@
Appl. Gen. Topol. 21, no. 2 (2020), 177-194

doi:10.4995/agt.2020.11369

c© AGT, UPV, 2020

Fixed points for fuzzy quasi-contractions in

fuzzy metric spaces endowed with a graph

Mina Dinarvand

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi Univer-

sity, 50 Taleghani Avenue, 15618, Tehran, Iran (dinarvand mina@yahoo.com)

Communicated by I. Altun

Abstract

In this paper, we introduce the notion of G-fuzzy H-quasi-contractions
using directed graphs in the setting of fuzzy metric spaces endowed
with a graph and we show that this new type of contraction generalizes
a large number of different types of contractions. Subsequently, we
investigate some results concerning the existence of fixed points for
this class of contractions under two different conditions in M-complete
fuzzy metric spaces endowed with a graph. Our main results of the work
significantly generalize many known comparable results in the existing
literature. Examples are given to support the usability of our results
and to show that they are improvements of some known ones.

2010 MSC: 54H25; 47H10; 05C40.

Keywords: fuzzy metric space; (!C)-graph; !G-fuzzy quasi-contraction; fixed
point.

1. Introduction and preliminaries

In attempt to model the real world problems, we have to deal with uncertain-
ties and vagueness of the data, tools or conditions in the form of constraints.
To deal with uncertainty, we need techniques other than classical ones wherein
some specific logic is required. Fuzzy set theory is one of the uncertainty ap-
proaches wherein topological structures are basic tools to develop mathematical
models compatible to concrete real life situations. Zadeh [21] considered the

Received 08 February 2019 – Accepted 24 June 2020

http://dx.doi.org/10.4995/agt.2020.11369


M. Dinarvand

nature of uncertainty in the behaviour of systems possessing fuzzy nature by
means of a fuzzy set.

In 1994, George and Veeramani [11] modified the concept of fuzzy metric
space introduced by Kramosil and Michálek [15].

Definition 1.1 (George and Veeramani [11]). A fuzzy metric space is a triple
(X, M, !) such that X is a nonempty set, ! is a continuous t-norm and M is a
fuzzy set on X × X × (0, ∞) satisfying the following conditions:
(FM1) M(x, y, t) > 0 for all x, y ∈ X and each t > 0;
(FM2) M(x, y, t) = 1 for all x, y ∈ X and each t > 0 if and only if x = y;
(FM3) M(x, y, t) = M(y, x, t) for all x, y ∈ X and each t > 0;
(FM4) M(x, z, t + s) ≥ M(x, y, t) ! M(y, z, s) for all x, y, z ∈ X and each

t, s > 0;
(FM5) M(x, y, ·) : (0, ∞) → [0, 1] is continuous.

If we replace (FM4) by

(NA) M(x, z, max{t, s}) ≥ M(x, y, t) ! M(y, z, s) for all x, y, z ∈ X and each
t, s > 0,

then the triple (X, M, !) is called a non-Archimedean fuzzy metric space. It
should be noted that any non-Archimedean fuzzy metric space is a fuzzy metric
space.

Example 1.2 ([11]). Let (X, d) be a metric space. Then the triple (X, Md, !)
is a fuzzy metric space, where a ! b = ab for all a, b ∈ [0, 1] and Md(x, y, t) =

t
t+d(x,y)

for all x, y ∈ X and each t > 0. We call this Md as the standard fuzzy
metric induced by the metric d. Even if we define a ! b = min{a, b} for all
a, b ∈ [0, 1], then the triple (X, Md, !) will be a fuzzy metric space.

Let (X, Md, !) be a fuzzy metric space. For t > 0, the open ball B(x, r, t)
with a center x ∈ X and a radius 0 < r < 1 is defined by

B(x, r, t) =
!
y ∈ X : M(x, y, t) > 1 − r

"
.

The collection {B(x, r, t) : x ∈ X, 0 < " < 1, t > 0} is a neighbourhood
system for the topology τ on X induced by the fuzzy metric M.

Lemma 1.3 ([9]). Let (X, M, !) be a fuzzy metric space. Then M is a contin-
uous function on X × X × (0, ∞).

