A natural selection of a graphic contraction transformation in fuzzy metric spaces

Abstract

In this paper, we study sufficient conditions to find a vertex v of a graph such that Tv is a terminal vertex of a path which starts from v, where T is a self graphic contraction transformation defined on the set of vertices. Some examples are presented to support the results proved herein. Our results widen the scope of various results in the existing literature.

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J. Nonlinear Sci. Appl., 11 (2018), 218–227
Research Article
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A natural selection of a graphic contraction transformation in fuzzy metric
spaces
Hanan Alolaiyana,∗, Naeem Saleemb, Mujahid Abbasc,d
aDepartment of Mathematics, King Saud University, Saudi Arabia.
bDepartment of Mathematics, University of Management and Technology, Lahore, Pakistan.
cDepartment of Mathematics, Government College University, Lahore, Pakistan.
dDepartment of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Communicated by C. Vetro
Abstract
In this paper, we study sufficient conditions to find a vertex vof a graph such that Tv is a terminal vertex of a path which
starts from v, where Tis a self graphic contraction transformation defined on the set of vertices. Some examples are presented
to support the results proved herein. Our results widen the scope of various results in the existing literature.
Keywords: Graphic contraction, fuzzy metric space, natural selection.
2010 MSC: 47H10, 47H04, 47H07, 54H25, 54C60.
c
2018 All rights reserved.
1. Introduction and preliminaries
Zadeh [20] introduced the notion of fuzzy sets, a new way to represent vagueness and uncertainties
in daily life. Kramosil and Michalek [14] introduced the notion of a fuzzy metric by using continuous t-
norms, which generalizes the concept of a probabilistic metric space to fuzzy situation. Moreover, George
and Veeramani ([5,6]) modified the concept of fuzzy metric spaces and obtained a Hausdorff topology
for this kind of fuzzy metric spaces.
Romaguera [17] introduced Hausdorff fuzzy metric on a set of nonempty compact subsets of a fuzzy
metric space.
In the sequel, the letters N,R+, and Rwill denote the set of natural numbers, the set of positive real
numbers, and the set of real numbers, respectively.
Following definitions and known results will be needed in the sequel.
∗Corresponding author
Email addresses: holayan@ksu.edu.sa (Hanan Alolaiyan), naeem.saleem2@gmail.com (Naeem Saleem),
mujahid.abbas@up.ac.za (Mujahid Abbas)
doi: 10.22436/jnsa.011.02.04
Received 2017-07-02

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 219
Definition 1.1 ([19]).A binary operation ∗: [0, 1]2−→ [0, 1]is called a continuous t-norm if
(1) ∗is associative and commutative;
(2) ∗: [0, 1]2−→ [0, 1]is continuous (it is continuous as a mapping under the usual topology on [0, 1]2);
(3) a∗1=afor all a∈[0, 1];
(4) a∗b6c∗dwhenever a6cand b6d.
Some basic examples of continuous t-norms are ∧(minimum t-norm), ·( product t-norm), and
∗L(Lukasiewicz t-norm), where, for all a,b∈[0, 1],
a∧b=min{a,b},a·b=ab,a∗Lb=max{a+b−1, 0}.
It is easy to check that ∗L6·6∧. In fact ∗6∧for all continuous t-norm ∗.
Definition 1.2 ([5,6]).Let Xbe a nonempty set and ∗a continuous t-norm. A fuzzy set Mon X×X×(0, ∞)
is said to be a fuzzy metric on Xif for any x,y,z∈Xand s,t > 0, the following conditions hold
(i) M(x,y,t)>0;
(ii) x=yif and only if M(x,y,t) = 1 for all t > 0;
(iii) M(x,y,t) = M(y,x,t);
(iv) M(x,z,t+s)>M(x,y,t)∗M(y,z,s)for all t,s > 0;
(v) M(x,y,·):(0, ∞)→(0, 1]is continuous.
