# Pata-Type Fixed-Point Theorems in Kaleva–Seikkala’s Type Fuzzy Metric Space

Authors: Ao-Lei Sima; Fei He; Ning Lu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The purpose of this paper is to generalize the fixed-point theorems for Banach–Pata-type contraction and Kannan–Pata-type contraction from metric spaces to Kaleva–Seikkala’s type fuzzy metric spaces. Moreover, two examples are given for the support of our results.


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## Body

## 1. Introduction and Preliminaries

In 1984, Kaleva and Seikkala [1] introduced the concept of fuzzy metric spaces for the first time, and they established some fixed-point theorems in a complete fuzzy metric space. This result was investigated by many authors from different points of view, see [2–10] and the references therein.Definition 1 (see [11]).
A mappingη:ℝ⟶0,1 is called a fuzzy real number, whose α− level set is denoted by ηα=t∈ℝ:ηt≥α if it satisfies two axioms:(1)
There existst0∈ℝ such that ηt0=1(2)
For eachα∈0,1,ηα=λα,ρα is a closed interval of ℝ, where −∞<λα≤ρα<+∞
The set of all such fuzzy real numbers is denoted byℱ. If η∈ℱ and ηq=0 whenever q<0, then η is called a nonnegative fuzzy real number, and by ℱ+, we mean the set of all nonnegative fuzzy real numbers. The notation 0¯ stands for the fuzzy number satisfying 0¯t=1 and 0¯t=0 if t≠0. Clearly, 0¯∈ℱ+. ℝ can be embedded in ℱ if a∈ℱ satisfies a¯q=0¯q−a.Definition 2 (see [1]).
LetX be a nonempty set and the mappings L,R:0,1×0,1⟶0,1 be symmetric, nondecreasing in both arguments, and satisfy L0,0=0,R1,1=1. Let d be the mapping X×X⟶ℱ+, and write dx,yα=λαx,y,ραx,y for all x,y∈X and all α∈0,1. The quadruple X,d,L,R is called a fuzzy metric space if the following axioms are satisfied:(D1)dx,y=0¯⟺x=y.(D2) for allx,y∈X, dx,y=dy,x.(D3) for allx,y,z∈X.(D3L) wheneverp≤λ1x,z, q≤λ1z,y, and p+q≤λ1x,y, dx,yp+q≥Ldx,zp,dz,yq.(D3R) wheneverp≥λ1x,z, q≥λ1z,y, and p+q≥λ1x,y, dx,yp+q≤Rdx,zp,dz,yq.Lemma 1 (see [6]).
LetX,d,L,R be a fuzzy metric space and dx,yt=λtx,y,ρtx,y for all t∈0,1, where x,y∈X are any two fixed elements. Then,(1)
limq⟶−∞dx,yq=limq⟶+∞dx,yq=0(2)
dx,yq is a left continuous and nonincreasing function for q∈λ1x,y,+∞(3)
ρtx,y is a left continuous and nonincreasing function for t∈0,1Lemma 2 (see [12]).
LetX,d,L,R be a fuzzy metric space, and suppose that(R-1)Ra,b≤maxa,b(R-2) for eacht∈0,1, there exists s∈0,t such that Rs,r<t for all r∈0,t(R-3)limt⟶0+Rt,t=0
Then, (R-1)⇒ (R-2) ⇒ (R-3).Lemma 3 (see [6]).
LetX,d,L,R be a fuzzy metric space. Then,(1)
(R-1) implies that, for eacht∈0,1,(1)ρtx,y≤ρtx,z+ρtz,y,for all x,y,z∈X.(2)
(R-2) implies that, for eacht∈0,1, there exists s=st∈0,t such that(2)ρtx,y≤ρsx,z+ρtz,y,for all x,y,z∈X.(3)
(R-3) implies that, for eacht∈0,1, there exists s=st∈0,t such that(3)ρtx,y≤ρsx,z+ρsz,y,for all x,y,z∈X.Definition 3 (see [6]).