Definition 1.4 (George and Veeramani [11]). Let (X, M, !) be a fuzzy metric
space.

(1) A sequence {xn} in X is said to be convergent to a point x ∈ X,
denoted by xn → x as n → ∞, 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.

(2) A sequence {xn} in X is a M-Cauchy sequence if and only if for all
" ∈ (0, 1) and t > 0, there exists n0 ∈ N such that M(xn, xm, t) > 1 − "
for all m, n ≥ n0.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 178



Quasi-contractions in fuzzy metric spaces with a graph

(3) An M-complete fuzzy metric space is a fuzzy metric space in which
every M-Cauchy sequence is convergent.

Definition 1.5 ([10]). Let (X, M, !) be a fuzzy metric space. A mapping
T : X → X is called t-uniformly continuous if for all r ∈ (0, 1), there exists
s ∈ (0, 1) such that

M(x, y, t) ≥ 1 − s implies M(Tx, Ty, t) ≥ 1 − r
for all x, y ∈ X and each t > 0.

Remark 1.6. If T is t-uniformly continuous, then it is uniformly continuous
for the uniformity generated by M and so it is continuous for the topology
deduced from M. For the details concerning a uniform structure in a fuzzy
metric space, the reader is directed to [10].

Fixed point theory is one of the most fruitful and effective tools in mathemat-
ics which has enormous applications within as well as outside the mathematics.
The paper of Grabiec [9] started the investigations concerning fixed point the-
ory in fuzzy metric spaces. Afterwards, Gregori and Sapena [10] introduced
the notion of fuzzy contractive mappings and gave some fixed point results in
fuzzy metric spaces.

Recently, Wardowski [20] introduced the following class of functions which
will be used densely in the sequel.

Denote by H the family of all the mappings η : (0, 1] → [0, ∞) satisfying the
following properties:

(H1) η transforms (0, 1] onto [0, ∞);
(H2) η is strictly decreasing (i.e. s < t implies η(s) > η(t) for all t, s ∈ (0, 1]).

It is worth mentioning that if η ∈ H, then η(1) = 0 and η is continuous.

Theorem 1.7 (Wardowski [20]). Let (X, M, !) be an M-complete fuzzy metric
space and suppose that T : X → X be a self-mapping satisfying

η
#
M(Tx, Ty, t)

$
≤ kη

#
M(x, y, t)

$
(1.1)

for all x, y ∈ X and each t > 0, where k ∈ (0, 1). Assume also that the following
assertions hold:

(a)
%k

i=1 M(x, Tx, ti) ∕= 0 for all x ∈ X, k ∈ N and any sequence {tn} ⊆
(0, ∞), tn ↓ 0;

(b) r ! s > 0 implies η(r ! s) ≤ η(r) + η(s) for all r, s ∈
!
M(x, Tx, t) : x ∈

X, t > 0
"
;

(c)
!
η(M(x, Tx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then T has a unique fixed point in x∗ ∈ X and for each x0 ∈ X the sequence
{T nx0} converges to x∗.

By considering a mapping η ∈ H of the form η(t) = 1
t
− 1 where t ∈ (0, 1],

the fuzzy contraction condition (1.1) reduces to the class of fuzzy contractive
mappings introduced by Gregori and Sapena [10].

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 179



M. Dinarvand

On the other hand, the most important graph theory approach to metric
fixed point theory introduced so far is attributed to Jachymski [13]. In this new
approach, the underlying metric space is equipped with a directed graph and
the Banach contraction is formulated in a graph language. Using this simple
but very interesting idea, Jachymski generalized several well known versions of
Banach contraction principle in metric spaces simultaneously and from various
aspects. We commence by reviewing some basic notions in graph theory which
will be used throughout this paper. The readers interested in this topic are
referred to [5, 8, 19] and references cited therein.

In an arbitrary (not necessarily simple) graph G, a link is an edge of G with
distinct ends and a loop is an edge of G with identical ends. Two or more links
of G with the same pairs of ends are called parallel edges of G.