The triplet (X,M,∗)is called a fuzzy metric space. Each fuzzy metric Mon Xgenerates Hausdorff
topology τMon Xwhose base is the family of open M-balls {BM(x,ε,t) : x∈X,ε∈(0, 1),t > 0}, where
BM(x,ε,t) = {y∈X:M(x,y,t)>1−ε}.
Note that a sequence {xn}converges to x∈X(with respect to τM) if and only if limn→∞M(xn,x,t) = 1
for all t > 0.
Since for each x,y∈X,M(x,y,·)is a nondecreasing function on (0, ∞)(see [7]). Moreover every fuzzy
metric space X(in the sense of George and Veeramani [5]) is metrizable, that is, there exists a metric
don Xwhich induces a topology that agrees with τM([8]). Conversely, if (X,d)is a metric space and
Md:X×X×(0, ∞)→(0, 1]is defined as follows:
Md(x,y,t) = t
t+d(x,y)
for all t > 0, then (X,Md,∧)is a fuzzy metric space, called the standard fuzzy metric space induced by
the metric d(see [5]). The topologies induced by the standard fuzzy metric and the corresponding metric
are the same ([9]).
A sequence {xn}in a fuzzy metric space Xis said to be a Cauchy sequence if for each ε∈(0, 1), there
exists n0∈Nsuch that M(xn,xm,t)>1−εfor all n,m>n0. A fuzzy metric space Xis complete ([6])
if every Cauchy sequence converges in X. A subset Aof Xis closed if for each convergent sequence {xn}
in Awith xn−→ x, we have x∈A. A subset Aof Xis compact if each sequence in Ahas a convergent
subsequence.
Definition 1.3 ([17]).A fuzzy metric Mis said to be continuous on X2×(0, ∞)if
lim
n→∞
M(xn,yn,tn) = M(x,y,t),
whenever {(xn,yn,tn)}is a sequence in X2×(0, ∞)which converges to a point (x,y,t)∈X2×(0, ∞).
Proposition 1.4 ([17]).Let (X,M,∗)be a fuzzy metric space. Then Mis a continuous function on X×X×(0, ∞).

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 220
Lemma 1.5 ([17]).Let (X,M,∗)be a fuzzy metric space. Then for each a∈X,B∈K(X)and t > 0, there is
b0∈Bsuch that M(a,B,t) = M(a,b0,t).
A sequence {tn}of positive real numbers is said to be s-increasing ([9]) if there exists n0∈Nsuch that
tm+1>tm+1
for all m>n0. In a fuzzy metric space (X,M,∧), an infinite product (compare [11]) is denoted by
M(x,y,t1)∧M(x,y,t2)∧···∧M(x,y,tn)∧···=∞
Y
i=1
M(x,y,ti)
for all x,y∈X.
Definition 1.6 ([18]).Let Ω={η: [0, 1]→[0, 1],ηis continuous, nondecreasing and η(t)> t for t∈(0, 1),
further η(t) = 1 if and only if t=1 or η(t) = 0 if and only if t=0, overall η(t)>t, for all t∈[0, 1]}.
Let Ψbe a collection of all continuous and decreasing functions ψ: [0, 1]→[0, 1]with ψ(t) = 0 if and
only if t=1.
A function ψ∈Ψis said to have property (α)if for all r,t > 0,
r∗t > 0, we have ψ(r∗t)6ψ(r) + ψ(t),
where ∗is any continuous t-norm.
Now, an example is provided to explain the property (α).
Example 1.7. Define the mapping ψ: [0, 1]→[0, 1]by ψ(t) = 1−t. Note that, it admits property (α)for
different continuous t-norms. Take ∗=∧. Suppose that (s∗t) = min{s,t}= (s∧t) = s6t, where s,t
∈R∪{0}. Then
ψ(s∧t) = ψ(s)6ψ(s) + ψ(t).
