# Fixed-Point Theorem for Multivalued Quasi-Contraction Maps in aV-Fuzzy Metric Space

Authors: Matthew Brijesh Sookoo; Sreedhara Rao Gunakala
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we introduce the concept of a set-valued or multivalued quasi-contraction mapping inV-fuzzy metric spaces. Using this new concept, a fixed-point theorem is established. We also provide an example verifying and illustrating the fixed-point theorem in action.


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## Body

## 1. Introduction and Preliminaries

The concept of a metric space has been a fascinating subject that allows distance measurement between two points. Over the years, many mathematicians have been trying to further generalize the notion of a metric space. Two such mathematicians were Mustafa and Sims. In their paper [1], they provided a way to interpret distance using three points instead of two points as was the case in metric spaces. They called their new approach a G-metric space.Definition 1.
The pairX,d is called a G-metric space if X is a nonempty set and d is a G-metric on X. That is, d:X3⟶0,∞ such that for all x1,x2,x3,a∈X, we have(i)
Gx1,x2,x3=0 if and only if x1=x2=x3(ii)
Gx1,x1,x2>0 with x1≠x2(iii)
Gx1,x1,x2≤Gx1,x2,x3 for all with x2≠x3(iv)
Gx1,x2,x3=Gx1,x3,x2=Gx2,x1,x3=Gx2,x3,x1=Gx3,x1,x2=Gx3,x2,x1(v)
Gx1,x2,x3≤Gx1,a,a+Ga,x2,x3
Sedghi et al. in [2] modified the G-metric space and introduced their version of interpretation of distance between three points which is called the S-metric space.Definition 2.
The pairX,d is called an S-metric space if X is a nonempty set and d is an S-metric on X. That is, d:X3⟶0,∞ such that for all x1,x2,x3∈X, we have(i)
dx1,x2,x3≥0(ii)
dx1,x2,x3=0⟺x1=x2=x3(iii)
dx1,x2,x3≤dx1,x1,a+dx2,x2,a+dx3,x3,a
In view of the work of Sedghi et al. in [2], Abbas et al. in [3] further generalized the S-metric space giving an interpretation of distance between n points.Definition 3.
The pairX,d is called an A-metric space if X is a nonempty set and d is an A-metric on X. That is, d:Xn⟶0,∞ such that for all a,xi∈X,i=1,2,…,n, we have(i)
dx1,x2,…,xn−1,xn≥0(ii)
dx1,x2,…,xn−1,xn=0⟺x1=x2=⋯=xn−1=xn(iii)
dx1,x2,…,xn−1,xn≤dx1,x1,…,x1,a+dx2,x2,…,x2,a+⋯+dxn−1,xn−1,…,xn−1,a+dxn,xn,…,xn,a
In 2016, Gupta and Kanwar in [4] used the same approach in the fuzzy setting to extend a fuzzy metric space which uses two points to a V-fuzzy metric space which uses n points. We now outline this sequel that leads up to V-fuzzy metric spaces.
A fuzzy set was first proposed in the 1960s by an electrical engineer named Lotfi Zadeh in [5]. It is an extension of the classical notion of a set.
It should be noted that a fuzzy set is always considered with respect to a nonempty setX called its base set. Each x∈X is assigned a membership grade 0≤Mx≤1. The formal definition of a fuzzy set is now given.Definition 4.
A fuzzy set is a pairX,M, where X is a nonempty set and M:X⟶0,1.
The valueMx is called the membership grade of x∈X or the degree of membership of x∈X. The nearer the value of Mx to unity, the higher the degree of membership of x∈X. Conversely, the nearer the value of Mx to zero, the lower the degree of membership of x∈X. A fuzzy set X,M is sometimes read as, M is a fuzzy set on X.Definition 5.
A t-norm∗ is a function, ∗:0,1×0,1⟶0,1 such that for all a,b,c,d∈0,1, the following are satisfied:(i)
a∗1=a (1 acts as the identity element)(ii)
a∗b=b∗a (symmetry)(iii)
a∗b≤c∗d whenever a≤c and b≤d (non-decreasing)(iv)
a∗b∗c=a∗b∗c (associative).