Let (X, M, !) be a fuzzy metric space and ∆(X) denotes the diagonal of the
Cartesian product X ×X. Consider a directed graph G such that the set V (G)
of the vertices of G coincides with X, i.e. V (G) = X, and the set E(G) of
the edges of G contains all loops, i.e. E(G) ⊇ ∆(X) (note that in general, G
can have uncountably many vertices). Suppose further that G has no parallel
edges. In this case, the graph G can be simply denoted by the ordered pair
G = (V (G), E(G)) = (X, E(G)). If G is such a graph, then it is said that the
fuzzy metric space (X, M, !) is endowed with the graph G.

By the notation G−1, it is meant the conversion of G as usual, i.e. a directed
graph obtained from G by reversing the directions of the edges of G, and by

the notation &G, it is always meant the undirected graph obtained from G by
ignoring the directions of the edges of G. Thus, it is clear that V (G−1) =

V ( &G) = V (G) = X and so

E(G−1) =
!
(x, y) ∈ X × X : (y, x) ∈ E(G)

"
and E( &G) = E(G) ∪ E(G−1).

It should be remarked that if both (x, y) and (y, x) belong to E(G), then we

will face with parallel edges in the graph &G. To avoid this problem, we delete
either the edge (x, y) or the edge (y, x) (but not both of them) from G and

consider the graph &G obtained from the remaining graph.
If (X, ≼) is a partially ordered set, then by comparable elements of (X, ≼), it

is meant two elements x, y ∈ X satisfying either x ≼ y or y ≼ x, and a mapping
T : X → X is called order-preserving whenever x ≼ y implies Tx ≼ Ty for all
x, y ∈ X.

In 1974, Ćirić [7] introduced quasi-contractions in metric spaces and gave
an example to show that this new contraction is a real generalization of some
well known linear contraction. The main purpose of the present work is to
formulate a G-fuzzy H-quasi-contraction which generalizes a large number of
contractions in fuzzy metric spaces endowed with a graph. We then investi-
gate some sufficient conditions which ensure the existence of fixed points for
such mappings on M-complete fuzzy metric spaces in the sense of George and
Veeramani endowed with a graph. The obtained results generalize many known
results in the recent literature. Some examples are given which illustrate the

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 180



Quasi-contractions in fuzzy metric spaces with a graph

value of the obtained results in comparison to some of the existing ones in
literature.

2. Main results

Suppose that (X, M, !) be a fuzzy metric space endowed with a graph G
and T : X → X be an arbitrary mapping. Throughout this section, we use the
letter CT to denote the set of all points x ∈ X such that (T mx, T nx) ∈ E( &G)
for all m, n ∈ N ∪ {0}.

Remark 2.1. Let (R, d) be the usual (Euclidean) metric space of all real numbers
and (R, Md, !) be the standard fuzzy metric space induced by d. Consider a
graph G given by V (G) = R and E(G) = {(x, x) : x ∈ R}. Define a mapping
T : R → R by the rule Tx = x + 2 for all x ∈ R. Now one can see easily that
CT = ∅.

Given x ∈ X and n ∈ N ∪ {0}, the n-th orbit of x under T is denoted by
O(x; n), i.e.

O(x; n) =
!
x, Tx, . . . , T nx

"
.

If A is a subset of X, then by δt(A), it is meant the diameter of A in X, i.e.

δt(A) = sup
!
η
#
M(x, y, t)

$
: x, y ∈ A

"
.

Motivated by Aleomraninejad et al. [1], we say that G is a (&C)-graph when-
ever the triple (X, d, G) has the following property:

If {xn} is a sequence in (X, d, G) such that xn → x ∈ X and (xn, xn+1) ∈
E( &G) for all n ∈ N, then there exists a subsequence {xnk} of {xn} such
that (xnk, x) ∈ E( &G) for all k ∈ N.

Now, we are ready to introduce the concept of G-fuzzy H-quasi-contractions
with respect to η ∈ H in fuzzy metric spaces endowed with a graph which is
inspired by [2, Definition 2.2] and [13, Definition 2.1].