Similarly, if min{s,t}=t6s, then we have ψ(s∧t) = ψ(s)6ψ(s) + ψ(t). This shows that property (α)
holds for minimum t-norm. If ∗=·, then (s∗t)=(s·t) = st, where s,t∈R∪{0}. Note that
ψ(s·t) = ψ(st)6ψ(s) + ψ(t).
Thus property (α)holds for a product t-norm. Suppose that ∗=∗L, that is, s∗Lt=max{s+t−1, 0}. If
s∗Lt=max{s+t−1, 0}=0, then ψ(s∗Lt) = ψ(0)6ψ(s) + ψ(t). If s∗Lt=max{s+t−1, 0}=s+t−1,
then we have
ψ(s∗Lt) = ψ(s+t−1)6ψ(s) + ψ(t),
which shows property (α)holds for Lukasiewicz t-norm.
On the other hand, the interplay between the preference relation of abstract objects of underlying
mathematical structure and fixed point theory is very strong and fruitful. This gives rise to an interesting
branch of nonlinear functional analysis called order oriented fixed point theory. This theory is studied
in the framework of a partially ordered sets along with appropriate mappings satisfying certain order
conditions and has many applications in economics, computer science and other related disciplines.
Existence of fixed points in partially ordered metric spaces was first investigated in 2004 by Ran and
Reurings [16], and then by Nieto and Lopez [15].
Recently, Azam [4] obtained coincidence points of mappings and relations satisfying certain contrac-
tive conditions in the setup of a metric space.
Jachymski [13] introduced a new approach in metric fixed point theory by replacing order structure
with a graph structure on a metric space. In this way, the results obtained in ordered metric spaces
are generalized (see also [12] and the reference therein); in fact, Gwodzdz-lukawska and Jachymski [10]

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 221
developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed
with a directed graph, further Abbas et al. obtained results using graphical contractions (for details see
[1–3]).
The following definitions and notations will be needed in the sequel.
Let Xbe any set and ∆denotes the diagonal of X×X. Let G(V,E)be a undirected graph such that
the set Vof its vertices is a subset of Xand Ethe set of edges of the graph which contains all loops, that
is, ∆⊆E. Also assume that the graph Ghas no parallel edges and, thus one can identify Gwith the pair
(V,E).
Motivated by the work in [4], we introduce the following concept of a natural selection of a transfor-
mation T:V→V.
If x,y∈V, then (x,y)denotes an edge between xand y.
If vertices xand yof a graph are connected by certain edges, we say there exists a path between xand
y. In this case, we denote [x,y]a path which starts from xand terminates at y(we call vertex ya terminal
vertex and vertex xa reference vertex). Set
Ex=The collection of all terminal vertices of edges starting from xand EX=∪x∈XEx.
A vertex w∈Vis called a natural selection of T:V→Vif Tw ∈Ew, that is, Tw is a terminal vertex of
[w,Tw].
Let xand ybe two vertices of a graph G. It is a common practice to assign a certain weight to each
edge of a graph. The positive real number obtained by calculating the distance between xand ycan be
used as a weight of an edge joining xand y.
In this paper, we assign a fuzzy weight M(x,y,t)(a number between 0 and 1) to an edge (x,y)at t,
where tis interpreted as a time. In this case we have a larger flexibility in choosing the weights specially
when one is uncertain or confused at a certain point of time in assigning a weight to an edge (x,y)at a
time t.
We establish an existence of a vertex vof a graph such that its image under a graphic transformation
satisfying certain contraction conditions becomes a terminal vertex of a path starting from v.
We give examples to support our results and to show that our results are potential generalization of
comparable results in the existing literature.
M(x,y,t) = 1 for all t > 0 if and only if a path [x,y]defines a loop. Define
D(Ex,Ey,t)=sup{M(u,v,t),u∈Ex,v∈Ey}.
2. Natural selection of graphic contractions
We start with the following result.