LetX,d,L,R be a fuzzy metric space and xn⊂X,x∈X. Then,(1)
xn is said to be convergent to x if limn⟶∞dxn,x=0¯, i.e., limn⟶∞ρtxn,x=0 for all t∈0,1(2)
xn is called a Cauchy sequence in X if limn,m⟶∞dxn,xm=0¯; equivalently, for any given ε>0 and t∈0,1, there exists N=Nε,t∈ℕ such that ρtxn,xm<ε whenever n,m≥N(3)
X,d,L,R is said to be complete if each Cauchy sequence in X is convergent to some point in XLemma 4 (see [6]).
LetX,d,L,R be a fuzzy metric space with (R-2). Then, for each t∈0,1, ρtx,y is continuous at x,y∈X×X.In 2011, Pata [13] extended the Banach contraction principle with weaker hypotheses than those of the Banach contraction principle in the complete metric space. Since then, several other fixed point results in the spirit of Pata have appeared, see [14–18]. In particular, Chakraborty and Samanta [14] proved a generalization of Kannan’s fixed-point theorem based on the result of Pata.Throughout the following,X,d will be a complete metric space, and X,d,L,R will be a complete fuzzy metric space. Fix an arbitrary point x0∈X, and we denote x=dx,x0 and xt=ρtx,x0 for all x∈X and t∈0,1. Also, ψ:0,1]⟶[0,∞ is an increasing function, continuous at zero, with ψ0=0. Given a function f:X⟶X.Theorem 1 (see [13]).
LetX,d be a complete metric space. Let Λ≥0,α≥1, and β∈0,α be fixed constants. If the inequality(4)dfx,fy≤1−εdx,y+Λεαψε1+x+yβ,is satisfied for every ε∈0,1 and all x,y∈X, then f has a unique fixed point z∈X.Theorem 2 (see [14]).
LetX,d be a complete metric space. Let Λ≥0, α≥1, and β∈0,α be fixed constants. If the inequality(5)dfx,fy≤1−ε2dx,fx+dy,fy+Λεαψε1+x+y+fx+fyβ,is satisfied for every ε∈0,1 and all x,y∈X, then f has a unique fixed point z∈X.In this paper, we prove two further extensions of Pata-type fixed theorems in complete Kaleva–Seikkala’s type fuzzy metric space using contractive condition of Banach type and Kannan type. Afterwards, the fixed-point theorems for the corresponding linear contraction are given as corollaries. Our theorems extend the main results of [13, 14]. Moreover, two nontrivial examples are given to illustrate our two theorems, and our examples show that these two theorems are independent to each other.


## 2. Main Results

Our results of this paper are stated as follows.Theorem 3.
LetX,d,L,R be a complete fuzzy metric space with (R-2). Let Λ≥0 and α≥1 be fixed constants. Let f:X⟶X be a mapping such that(6)ρtfx,fy≤1−ερtx,y+Λεαψε1+xt+yt+fxt+fytα,for every ε∈0,1, t∈0,1 and all x,y∈X. Then, f has a unique fixed point z∈X.Proof.
Starting fromx0, construct a sequence xn such that xn=fxn−1=fnx0. For all t∈0,1, we denote Cnt=xnt. If xn0=xn0+1 for some n0, then xn0 is a fixed point of f. Thus, we always assume that xn≠xn+1 for all n∈ℕ.
In order to prove this theorem, we divide into the following five steps:Step 1.
We show that the sequenceρtxn+1,xn is decreasing. Clearly, suppose that t∈0,1 and ε=0 in (6), we obtain(7)ρtxn+1,xn≤ρtxn,xn−1≤⋯≤ρtx1,x0.Step 2.
We prove that the sequenceCnt is bounded. For any t∈0,1, by Lemma 3, there exists s=st∈0,t such that(8)ρtx,y≤ρsx,z+ρtz,y,for all x,y,z∈X. Then, from Step 1, we have(9)Cnt=ρtxn,x0≤ρsxn,xn+1+ρsx1,x0+ρtxn+1,x1≤ρtxn+1,x1+2ρsx1,x0≤1−ερtxn,x0+Λεαψε1+xnt+x0t+xn+1t+x1tα+2ρsx1,x0=1−ερtxn,x0+Λεαψε1+Cnt+Cn+1t+C1tα+2ρsx1,x0.