Definition 2.2. Let (X, M, !) be a fuzzy metric space endowed with a graph
G and T : X → X be a mapping. We say T is a G-fuzzy H-quasi-contraction
with respect to η ∈ H if
(FQ1) T preserves the edges of G, i.e. (x, y) ∈ E(G) implies (Tx, Ty) ∈ E(G)

for all x, y ∈ X;
(FQ2) there exists λ ∈ (0, 1) such that

η
#
M(Tx, Ty, t)

$
≤ λ max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"

for all x, y ∈ X with (x, y) ∈ E(G) and each t > 0.
If T is a G-fuzzy H-quasi-contraction with respect to η ∈ H, then we call λ in
(FQ2) a quasi-contractive constant of T .

We now give some examples of G-fuzzy H-quasi-contractions with respect
to η ∈ H in fuzzy metric spaces endowed with a graph.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 181



M. Dinarvand

Example 2.3. Suppose that (X, M, !) is a fuzzy metric space endowed with
a graph G and x0 be a point in X. It is elementary to check that the constant

mapping x
T.→ x0 is a G-fuzzy H-quasi-contraction with respect to η ∈ H

with arbitrary quasi-contractive constant λ ∈ (0, 1) since E(G) contains all the
loops. So the cardinality of the set of all G-fuzzy H-quasi-contractions defined
on a fuzzy metric space (X, M, !) endowed with a graph G is no less than the
cardinality of X.

Example 2.4. Suppose that (X, M, !) is a fuzzy metric space and T : X → X
is a fuzzy H-quasi-contraction with respect to η ∈ H in the sense that there
exists λ ∈ (0, 1) such that

η
#
M(Tx, Ty, t)

$
≤ λ max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
(2.1)

for all x, y ∈ X and any t > 0. Define a graph G0 by V (G0) = X and
E(G0) = X × X, i.e. G0 is the complete graph whose vertex set coincides with
X. Obviously, T preserves the edges of G0 and (2.1) guarantees that T satisfies
(FQ2) for the complete graph G0. Thus, T is a G0-fuzzy H-quasi-contraction
with respect to η ∈ H with the quasi-contractive constant λ ∈ (0, 1). Hence,
G0-fuzzy H-quasi-contractions with respect to η ∈ H on fuzzy metric spaces en-
dowed with the graph G0 are precisely the fuzzy H-quasi-contractions with re-
spect to η ∈ H on fuzzy metric spaces. Therefore, G-fuzzy H-quasi-contractions
with respect to η ∈ H are a generalization of fuzzy H-quasi-contractions with
respect to η ∈ H from fuzzy metric spaces to fuzzy metric spaces endowed with
a graph. As stated before, the concept of quasi-contractions in metric spaces
initiated by Ćirić [7] in 1974. Moreover, Rhoades [18] showed that Ćirić’s con-
tractive condition is one of the most general contractive definitions in metric
spaces and includes a large number of different types of contractions.

Example 2.5. Let (X, ≼) be a partially ordered set and (X, M, !) be a fuzzy
metric space. Consider the poset graphs G1 and G2 by

V (G1) = X and E(G1) =
!
(x, y) ∈ X × X : x ≼ y

"
,

and

V (G2) = X and E(G2) =
!
(x, y) ∈ X × X : x ≼ y ∨ y ≼ x

"
.

A mapping T : X → X preserves the edges of G1 if and only if T is order-
preserving, and T satisfies (FQ2) for the graph G1 if and only if there exists
λ ∈ (0, 1) such that

η
#
M(Tx, Ty, t)

$
≤ λ max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
(2.2)

for all comparable elements x, y ∈ X and any t > 0, where η ∈ H. Moreover,
T preserves the edges of G2 if and only if T maps comparable elements of
(X, ≼) onto comparable elements, and T satisfies (FQ2) for the graph G2 if
and only if (2.2) holds for all comparable elements x, y ∈ X and any t > 0.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 182



Quasi-contractions in fuzzy metric spaces with a graph

Hence, if T is a G1-fuzzy H-quasi-contraction with respect to η ∈ H, then T is
a G2-fuzzy H-quasi-contraction with respect to η ∈ H. Therefore, G-fuzzy H-
quasi-contractions with respect to η ∈ H are a generalization of ordered fuzzy
H-quasi-contractions with respect to η ∈ H from fuzzy metric spaces equipped
with a partial order to fuzzy metric spaces endowed with a graph.