Theorem 2.1. Let T:V→V.If there exists a functions ψ∈Ψhaving property (α)such that
ψ(M(Tx,Ty,t)) 6kψ(D(Ex,Ey,t)) (2.1)
holds for all vertices x,y∈Vand 0< k < 1, then there exists w∈Vsuch that Tw ∈Ewprovided that T(V)⊆
EXand EXis complete subspace of fuzzy metric space (X,M,∗).
Proof. Let x0be an arbitrarily fixed vertex of graph G(V,E)(for simplicity G). We shall construct sequences
of vertices {xn}⊂V,{yn}⊂EXas follows: Let y1=Tx0. Since T(V)⊆EX, we can choose a vertex x1in
Vsuch that y1∈Ex1. Let y2=Tx1. If ψ(D(Ex0,Ex1,t) = 0, then we have Tx0=Tx1which implies that
y2∈Ex1and hence x1becomes the required vertex of G. If ψ(D(Ex0,Ex1,t)6=0, then by inequality (2.1),
we have
ψ(M(Tx0,Tx1,t)) 6kψ(D(Ex0,Ex1,t)) 6=0.

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 222
Choose another vertex x2in Vsuch that y2∈Ex2. If ψ(D(Ex1,Ex2,t) = 0, then x2is the required vertex in
V. If ψ(D(Ex1,Ex2,t)6=0, then by inequality (2.1), we obtain that
ψ(M(Tx1,Tx2,t)) 6kψ(D(Ex1,Ex2,t)6=0.
Continuing this way, we can obtain two sequences of vertices {xn}⊂Vand {yn}⊂EXsuch that yn=
Txn−1,yn∈Exnand it satisfies:
ψ(M(yn,yn+1,t)) 6kψ(D(Exn−1,Exn,t)6=0, n=1, 2, 3, . . . .
Since yn−1∈Exn−1,yn∈Exn, we have
D(Exn−1,Exn,t)>M(yn−1,yn,t),
which further implies that
ψ(M(yn,yn+1,t)) 6kψ(M(yn−1,yn,t)).
Note that
ψ(M(yn,yn+1,t)) 6kψ(M(yn−1,yn,t))
6k2ψ(M(yn−2,yn−1,t))
.
.
.
6knψ(M(y0,y1,t)),n=1, 2, 3, . . . .
That is,
ψ(M(yn,yn+1,t)) 6knψ(M(y0,y1,t)).
On taking limit as n→∞on both sides of the above inequality, we have
lim
n→∞
ψ(M(yn,yn+1,t)) = 0.
Now, we show that {yn}is a Cauchy sequence. Suppose that there exist some n0∈Nwith m>n>n0
such that
ψ(M(yn,ym,t)) = ψ(M(yn,ym,
m−1
X
i=n
ait),
where {ai}is a decreasing sequence of positive real numbers satisfying Pm−1
i=nai=1. Thus
M(yn,ym,t) = M(yn,ym,
m−1
X
i=n
ait)
>M(yn,yn+1,ant)∗M(yn+1,yn+2,an+1t)∗· ·· ∗M(ym−1,ym,am−1t).
Further, we obtain that
ψ(M(yn,ym,t))
=ψ(M(yn,ym,
m−1
X
i=n
ait))
6ψ[M(yn,yn+1,ant)∗M(yn+1,yn+2,an+1t)∗· ·· ∗M(ym−1,ym,am−1t)]
6ψ(M(yn,yn+1,ant)) + ψ(M(yn+1,yn+2,an+1t)) + ···+ψ(M(ym−1,ym,am−1t))
6knψ(M(y0,y1,ant)) + kn+1ψ(M(y0,y1,an+1t)) + ···+km−1ψ(M(y0,y1,am−1t)).