Additionally,∗ is said to be a continuous t-norm if ∗ is a t-norm and for all sequences xn and yn in 0,1, where n∈ℕ, we have that the limit of these sequences exist and(1)limn⟶∞xn∗yn=limn⟶∞xn∗limn⟶∞yn.
More specifically,∗ is called left continuous if for each y∈0,1,(2)limn⟶∞xn∗y=limn⟶∞xn∗y.
Right continuity is analogously defined.Definition 6.
The tripleX,M,∗ is called a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm, and M is a fuzzy set on X2×0,∞ satisfying the following for each x,y,z∈X and t,s>0:(i)
Mx,y,t>0(ii)
Mx,y,t=1⟺x=y(iii)
Mx,y,t=My,x,t(iv)
Mx,y,t∗My,z,s≤Mx,z,t+s(v)
limt⟶∞Mx,y,t=1, for all x,y∈X(vi)
Mx,y:0,∞⟶0,1 is continuousRemark 1 (see [6]).
Informally, we can think ofMx,y,t as the degree of nearness between x and y with respect to t.
In 2010, Sun and Yang in [7] made the first step in generalizing the fuzzy metric space. They called it the Q-fuzzy metric space, and they proved several fixed-point theorems in this space. We define the Q-fuzzy metric space as follows.Definition7 (see [7]).
The tripletX,Q,∗ is called a Q-fuzzy metric space if ∗ be a continuous t-norm and Q is a fuzzy set on X3×0,∞ satisfying the following for each x⋅y,z,a∈X and s,t>0:(i)
Qx,x,y,t>0 with x≠y(ii)
Qx,x,y,t≥Qx,y,z,t with y≠z(iii)
Qx,y,z,t=1⟺x=y=z(iv)
Qx,y,z,t=Qx,z,y,t=Qy,x,z,t=Qz,x,y,t=Qy,z,x,t=Qz,y,x,t(v)
Qx,a,a,t∗Qa,y,z,s≤Qx,y,z,t+s(vi)
Qx,y,z,⋅:0,∞⟶0,1 is continuous
At this point, Gupta and Kanwar was ready to provide an interpretation of fuzzy metric spaces usingn points. They called it the V-fuzzy metric space.Definition8 (see [4]).
The tripletX,V,∗ is called a V-fuzzy metric space if ∗ be a continuous t-norm and V is a fuzzy set on Xn×0,∞ satisfying the following for each xi,y,a∈X and s,t>0:(i)
Vx,x,…,x,y,t>0 , x≠y(ii)
Vx1,x1,…,x1,x2,t≥Vx1,x2,x3,…,xn,t, x2≠x3≠…≠xn(iii)
Vx1,x2,x3,…,xn,t=1⟺x1=x2=x3=…=xn(iv)
Vx1,x2,x3,…,xn,t=Vpx1,x2,x3,…,xn,t where px1,x2,x3,…,xn is a permutation of x1,x2,x3,…,xn(v)
Vx1,x2,x3,…,xn−1,a,t∗Va,a,a,…,a,xn,s≤Vx1,x2,x3,…,xn−1,xn,t+s(vi)
Vx1,x2,x3,…,xn,t=1 as t⟶∞(vii)
Vx1,x2,x3,…,xn,⋅:0,∞⟶0,1 is continuousLemma 1 (see [4]).
LetX,V,∗ be a V-fuzzy metric space. Then, Vx1,x2,x3,…,xn,⋅ is nondecreasing. That is, for some 0<s<t, we have(3)Vx1,x2,x3,…,xn,s≤Vx1,x2,x3,…,xn,t.Proof.
Sinces<t, we have t−s>0. Now,(4)Vx1,x2,x3,…,xn,s∗Vxn,xn,xn,…,xn,t−s≤Vx1,x2,x3,…,xn,t.
Hence, for all0<s<t, we have(5)Vx1,x2,…,xn,s≤Vx1,x2,…,xn,t.Lemma 2 (see [4]).
