxn,t1a,S∗fx,fx,gx,ϕt.
By (86), there exists a positive integer N3 such that, for ∀n>N3,(89)S∗fx,fx,fxn,η1>S∗fx,fx,gx,ϕt,S∗fx,fx,fxn,η2>S∗fx,fx,gx,ϕt,S∗gxn,gxn,fxn,t1a>S∗fx,fx,gx,ϕt.
Then, we have(90)S∗fx,fx,gx,ϕt>S∗fx,fx,gx,ϕt,which is a contradiction. Therefore, S∗fx,fx,gx,ϕt=1, ∀t>0, which implies fx=gx=u.
Using the similar argument to that in the proof of Theorem6, we can obtain that u is a unique common fixed point of f and g.



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*Source: 1712486-2020-05-31.xml*