SinceCn+1t≤ρsxn+1,xn+ρtxn,x0≤ρsx1,x0+ρtxn,x0, we have(10)Cnt≤1−εCnt+Λεαψε1+2Cnt+C1t+ρsx1,x0α+2ρsx1,x0≤1−εCnt+Λεαψε1+2Cnt+2ρsx1,x0α+2ρsx1,x0.
Hence,(11)εCnt−2ρsx1,x0≤Λεαψε1+2Cnt+2ρsx1,x0α.
Suppose thatCnt is unbounded. Then, there exist t0∈0,1, s0∈0,t0, and a subsequence Cnkt0 of Cnt0 such that Cnkt0⟶∞k⟶∞ and Cnkt0≥1+2ρs0x0,x1. Let(12)ε=εk=1+2ρs0x1,x0Cnkt0.
Then, we get(13)1+2ρs0x1,x0Cnkt0Cnkt0−2ρs0x1,x0≤Λεkαψεk1+2Cnkt0+2ρs0x1,x0α=Λ1+2ρs0x1,x01+2Cnkt0+2ρs0x1,x0Cnkt0αψεk,which implies that(14)1≤Λ1+2ρs0x1,x01+2Cnkt0+2ρs0x1,x0Cnkt0αψεk.
Lettingk⟶∞ in (14), we have ψεk⟶0 and(15)Λ1+2ρs0x1,x0α1+C1t0+2ρs0x1,x0Cnkt0+2αψεk⟶0,which contradict (14). Thus, Cnt is bounded, that is, for any t∈0,1, there exists a constant Mt>0 such that xnt=Cnt≤Mt.Step 3.
We shall show that(16)limn⟶∞ρtxn+1,xn=0t∈0,1.
Note thatρtxn+1,xn is a decreasing and bounded sequence. So, assume that limn⟶∞ρt0xn+1,xn=a for some a>0 and t0∈0,1. By (6), we obtain that(17)ρt0xn+1,xn=ρt0fxn,fxn−1≤1−ερt0xn,xn−1+Λεαψε1+xnt0+xn−1t0+xn+1t0+xnt0α≤1−ερt0xn,xn−1+Λεαψε1+4Mt0α.
Lettingn⟶∞, we have(18)a≤Λεα−1ψε1+4Mt0α.
Note that(19)Λεα−1ψε1+4Mt0α⟶0asε⟶0.
Then, we can seea≤0, which contradicts our assumption.Step 4.
We show thatxn is a Cauchy sequence. Suppose not, choose δ>0 and t0∈0,1, and then there exist subsequences xnk and xmk of xn with k<mk<nk such that(20)ρt0xmk,xnk>δandρt0xmk,xnk−1≤δ.
By Lemma3, there exists s0∈0,t0 such that(21)δ<ρt0xmk,xnk≤ρt0xmk,xnk−1+ρs0xnk−1,xnk≤ρs0xnk−1,xnk+δ.
Puttingn⟶∞, we get ρt0xmk,xnk⟶δ. Similarly, we can see that(22)ρt0xnk+1,xmk+1≤ρt0xnk,xmk+ρs0xnk,xnk+1+ρs0xmk,xmk+1,(23)ρt0xnk+1,xmk+1≥ρt0xnk,xmk−ρs0xnk,xnk+1−ρs0xmk,xmk+1.
Passing to the limit asn⟶∞ in (22) and (23), we have ρt0xnk+1,xmk+1⟶δ. By (6), we get(24)ρt0xnk+1,xmk+1=ρt0fxnk,fxmk≤1−ερt0xnk,xmk+Λεαψε1+xnkt0+xmkt0+xnk+1t0+xmk+1t0α≤1−ερt0xnk,xmk+Λεαψε1+4Mt0α.