From now on, we assume that the graphs G0, G1 and G2 are as defined in
Examples 2.4 and 2.5.

Remark 2.6. In the definitions of (&C)-graph and the set CT , let’s set G the
special graphs G0, G1 and G2. Then we obtain the following special cases:

• The set CT related to the complete graph G0 coincides with X and G0
is a (&C)-graph.

• If ≼ is a partial order on X, then the set CT related to the graph G1
(and also G2) consists of all points x ∈ X whose every two iterates
under T are comparable elements of X. Moreover, G1 (and also G2)

is a (&C)-graph whenever the quadruple (X, M, !, ≼) has the following
property:
(∗): If {xn} is a sequence in (X, M, !) converging to a point x ∈

X whose successive terms are pairwise comparable elements of
(X, ≼), then there exists a subsequence of {xn} whose terms and
x are comparable elements of (X, ≼).

Example 2.7. Let X = [0, 1] and ! be the usual product. Then (X, M, !) is
a fuzzy metric space, where

M(x, y, t) =

'
t

t + 1

(|x−y|

for all x, y ∈ X and each t > 0. Define a self-mapping T : X → X by the
formula

Tx =

)
1
9
, x = 0,

1
3
, 0 < x ≤ 1.

We show that T is not a G0-fuzzy H-Banach contraction with respect to η ∈ H
on X. Arguing by contradiction, we suppose that there exists η ∈ H such that

η
#
M(Tx, Ty, t)

$
≤ λη

#
M(x, y, t)

$

for all x, y ∈ X and each t > 0, where λ ∈ (0, 1) is a constant. Now, by
taking the points x = 0, 0 < y ≤ 1 and t = 1 in the above inequality, we get
η((1

2
)

2
9 ) ≤ λη((1

2
)y) and so

η
##1

2

$ 2
9
$
≤ λ lim

y→0+
η
##1

2

$y$
= λη(1) = 0,

which gives a contradiction.
On the other hand, from the equality

ln M(
1

9
,
1

3
, t) =

2

3
ln M(0,

1

3
, t) for all t > 0,

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 183



M. Dinarvand

it immediately follows that for all x, y ∈ X and each t > 0,

η
#
M(Tx, Ty, t)

$
≤

2

3
max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
,

by considering a mapping η ∈ H of the form η(s) = ln(1
s
) for s ∈ (0, 1].

Therefore, T is a G0-fuzzy H-quasi-contraction with respect to η ∈ H with the
quasi-contractive constant λ = 2

3
.

Remark 2.8. Suppose that (X, M, !) is a fuzzy metric space endowed with a
graph G and T : X → X is a G-fuzzy H-Banach contraction with respect to
η ∈ H in the sense that T preserves the edges of G and there exists α ∈ (0, 1)
such that

η
#
M(Tx, Ty, t)

$
≤ αη

#
M(x, y, t)

$

for all x, y ∈ X with (x, y) ∈ E(G) and any t > 0. If (x, y) ∈ E(G), then

η
#
M(Tx, Ty, t)

$
≤ αη

#
M(x, y, t)

$

≤ α max
!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
.

Therefore, T satisfies (FQ2) and so T is a G-fuzzy H-quasi-contraction with
respect to η ∈ H. Hence every G-fuzzy H-contraction with respect to η ∈ H is
a G-fuzzy H-quasi-contraction with respect to η ∈ H.