(2.2)

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 223
If
max{ψ(M(y0,y1,ant)),ψ(M(y0,y1,an+1t)),···,ψ(M(y0,y1,am−1t))}=ψ(M(y0,y1,bt))
for some b∈{ai:n6i6m−1}, then the inequality (2.2) becomes
ψ(M(yn,ym,t)) 6knψ(M(y0,y1,ant)) + kn+1ψ(M(y0,y1,an+1t)) + ···+km−1ψ(M(y0,y1,am−1t))
6knψ(M(y0,y1,bt)) + kn+1ψ(M(y0,y1,bt)) + ···+km−1ψ(M(y0,y1,bt))
6(kn+kn+1+···+km−1)ψ(M(y0,y1,bt))
6kn(1+k+···+km−n−1)ψ(M(y0,y1,bt))
6kn
1−kψ(M(y0,y1,bt)),
that is, for all n∈N,
ψ(M(yn,ym,t)) 6kn
1−kψ(M(y0,y1,bt)).
On taking limit as n→∞on both sides of the above inequality, we have
06lim
n,m→∞
ψ(M(yn,ym,t)) 60.
By continuity of ψ, we obtain that
lim
n,m→∞
M(yn,ym,t) = 1.
Hence {yn}is a Cauchy sequence in EX. Next we assume that there exists a vertex zin EXsuch that
limn→∞M(yn,z,t) = 1. Moreover, z∈Ewfor some w∈X. Also,
M(z,Tw,t)>M(z,yn+1,t
2)∗M(yn+1,Tw,t
2).
As ψ∈Ψ, so we have
ψ(M(z,Tw,t)) 6ψ[M(z,yn+1,t
2)∗M(yn+1,Tw,t
2)]
6ψ[M(z,yn+1,t
2)] + ψ[M(yn+1,Tw,t
2)]
=ψ[M(z,yn+1,t
2)] + ψ[M(Txn,Tw,t
2)]
6ψ[M(z,yn+1,t
2)] + kψ[D(Exn,Ew,t
2)]
6ψ[M(z,yn+1,t
2)] + kψ[M(yn,z,t
2)].
On taking limit as n→∞, we have z=Tw, that is, T w ∈Ew.
Example 2.2. Let V=Q∪Q0=Rand M:X×X×(0, ∞)→(0, 1]be the fuzzy metric defined by
M(x,y,t) = t
t+d(x,y), where dis the usual metric on X. Suppose that ψ(t)=1−tfor all t∈(0, 1).
Define the mapping T:R→Rby
Tx =2, if x∈Q,
0, if x∈Q0.
If x,y∈Q,Tx =Ty =2 and Ex=Ey= [0, 4]. Note that
ψ(M(Tx,Ty,t)) = ψ(1)6kψ(D(Ex,Ey,t)) = kψ(1) = 0.
If x,y∈Q0, then Tx =Ty =0 and Ex=Ey= [7, 9]. In this case, we have
ψ(M(Tx,Ty,t)) = ψ(1)6kψ(D(Ex,Ey,t)) = kψ(1) = 0.
If x∈Qand y∈Q0or x∈Q0and y∈Q, then Ex= [0, 4]and Ey= [7, 9]or Ex= [7, 9]and Ey= [0, 4].

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 224
For, k>3
4, we have
ψ(M(Tx,Ty,t)) = ψ(t
t+2) = 1−t
t+26kψ(D(Ex,Ey,t)).
Also, T(V) = {0, 2}⊂Ex=[0, 4]∪[7, 9]. Thus all the conditions of Theorem (2.1) are satisfied. However,
Theorem 3.1 in [4] does not hold in last case.
Theorem 2.3. Let T:V→Vwith T(V)⊆EXand EXa complete subspace of fuzzy metric space (X,M,∗). If
there exists η∈Ωand k∈(0, 1)such that for all vertices x,y∈V,we have
M(Tx,Ty,kt)>η(D(Ex,Ey,t)), (2.3)
then there exists a vertex w∈Vsuch that Tw ∈Ewprovided that for each ε > 0and an s-increasing sequence
{tn},there exists n0in N0such that Q∞
n>n0M(x,y,tn)>1−εfor all n>n0.