Takingn⟶∞ in (2), we obtain that(25)δ≤1−εδ+Λεαψε1+4Mt0α,δ≤Λεα−1ψε1+4Mt0α.
Note thatΛεα−1ψε1+4Mt0α⟶0ε⟶0, a contradiction. Thus, xn is a Cauchy sequence. Since X is complete, there exists z∈X such that xn⟶zn⟶∞.Step 5.
We prove thatz is the unique fixed point for mapping f.
First, we show thatfz=z. By (6), for all t∈0,1, there exists s∈0,t, and we have(26)ρtfz,z≤ρsxn+1,z+ρtfz,xn+1≤1−ερtz,xn+Λεαψε1+zt+xnt+fzt+xn+1tα+ρsxn+1,z≤1−ερtz,xn+Λεαψε1+4Mtα+ρsxn+1,z.
Lettingn⟶∞, we get ρtfz,z=0. Hence, fz=z.
Next, we prove the uniqueness ofz. Assume that z′ is another fixed point for f. For each t∈0,1, there exists s∈0,t such that(27)ρtz,z′=ρtfz,fz′≤1−ερtz,z′+Λϵαψε1+zt+z′t+fzt+fz′tα≤1−ερtz,z′+Λϵαψε1+4Mtα.
Thus, we haveρtz,z′≤Λεα−1ψε1+4Mtα⟶0 as ε⟶0, which implies z=z′. Therefore, f has a unique fixed point z∈X.Remark 1.
From the proof of Theorem3, we can see that, to keep the sequence xn converge to the fixed point, the range of ε in (6) can be limited from 0,1 to 0,γ for some given constant γ∈0,1.
By Remark1, letting γ=λ, we can deduce the following corollary, which is the Banach contraction principle in fuzzy metric spaces.Corollary 1 (see Corollary 5.3, [6]).
LetX,d,L,R be a complete fuzzy metric space with (R-2) and f:X⟶X be a mapping. If there exists λ∈0,1 such that(28)ρtfx,fy≤λρtx,y,for all t∈0,1 and x,y∈X, then f has a unique fixed point z∈X.Theorem4 generalizes the result in [14] to fuzzy metric spaces.Theorem 4.
LetX,d,L,R be a complete fuzzy metric space with (R-2). Let Λ≥0 and α≥1 be fixed constants. Let f:X⟶X be a mapping such that(29)ρtfx,fy≤1−ε2ρtx,fx+ρty,fy+Λεαψε1+xt+yt+fxt+fytα,for every ε∈0,1, t∈0,1, and all x,y∈X. Then, f has a unique fixed point z∈X.Proof.
Starting fromx0, construct a sequence xn such that xn=fxn−1=fnx0. For any t∈0,1, we denote Cnt=xnt. If xn0=xn0+1 for some n0, then xn0 is a fixed point of f. Thus, we always assume that xn≠xn+1 for all n∈ℕ. Then, the proof is divided into the following five steps:Step 1.
We show that the sequenceρtxn,xn+1 is decreasing. Suppose that t∈0,1 and ε=0 in (29), we obtain(30)ρtxn,xn+1≤12ρtxn−1,xn+ρtxn,xn+1.
Thus, we get(31)ρtxn+1,xn≤ρtxn,xn−1≤⋯≤ρtx1,x0.Step 2.
The sequenceCnt is bounded. For each t∈0,1, there exists s=st∈0,t, and(32)Cnt=ρtxn,x0≤ρsxn,xn+1+ρsx1,x0+ρtxn+1,x1≤ρtxn+1,x1+2ρsx1,x0≤1−ε12ρtxn,xn+1+ρtx1,x0+Λεαψε1+xnt+x0t+xn+1t+x1tα+2ρsx1,x0≤1−ερtx1,x0+Λεαψε1+Cnt+Cn+1t+C1tα+2ρsx1,x0.
Lettingε=0 in (32), there exists a constant Mt>0 such that Cnt≤ρtx0,x1=Mt.Step 3.