Remark 2.9. Suppose that (X, M, !) is a fuzzy metric space endowed with a
graph G and T : X → X is a G-fuzzy H-Kannan contraction with respect to
η ∈ H in the sense that T preserves the edges of G and there exists α ∈ (0, 1

2
)

such that

η
#
M(Tx, Ty, t)

$
≤ α

#
η
#
M(x, Tx, t)

$
+ η

#
M(y, Ty, t)

$$

for all x, y ∈ X with (x, y) ∈ E(G) and any t > 0 (see [14] for the definition in
metric spaces). If (x, y) ∈ E(G), then

η
#
M(Tx, Ty, t)

$
≤ α

#
η
#
M(x, Tx, t)

$
+ η

#
M(y, Ty, t)

$$

≤ 2α max
!
η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$"

≤ 2α max
!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
.

Therefore, T satisfies (FQ2) and so T is a G-fuzzy H-quasi-contraction with
respect to η ∈ H. Hence every G-fuzzy H-Kannan contraction with respect to
η ∈ H is a G-fuzzy H-quasi-contraction with respect to η ∈ H.

Remark 2.10. Suppose that (X, M, !) is a fuzzy metric space endowed with a
graph G and T : X → X is a G-fuzzy H-Chatterjea contraction with respect to

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 184



Quasi-contractions in fuzzy metric spaces with a graph

η ∈ H in the sense that T preserves the edges of G and there exists α ∈ (0, 1
2
)

such that

η
#
M(Tx, Ty, t)

$
≤ α

#
η
#
M(x, Ty, t)

$
+ η

#
M(y, Tx, t)

$$

for all x, y ∈ X with (x, y) ∈ E(G) and any t > 0 (see [6] for the definition in
metric spaces). If (x, y) ∈ E(G), then an argument similar to that appeared in
Remark 2.9 establishes that

η
#
M(Tx, Ty, t)

$
≤ 2α max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
, η
#
M(y, Ty, t)

$
,

η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
.

Therefore, T satisfies (FQ2) and so T is a G-fuzzy H-quasi-contraction with
respect to η ∈ H. Hence every G-fuzzy H-Chatterjea contraction with respect
to η ∈ H is a G-fuzzy H-quasi-contraction with respect to η ∈ H.

Remark 2.11. Suppose that (X, M, !) is a fuzzy metric space endowed with

a graph G and T : X → X is a G-fuzzy H-Ćirić-Reich-Rus contraction with
respect to η ∈ H in the sense that T preserves the edges of G and there exist
α, β, γ ≥ 0 with α + β + γ < 1 such that

η
#
M(Tx, Ty, t)

$
≤ αη

#
M(x, y, t)

$
+ βη

#
M(x, Tx, t)

$
+ γη

#
M(y, Ty, t)

$

for all x, y ∈ X with (x, y) ∈ E(G) and any t > 0 (see [17] for the definition in
metric spaces). If (x, y) ∈ E(G), then an argument similar to that appeared in
Remark 2.9 establishes that

η
#
M(Tx, Ty, t)

$
≤ (α + β + γ) max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
,

η
#
M(y, Ty, t)

$
, η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
.

Therefore, T satisfies (FQ2) and so T is a G-fuzzy H-quasi-contraction with
respect to η ∈ H. Hence every G-fuzzy H-Ćirić-Reich-Rus contraction with
respect to η ∈ H is a G-fuzzy H-quasi-contraction with respect to η ∈ H.

Remark 2.12. Suppose that (X, M, !) is a fuzzy metric space endowed with
a graph G and T : X → X is a G-fuzzy H-Hardy-Rogers contraction with
respect to η ∈ H in the sense that T preserves the edges of G and there exist
α, β, γ, δ, θ ≥ 0 with α + β + γ + δ + θ < 1 such that

η
#
M(Tx, Ty, t)

$
≤ αη

#
M(x, y, t)

$
+ βη

#
M(x, Tx, t)

$
+ γη

#
M(y, Ty, t)

$

+ δη
#
M(x, Ty, t)

$
+ θη

#
M(y, Tx, t)

$

for all x, y ∈ X with (x, y) ∈ E(G) and any t > 0 (see [12] for the definition in
metric spaces). If (x, y) ∈ E(G), then an argument similar to that appeared in
Remark 2.9 establishes that

η
#
M(Tx, Ty, t)

$
≤ (α + β + γ + δ + θ) max

!
η
#
M(x, y, t)

$
, η
#
M(x, Tx, t)

$
,

η
#
M(y, Ty, t)

$
, η
#
M(x, Ty, t)

$
, η
#
M(y, Tx, t)

$"
.