LetX,V,∗ be a V-fuzzy metric space. If for all t>0 and x1,x2,…,xn∈X with k∈0,1, we have(6)Vx1,x2,…,xn,kt≥Vx1,x2,…,xn,t.
Then,x1=x2=⋯=xn.We now examine convergence and Cauchy sequences inV-fuzzy metric spaces.Definition9 (see [4]).
A sequencexn in a V-fuzzy metric space X,V,∗ is said to be convergent and converges to x∈X if for each t>0 and 0<ϵ<1, there exist N∈ℕ such that(7)Vxi,xi,xi,…,xi,x,t>1−ϵ,foralli≥N.
That is,(8)Vxi,xi,xi,…,xi,x,t⟶1asi⟶∞.Definition10 (see [4]).
A sequencexn in a V-fuzzy metric space X,V,∗ is said to be Cauchy if for each t>0 and 0<ϵ<1, there exist N∈ℕ such that(9)Vxi,xi,xi,…,xi,xj,t>1−ϵfor alli,j≥N.
That is,(10)Vxi,xi,xi,…,xi,xj,t⟶1asi,j⟶∞.Definition11 (see [4]).
AV-fuzzy metric space X,V,∗ is said to be complete if every Cauchy sequence in X is convergent in X.


## 2. Fixed-Point Theorem inV-Fuzzy Metric Spaces

In this section, we extend the concepts in the existing literature. We introduce the concept of multivalued quasi-contraction maps inV-fuzzy metric spaces, and we also give a fixed-point theorem. We begin with the following concepts in the V-fuzzy setting.Notation 1.
We denote the set of all nonempty closed and bounded subsets ofX in a V-fuzzy metric space by CBVX.Definition 12.
LetA1,A2,…,An⊆CBVX and t>0. Then,(11)Vx,A2,A3,…,An,t=infVx,a2,…,an,t:a2∈A2,a3∈A3,…,an∈An.Lemma 3.
LetX,V,∗ be a V-fuzzy metric space. If for all t>0 and x∈X with k∈0,1 and A⊆CBVX, we have(12)Vx,A,…,A,kt≥Vx,A,…,A,t.
Then,x∈A.Proof.
Assume for a contraction that(13)x∉A.
Lety∈A. Then, Vx,y,…,y,kt≥Vx,y,…,y,t. This implies x=y∈A by Lemma 2. This contradicts equation (13). Therefore, our original assumption is false. Hence, x∈A.Definition 13.
LetA1,A2,…,An∈CBVX and t>0. The Hausdorff V-fuzzy metric or Hausdorff V-fuzzy metric distance is denoted by VHA1,A2,…,An,t and is defined by(14)VHA1,A2,…,An,t=maxsupx∈A1Vx,A2,…,An,t,supx∈A2VA1,x,A3,…,An,t,⋮supx∈AnVA1,A2,…,x,t,where(15)Vx,A2,A3,…,An,t=infVx,a2,…,an,t:a2∈A2,a3∈A3,…,an∈An,VA1,x,A3,…,An,t=infVa1,x,a3,…,an,t:a1∈A1,a3∈A3,…,an∈An,⋮VA1,…,An−1,x,t=infVa1,…,an−1,x,t:a1∈A1,…,an−1∈An−1.Remark 2.
Informally, we can think of the HausdorffV-fuzzy metric as the greatest degree of nearness from a point in one set to the closest point in the other sets with respect to t.Definition 14.
LetX,V,∗ be a V-fuzzy metric space. The mapping T:X⟶CBVX is said to be a q-multivalued quasi-contraction if there exist 0≤q<1 such that for all ai∈X,i=1,2,…,n,(16)VHTa1,Ta2,…,Tan,t≤q⋅maxVa1,a2,…,an,t,Va1,Ta1,Ta1,…,Ta1,t,Va1,Ta2,Ta3,…,Tan,t,Va2,Ta2,Ta2,…,Ta2,t,Va2,Ta1,Ta3,…,Tan,t,⋮,Van,Tan,Tan,…,Tan,t,Van,Ta1,Ta2,…,Tan−1,t.