We shall prove thatlimn⟶∞ρtxn,xn+1=0 for each t∈0,1. For any t∈0,1, we obtain(33)ρtxn,xn+1=ρtfxn−1,fxn≤1−ε12ρtxn−1,xn+ρtxn,xn+1+Λεαψε1+xn−1t+xnt+xnt+xn+1tα≤1−ε12ρtxn−1,xn+ρtxn,xn+1+Λεαψε1+4Mtα.
Sinceρtxn,xn+1 is a decreasing and bounded sequence, assume that limn⟶∞ρtxn,xn+1=a for some a>0. Letting n⟶∞ in (2), we have(34)a≤1−ε12⋅2a+Λεαψε1+4Mtα≤1−εa+Λεαψε1+4Mtαa≤Λεα−1ψε1+4Mtα.
From(35)Λεα−1ψε1+4Mtα⟶0ε⟶0,we see a≤0, a contradiction. For each t∈0,1, we get ρtxn,xn+1⟶0n⟶∞.Step 4.
We show thatlimn⟶∞ρtxn,xm=0 for each t∈0,1 and all n,m∈ℕ. For any t∈0,1, there exists s=st∈0,t, and we have(36)ρtxn,xm=ρtfxn−1,fxm−1≤1−ε12ρtxn,xn+1+ρtxm,xm+1+Λεα−1ψε1+xn−1t+xm−1t+xnt+xmtα≤1−ε12ρtxn,xn+1+ρtxm,xm+1+Λεα−1ψε1+4Mtα.
Puttingn⟶∞ and ε⟶0 in (36), we get ρtxn,xm⟶0.
Therefore,xn is a Cauchy sequence. Since X is complete, there exists z∈X such that xn⟶zn⟶∞.Step 5.
We prove thatz is the unique fixed point for mapping f.
First, we show thatfz=z. Using (29), for any t∈0,1, there exists s∈0,t, and we have(37)ρtfz,z≤ρsxn+1,z+ρtfz,xn+1≤1−ε12ρtz,fz+ρtxn,xn+1+Λεαψε1+zt+xnt+fzt+xn+1tα+ρsxn+1,z≤1−ε12ρtz,fz+ρtxn,xn+1+Λεαψε1+4Mtα+ρsxn+1,z.
By taking limits asn⟶∞ and ε⟶0 in the inequality above, we get ρtfz,z≤1/2ρtfz,z. Hence, ρtfz,z=0 which implies fz=z.
Finally, we prove the uniqueness ofz. Assume that z′ is another fixed point of f. For any t∈0,1, there exists s∈0,t, and we get(38)ρtz,z′=ρtfz,fz′≤1−ε12ρtz,fz+ρtz′,fz′+Λϵαψε1+zt+z′t+fzt+fz′tα≤1−ε12ρtz,z+ρtz′,z′+Λϵαψε1+4Mtα≤Λϵαψε1+4Mtα.
Passing to the limit asε⟶0 in (38), we have ρtz,z′=0. Thus, z=z′. Therefore, f has a unique fixed point z∈X.
Following the arguments in Remark1, from Theorem 4, we can obtain the Corollary 2.Corollary 2 (see Corollary 5.3, [6]).
LetX,d,L,R be a complete fuzzy metric space with (R-2) and f:X⟶X be a mapping. If there exists λ∈0,1/2 such that(39)ρtfx,fy≤λρtx,fx+ρty,fy,for all t∈0,1 and x,y∈X, then f has a unique fixed point z∈X.Now, we construct two examples to illustrate Theorems3 and 4, respectively.Example 1.
LetX=1/2n:n∈ℕ∪0∪0, La,b=mina,b, and Ra,b=maxa,b. Define the fuzzy metric d:X×X⟶ℱ+ by(40)dx,y=0¯q−x−y=1,q=x−y,0,q≠x−y.