Proof. Let x0be an arbitrarily fixed vertex of graph G(V,E)(for simplicity G). We shall construct sequences
of vertices {xn}⊂V,{yn}⊂EXas follows: Let y1=Tx0. Since T(V)⊆EX, we can choose a vertex x1in
Vsuch that y1∈Ex1. Let y2=Tx1. If η(D(Ex0,Ex1,t) = 1, then we have Tx0=Tx1which implies that
y2∈Ex1and hence x1becomes the required vertex of G. If η(D(Ex0,Ex1,t)6=1, then by inequality (2.3),
we have
M(Tx0,Tx1,t)>η(D(Ex0,Ex1,t)) 6=1.
Choose another vertex x2in Vsuch that y2∈Ex2. If η(D(Ex1,Ex2,t) = 1, then x2is the required vertex in
V. If η(D(Ex1,Ex2,t)6=1, then by inequality (2.3), we obtain that
M(Tx1,Tx2,kt)>η(D(Ex1,Ex2,t)) 6=1.
Continuing this way, we can obtain two sequences of vertices {xn}⊂Vand {yn}⊂EXsuch that yn=
Txn−1,yn∈Exnand it satisfies:
M(yn,yn+1,kt)>η(D(Exn−1,Exn,t)) 6=1, n=1, 2, 3, . . . .
As yn∈Exnand yn+1∈Exn+1, we have
D(Exn,Exn+1,t)>M(yn,yn+1,t).
Thus
M(yn,yn+1,kt)>η(M(yn−1,yn,t))
implies that
M(yn,yn+1,t)>η(M(yn−1,yn,t
k)) >M(yn−1,yn,t
k).
Continuing this way, we have
M(yn,yn+1,t)>M(yn−1,yn,t
k)>M(yn−2,yn−1,t
k2)>···>M(y0,y1,t
kn).
Let t > 0, ε > 0, m,n∈Nsuch that m>nand hi=1
i(i+1)for i∈{n,n+1, . . . , m−1}. As
hn+hn+1+···+hm−1<1, we have
M(yn,ym,t)>M(yn,ym,(hn+hn+1+···+hm−1)t)
>M(yn,yn+1,hnt)∗M(yn+1,yn+2,hn+1t)∗· ·· ∗M(ym−1,ym,hm−1t)
>M(y0,y1,hn
knt)∗M(y0,y1,hn+1
kn+1t)∗· ·· ∗M(y0,y1,hm−1
km−1t)

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 225
=M(y0,y1,1
n(n+1)knt)∗M(y0,y1,1
(n+1)(n+2)kn+1t)∗· ·· ∗M(y0,y1,1
m(m−1)km−1t)
>∞
Y
i=n
M(y0,y1,t
i(i+1)ki) = ∞
Y
i=n
M(y0,y1,ti),
where ti=t
i(i+1)ki. Since limn→∞(tn+1−tn) = ∞, therefore {ti}is an s-increasing sequence. Conse-
quently, there exists n0∈N, such that for each ε > 0, we have Q∞
n=1M(y0,y1,tn)>1−εfor all n>n0.
Hence M(yn,ym,t)>1−εfor all n,m>n0. Thus {yn}is a Cauchy sequence in EX. Next we assume that
there exists a vertex zin EXsuch that limn→∞M(yn,z,t) = 1. Moreover, z∈Ewfor some w∈V. Now,
M(z,Tw,t)>M(z,yn+1,(1−k)t)∗M(yn+1,Tw,kt)
implies
M(z,Tw,t)>M(z,yn+1,(1−k)t)∗M(yn+1,Tw,kt)
>M(z,yn+1,(1−k)t)∗M(Txn,Tw,kt)]
>M(z,yn+1,(1−k)t)∗η[D(Exn,Ew,kt)]
>M(z,yn+1,(1−k)t)∗η[M(yn,z,kt)].