## 3. Main Result

We now proceed to give a fixed-point theorem which is our main result.Theorem 1.
LetX,V,∗ be a complete V-fuzzy metric space. If the mapping T:X⟶CBVX is a q-multivalued quasi-contraction, then T has a fixed point in X. That is, there exists u∈X such that u∈Tu.Proof.
T is given to be a q-multivalued quasi-contraction. This implies that there exists 0≤q<1 such that for all ai∈X,i=1,2,…,n,(17)VHTa1,Ta2,…,Tan,t≤q⋅maxVa1,a2,…,an,t,Va1,Ta1,Ta1,…,Ta1,t,Va1,Ta2,Ta3,…,Tan,t,Va2,Ta2,Ta2,…,Ta2,t,Va2,Ta1,Ta3,…,Tan,t,⋮,Van,Tan,Tan,…,Tan,t,Van,Ta1,Ta2,…,Tan−1,t.
It is clear that for somea1∈A1, with a2∈A2,a3∈A3,…,an∈An, we have Va1,a2,…,an,t≤VHA1,A2,…,An,t. Using this fact and setting a1∈Ta0, there exists a2∈Ta1,…,an∈Tan−1, Inequality 2 becomes(18)Va1,a2,…,an,t≤VHTa0,Ta1,…,Tan−1,t≤q⋅maxVa0,a1,a2,…,an−1,t,Va0,Ta0,Ta0,…,Ta0,t,Va0,Ta1,Ta2,…,Tan−1,t,Va1,Ta1,Ta1,…,Ta1,t,Va1,Ta0,Ta2,…,Tan−1,t,⋮Van−1,Tan−1,Tan−1,…,Tan−1,t,Van−1,Ta0,Ta1,…,Tan−2,t.
Similarly settinga2∈Ta1, there exists a3∈Ta2,…,an+1∈Tan, Inequality 2 becomes(19)Va2,a3,…,an+1,t≤VHTa1,Ta2,…,Tan,t≤q⋅maxVa1,a2,a3,…,an,t,Va1,Ta1,Ta1,…,Ta1,t,Va1,Ta2,Ta3,…,Tan,t,Va2,Ta2,Ta2,…,Ta2,t,Va2,Ta1,Ta3,…,Tan,t,⋮Van,Tan,Tan,…,Tan,t,Van,Ta1,Ta1,…,Tan−1,t.
Continuing in this manner by induction, we obtain a sequenceakk=0∞ such that(20)Vak,ak+1,…,ak+n−1,t≤VHTak−1,Tak,…,Tak+n−2,t≤q⋅maxVak−1,ak,ak+1,…,ak+n−2,t,Vak−1,Tak−1,Tak−1,…,Tak−1,t,Vak−1,Tak,Tsk+1,…,Tak+n−2,t,Vak,Tak,Tak,…,Tak,t,Vak,Tak−1,Tak+1,…,Tak+n−2,t,⋮Vak+n−2,Tak+n−2,Tak+n−2,…,Tak+n−2,t,Vak+n−2,Tak−1,Tak,…,Tak+n−3,t.
We now show thatakk=0∞ is Cauchy. If i=j, we get Vai,ai,…,ai,aj,t=1>1−ϵ, where ϵ∈0,1, and therefore ak will be trivially Cauchy. Thus, without loss of generality assume i<j and i≠j. We have(21)Vai,ai,…,ai,aj,t≤VHTai−1,Tai−1,…,Tai−1,Taj−1,t≤q⋅maxVai−1,ai−1,…,ai−1,aj−1,tVai−1,Tai−1,…,Tai−1,Tai−1,tVai−1,Tai−1,…,Tai−1,Taj−1,tVai−1,Tai−1,…,Tai−1,Tai−1,tVai−1,Tai−1,…,Tai−1,Taj−1,t⋮Vaj−1,Taj−1,…,Taj−1,Taj−1,tVaj−1,Tai−1,…,Tai−1,Tai−1,t,=q⋅maxVai−1,ai−1,…,ai−1,aj−1,tVai−1,Tai−1,…,Tai−1,Tai−1,tVai−1,Tai−1,…,Tai−1,Taj−1,tVaj−1,Taj−1,…,Taj−1,Taj−1,tVaj−1,Tai−1,…,Tai−1,Tai−1,t.