LetT:X⟶X be a map defined by Tx=1/2x for x∈X. Then, the following hold:(a)
X,d,L,R is a complete fuzzy metric with (R-2).(b)
(6) is satisfied for x0=1, Λ=1, α=1, and ψε=2ε.(c)
T has the unique fixed point x=0.(d)
(29) is not satisfied. So, Theorem 4 cannot be verified by this example.Proof.
First, it is easy to see thatX,d,L,R is a complete fuzzy metric space, and Ra,b=maxa,b satisfies (R-2). Note that Tx=x implies that x=0∈X; then, conclusion (c) is true. Next, we prove conclusions (b) and (d), respectively.(b)
Lettingx0=1, Λ=1, α=1, β=1, and ψε=2ε, we show that (6) holds for all ε∈0,1, t∈0,1 and x,y∈X.
For anyt∈0,1, we can see that ρtx,y=x−y. Then, we consider the following two cases:Case 1.
Ifε≤1/2, then 1−ε≥1/2. For any x,y∈X, we have(41)ρtTx,Ty=12x−12y=12x−y≤1−ερtx,y≤1−ερtx,y+εψε1+xt+yt+Txt+Tyt.Case 2.
Ifε>1/2, then ψε>1. Note that(42)ρtx,y≤ρtx,x0+ρ0x0,y=xt+yt.
Then, for anyx,y∈X, we have(43)ρtTx,Ty=12x−12y≤x−y=1−ερtx,y+ερtx,y≤1−ερtx,y+εψεxt+yt≤1−ερtx,y+εψε1+xt+yt+Txt+Tyt.
From Cases1 and 2, we see that (6) holds for all ε∈0,1. In fact, x=0 is the unique fixed point for map T. So, Theorem 3 is verified.(d)
We show that (29) is not satisfied.
Lettingε=0, by (29), we deduce that(44)ρtTx,Ty≤ρtx,Tx+ρty,Ty2.
However, letx0=0 and y0=1. For any t∈0,1, we have(45)ρtTx0,Ty0=0−12>12⋅12=ρtx0,Tx0+ρty0,Ty02.
So, this example cannot verify Theorem4, and we prove conclusion (d).Example 2.
LetX=6,−5,4,−3,2,−1,0, La,b=mina,b, and Ra,b=maxa,b. Let d:X×X⟶ℱ+ be a fuzzy metric such that(46)dx,yq=0,q<0,1,q=0,x−yx−y+q,q>0.
LetT:X⟶X be a map such that(47)Tx=−x+1,x>0,x−1,x<0,0,x=0,for x∈X. Then, the following hold:(a)
X,d,L,R is a complete fuzzy metric with (R-2).(b)
(29) is satisfied for x0=1, Λ=1, α=1, and ψε=6ε.(c)
T has the unique fixed point x=0.(d)
(6) is not satisfied. So, this example cannot verify Theorem 3.Proof.
Clearly, we can see thatx=0 is the unique fixed point for map T, and conclusion (c) is true. Now, we prove conclusions (a), (b), and (d), respectively.(a)
We show thatX,d,L,R is a complete fuzzy metric space with (R-2).
Clearly, for anyx,y∈X, dx,yq is a fuzzy number, and (D1) and (D2) in Definition 2 are satisfied. Note that Ra,b=maxa,b satisfies (R-2). So, it is sufficient to prove that (D3) holds.
For anyx,y,z∈X, it is obvious that λ1x,z=λ1z,y=λ1x,y=0. To see (D3L), let us consider the following two cases:(i)
Ifp<0 or q<0, then dx,zp=0 or dz,yq=0, leading to that(48)mindx,zp,dz,yq=0.
So, (D3L) is satisfied.(ii)
Ifp=0 and q=0, then p+q=0. So, dx,zp=dz,yq=dx,yp+q=1, leading to that(49)dx,yp+q=mindx,zp,dz,yq.