On taking limit as n→∞, we have z=Tw and hence T w ∈Ew.
Example 2.4. Let V=Q∪Q0=Rand M:X×X×(0, ∞)→(0, 1]be the fuzzy metric defined by
M(x,y,t) = t
t+d(x,y), where dis the usual metric on X. Define the transformation T:R→Rby
Tx =1, if x∈Q,
0, if x∈Q0.
Suppose that η(t)=√tfor all t∈(0, 1). If x,y∈Q, then Tx =Ty =1 and Ex=Ey= [0, 4]. Also,
M(Tx,Ty,kt) = 1>η(D(Ex,Ey,t)),
when x,y∈Q0. Then Tx =Ty =0, and Ex=Ey= [7, 9]. In this case, we have
M(Tx,Ty,kt) = 1>η(D(Ex,Ey,t)).
If x∈Qand y∈Q0or x∈Q0and y∈Q, then Ex= [0, 4]and Ey= [7, 9]or Ex= [7, 9]and Ey= [0, 4]. For
k>3
4, we have
M(Tx,Ty,3
4t) =
3
4t
3
4t+1>η(D(Ex,Ey,t)).
Note that T(V) = {0, 1}⊂Ex=[0, 4]∪[7, 9]. Thus all the conditions of Theorem (2.1) are satisfied.
In the next theorem, we prove the existence of a unique coincidence point of a pair of mappings under
a contractive condition.
Theorem 2.5. Let T,S:V→Vbe continuous mapping with T(V)⊆S(V)⊆EXand EXa complete subspace of
fuzzy metric space (X,M,∗). If there exists η∈Ωand k∈(0, 1)such that for all vertices x,y∈V,we have
M(Tx,Ty,kt)>η(M(Sx,Sy,t)),
then there exists a vertex w∈Vsuch that Tw,Sw ∈Ewprovided that for each ε > 0and an s-increasing sequence
{tn},there exists n0in N0such that Q∞
n>n0M(x,y,tn)>1−εfor all n>n0.Moreover, if either Tor Sis
injective, then the vertex wis unique.

H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., 11 (2018), 218–227 226
Proof. By Theorem (2.3) and yn=Txn−1∈T(V)⊆S(V)⊆EX, then there exists ynsuch that Txn−1=
Syn∈S(V), we obtain limn→∞yn=w∈Vsuch that Tw =Sw, where,
Sw =lim
n→∞
Syn=lim
n→∞
Txn−1=Tw,x0∈V.
For uniqueness, assume that w1,w2∈V,w16=w2,Tw1=Sw1, and Tw2=Sw2. Then M(Tw1,T w2,kt)>
η(M(Sw1,Sw2,t)). If Sor Tis injective, then
M(Sw1,Sw2,t)6=1,
and
M(Sw1,Sw2,t)> M (Sw1,Sw2,kt)=M(T w1,T w2,kt)>η(M(Sw1,Sw2,t))>M(Sw1,Sw2,t),
which is a contradiction.
Theorem 2.6. Let T,S:V→Vare continuous mappings with T(V)⊆S(V)⊆EXand EXis a complete subspace
of fuzzy metric space (X,M,∗). If there exists a k∈(0, 1)such that for all x,y∈V,we have
M(Tx,Ty,kt)>η(M(Sx,Sy,t)),
where η∈Ω,then Sand Thave a coincidence point in X. Moreover, if either Tor Sis injective, then Tand Shave
a unique coincidence point in X.
Proof. This uniqueness can be proved on the same lines as in proof of Theorem 2.5.
Remark 2.7.If in the above theorem we choose X=Yand R=I(the identity mapping on X), we obtain
the Banach contraction theorem.
Acknowledgment
The authors extend their appreciation to the International Scientific partnership program (ISPP) at
King Saud University for funding this research work through ISPP#0034.
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