Now, we consider the five cases.Case 1.
If(22)maxVai−1,ai−1,…,ai−1,aj−1,tVai−1,Tai−1,…,Tai−1,Tai−1,tVai−1,Tai−1,…,Tai−1,Taj−1,tVaj−1,Taj−1,…,Taj−1,Taj−1,tVaj−1,Tai−1,…,Tai−1,Tai−1,t=Vai−1,ai−1,…,ai−1,aj−1,t.
Then, asi,j⟶∞ and using the fact that q∈0,1, we have(23)1≥Vai−1,ai−1,…,ai−1,aj−1,t≥1qVai,ai,…,ai,aj,t≥1q2Vai+1,ai+1,…,ai+1,aj+1,t≥⋯≥1qs+1Vai+s,ai+s,…,ai+s,aj+s,t,s∈ℕ≥⋯≥1.
This implies thatVai,ai,…,ai,aj,t⟶1 as i,j⟶∞. Therefore, akk=0∞ is Cauchy.Case 2.
If(24)maxVai−1,ai−1,…,ai−1,aj−1,tVai−1,Tai−1,…,Tai−1,Tai−1,tVai−1,Tai−1,…,Tai−1,Taj−1,tVaj−1,Taj−1,…,Taj−1,Taj−1,tVaj−1,Tai−1,…,Tai−1,Tai−1,t=Vai−1,Tai−1,…,Tai−1,Tai−1,t,then as i,j⟶∞ and using the fact that q∈0,1, we have(25)1≥Vai−1,Tai−1,…,Tai−1,Tai−1,t≥1qVai,ai,…,ai,aj,t≥1q2Vai+1,ai+1,…,ai+1,aj+1,t≥⋯≥1qs+1Vai+s,ai+s,…,ai+s,aj+s,t,s∈ℕ≥⋯≥1.
This implies thatVai,ai,…,ai,aj,t⟶1 as i,j⟶∞. Therefore, akk=0∞ is Cauchy.
The other three cases can be done in a similar manner.
SinceX,V,∗ is complete, there exists u∈X such that(26)Van,an,…,an,u,t⟶1asn⟶∞.
That is,an⟶u as n⟶∞. Now, let p∈0,1. Then,(27)Vu,Tu,…,Tu,t≤1=Vu,u,…,u,pt=limn⟶∞Van,an,…,an,pt≤limn⟶∞Van,Tan−1,…,Tan−1,ptsincean∈Tan−1.=Vu,Tu,…,Tu,pt.
Therefore, by Lemma3, we have that u∈Tu. Hence, u is a fixed point of T in X.Example 1.
LetX,d be an A-metric space, and let ∗ be a continuous t-norm defined by a∗b=ab. If for all t>0 and ∈X, let(28)Vx1,x2,…,xn,t=tt+dx1,x2,…,xn.
Now,t>0 and dx1,x2,…,xn≥0; therefore, Vx1,x2,…,xn,t>0; hence, condition i of Definition 8 holds.
Also, sincedx1,x1,…,x1,x2≤dx1,x2,…,xn. Then,(29)t+dx1,x1,…,x1,x2≤t+dx1,x2,…,xn.
Therefore,(30)tt+dx1,x1,…,x1,x2≥tt+dx1,x2,…,xn.
That is,Vx1,x1,…,x1,xn,t≥Vx1,x2,…,xn,t.
For conditioniii of Definition 8, suppose Vx1,x2,…,xn,t=t/t+dx1,x2,…,xn=1, then t=t+dx1,x2,…,xn, this implies dx1,x2,…,xn=0, Hence, x1=x2=⋯=xn.