Thus, (D3L) holds. To see (D3R), we consider the following two cases:Case a1. Suppose that there are at least two points inx,y,z which are equal. For any p≥0 and q≥0,(i)
Ifz=x or z=y, without loss of generality, let z=x, then dz,y=dx,y. Since dx,yq is decreasing as q≥0, we have(50)maxdx,zp,dz,yq≥dz,yq=dx,yq≥dx,yp+q.(ii)
Ifx=y, then we have(51)p+q=0⇒p=0,q=0⇒dx,yp+q=1=maxdx,zp,dz,yq.p+q>0⇒dx,yp+q=0≤maxdx,zp,dz,yq.Case a2. Suppose thatx≠y, x≠z, and y≠z. For any p≥0 and q≥0, denote s=dx,zp and t=dz,yq. Without loss of generality, let 0<s≤t. Then, we have(52)p=x−z1−ss,q=z−y1−tt.
Sincedx,yq is decreasing as q≥0 and 1−t/t is decreasing, we deduce that(53)maxdx,zp,dz,yq=t=dx,yx−y1−tt≥dx,yx−z+z−y1−tt≥dx,yx−z1−ss+z−y1−tt=dx,yp+q.
From Cases a1 and a2, we conclude that (D3R) holds for allx,y,z∈X.
Note thatX is a finite set. Then, X,d,L,R is complete, and the proof of conclusion (a) is completed.(b)
We prove (29) for x0=1, Λ=1, α=1, β=1, and ψε=6ε.
For anyt∈0,1, we can see that ρtx,y=x−y1−t/t. Then, we consider the following two cases:Case b1. Ifε≤1/12, it is easy to see that, for any x≠0, we have Tx=x−1 and x⋅Tx≤0. So, we can obtain that x−Tx=x+Tx and(54)Tx−Ty≤Tx+Ty=Tx+x−12+Ty+y−12=x−Tx+y−Ty2−1for all x,y≠0. If y=0, then we have(55)Tx−Ty=Tx=Tx+x−12=x−Tx+y−Ty2−12.
From the above two equalities, by calculation, we obtain that, for anyx,y∈X,(56)ρtTx,Ty=Tx−Ty1−tt≤1011⋅x−Tx1−t/t+y−Ty1−t/t2≤1−ερtx,Tx+ρty,Ty2≤1−ερtx,Tx+ρty,Ty2+εψε1+xt+yt+Txt+Tyt.Case b2. Ifε>1/12, then we have ψε>1/2. Note that(57)ρtx,Tx+ρty,Ty≤ρtx,x0+ρtx0,Tx+ρty,x0+ρtx0,Ty=xt+yt+Txt+Tyt.
Then, for anyx,y∈X, we have(58)ρtTx,Ty=Tx−Ty1−tt≤x−Tx1−t/t+y−Ty1−t/t2=1−ερtx,Tx+ρty,Ty2+ερtx,Tx+ρty,Ty2≤1−ερtx,Tx+ρty,Ty2+ε12xt+yt+Txt+Tyt≤1−ερtx,Tx+ρty,Ty2+εψε1+xt+yt+Txt+Tyt.
From Cases b1 and b2, we see that (29) holds for all ε∈0,1. In fact, x=0 is the unique fixed point for map T. So, Theorem 4 is verified.(d)
We show that (6) is not satisfied for map T in X,d,L,R.
Finally, we show that (6) is not satisfied for map T in X,d,L,R. If ρtTx,Ty=ρtx,y for some x,y∈X and some t∈0,1, then by (6), we deduce that, for any ε>0,(59)ρtx,y≤Λεαψε1+xt+yt+fxt+fytβ.
Lettingε→0, we have ψε→0, which implies that x=y. However, for x0=6 and y0=4, we have Tx0=−5 and Ty0=−3 and(60)ρtTx0,Ty0=2⋅1−tt=ρtx0,y0for all t∈0,1, which is a contradiction.Remark 2.
From Examples1 and 2, we can see that Theorems 3 and 4 are independent to each other.
Recently, Jamshaid et al. [19] investigated the fuzzy fixed points of fuzzy mappings via F-contractions. On the basis of their work, a natural question for multivalued mappings can be raised as follows:Question 1.
Is the multivalued case of Theorem3 true?



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*Source: 6185894-2020-02-28.xml*