Conversely, ifx1=x2=⋯=xn, then dx1,x2,…,xn=0; thus, Vx1,x2,…,xn,t=1.
Since theA-metric space is symmetric, condition iv of Definition 8 readily follows.
Now, we show conditionv of Definition 8(31)Vx1,x2,…xn−1,a,t∗Va,a,a,…,a,xn,s=tt+dx1,x2,…,xn−1,a∗ss+da,a,a,…,a,xn=tt+dx1,x2,…,xn−1,ass+da,a,a,…,a,xn=11+dx1,x2,…,xn−1,a/t11+da,a,a,…,a,xn/s≤11+dx1,x2,…,xn−1,a/t+s11+da,a,a,…,a,xn/t+s≤11+dx1,x2,…,xn−1,a+da,a,a,…,a,xn/t+s≤t+st+s+dx1,x2,…,xn−1,xn=Vx1,x2,…,xn−1,xn,t+s.
This is conditionv of Definition 8.
Note that(32)1dx1,x2,…,xn−1,xn≥1dx1,x2,…,xn−1,a+da,a,a,…,a,xn.
Next, using L’Hopital’s rule,(33)limt⟶∞Vx1,x2,x3,…,xn,t=limt⟶∞tt+dx1,x2,x3,…,xn=limt⟶∞1=1.
This isvi of Definition 8.
Finally, to show continuity, letan be a sequence in 0,∞. Then,(34)limn⟶∞Vx1,x2,x3,…,xn,an=limn⟶∞anan+dx1,x2,x3,…,xn=limn⟶∞anlimn⟶∞an+dx1,x2,x3,…,xn=Vx1,x2,x3,…,xn,limn⟶∞an.
Hence, the tripletX,V,∗ is a V-fuzzy metric space. Let d:Xn⟶ℝ be an A-metric space such that(35)dx1,x2,…,xn−1,xn=maxx1−x2,x1−x3,…,x1−xnx2−x3,x2−x4,…,x2−xnx3−x4,x3−x5,…,x3−xn⋮xn−1−xn.
LetX=0,1 and define T:X⟶CBVX by(36)Tx=18,13,x∈0,112,x=1.
Ifx1=x2=⋯=xn, then Tx1=Tx2=⋯=Txn and this implies that VHTx1,Tx2,…,Txn=0 and Inequality (17) holds.
Ifxi≠xj for some i≠j, then VHTx1,Tx2,…,Txn,t=t/t+3/8.
Also,(37)tt+3/8=VHTx1,Tx2,…,Txn,t≤t/t+3/8t/t+1/8tt+1/8=q⋅V1,18,…,18≤q⋅maxdx1,x2,…,xn,tdx1,Tx1,Tx1,…,Tx1,tdx1,Tx2,Tx3,…,Txn,tdx2,Tx2,Tx2,…,Tx2,tdx2,Tx1,Tx3,…,Txn,t⋮dxn,Txn,Txn,…,Txn,tdxn,Tx1,Tx2,…,Txn−1,t.
Therefore, Inequality (17) is satisfied. Here, we choose q=t/t+3/8/t/t+1/8,t>0. The fixed points of the mapping T are 3/8 and 1/8.Remark 3.
Letn=2 in Definition 8, we get the fuzzy metric space definition.
Letn=2 in Definition 14 and also let CBX be the set of closed and bounded subsets of X we get the following definition and corollary.Definition 15.
LetX,M,∗ be a fuzzy metric space. The multivalued map T:X⟶CBX is said to be a q-multivalued quasi-contraction, if there exist q∈0,1 such that for all ai∈X,i=1,2,(38)dHTa1,Ta2≤q.maxda1,a2,da1,Ta1,da2,Ta2,da1,Ta2,da2,Ta1.Corollary 1.
LetX,M,∗ be a complete fuzzy metric space. If the mapping T:X⟶CBX is a q-multivalued quasi-contraction where 0≤q<1, then T has a fixed point in X. That is, there exist u∈X such that u∈Tu.



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