Therefore, T satisfies (FQ2) and so T is a G-fuzzy H-quasi-contraction with
respect to η ∈ H. Hence every G-fuzzy H-Hardy-Rogers contraction with
respect to η ∈ H is a G-fuzzy H-quasi-contraction with respect to η ∈ H.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 185



M. Dinarvand

Note that every fuzzy H-contractive mapping (1.1) due to Wardowski [20]
is t-uniformly continuous. In the next example, we see that a G-fuzzy H-quasi-
contraction with respect to η ∈ H need not be even continuous.

Example 2.13. Let (R+, d) be the usual (Euclidean) metric space of all non-
negative real numbers and (R+, Md, !) be the standard fuzzy metric space
induced by d. Consider a graph G given by V (G) = R+ and

E(G) = ∆ ∪
!
(x, y) ∈ X × X : x, y ∈ Q ∩ R+ with x ≤ y

"
,

where Q is the set of all rational numbers. Define a mapping T : R+ → R+ by
the rule

Tx =

)
x
2
, x ∈ R+ ∩ Q,

0, otherwise.

Then T is a G-fuzzy H-quasi-contraction with respect to η ∈ H with the
quasi-contractive constant λ = 1

2
. Obviously, T is only continuous at zero. In

particular, T is not continuous on the whole R+.

The following proposition is an immediate consequence of the definition of G-
fuzzy H-quasi-contractions with respect to η ∈ H and gives a simple procedure
to construct new G-fuzzy H-quasi-contractions with respect to η ∈ H from
older ones.

Proposition 2.14. Let (X, M, !) be a fuzzy metric space endowed with a graph
G and T : X → X be a mapping.

(1) If T preserves the edges of G, then T preserves the edges of G−1 (resp.
&G);

(2) If T satisfies (FQ2) for the graph G, then T satisfies (FQ2) for the

graph G−1 (resp. &G);
(3) If T is a G-fuzzy H-quasi-contraction with respect to η ∈ H with a

quasi-contractive constant λ ∈ (0, 1), then T is a G−1-fuzzy H-quasi-
contraction (resp. &G-fuzzy H-quasi-contraction) with respect to η ∈ H
with a quasi-contractive constant λ.

To prove the existence of a fixed point for a G-fuzzy H-quasi-contraction
with respect to η ∈ H, we make use of the following useful lemmas.

Lemma 2.15. Let (X, M, !) be a fuzzy metric space endowed with a graph G
and T : X → X be a G-fuzzy H-quasi-contraction with respect to η ∈ H with a
quasi-contractive constant λ ∈ (0, 1). Then

η
#
M(T ix, T jx, t)

$
≤ λδt

#
O(x; n)

$
i, j = 1, . . . , n

for all x ∈ CT and each t > 0 and any n ∈ N.

Proof. Suppose that x ∈ CT and n ∈ N be given. If i and j are arbitrary
positive integers no more than n, then (T i−1x, T j−1x) ∈ E( &G). According to
Proposition 2.14, T is also a &G-fuzzy H-quasi-contraction with respect to η ∈ H

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 186



Quasi-contractions in fuzzy metric spaces with a graph

with a quasi-contractive constant λ ∈ (0, 1). In particular, T satisfies (FQ2)
for the graph &G. Hence, we have

η
#
M(T ix, T jx, t)

$
= η

#
M(TT i−1x, TT j−1x, t)

$

≤ λ max
!
η
#
M(T i−1x, T j−1x, t)

$
, η
#
M(T i−1x, T ix, t)

$
,

η
#
M(T j−1x, T jx, t)

$
, η
#
M(T i−1x, T jx, t)

$
,

η
#
M(T j−1x, T ix, t)

$"

≤ λδt
#
O(x; n)

$

for all t > 0. □

Example 2.16. Consider the set R of real numbers with the usual (Euclidean)
metric and the standard fuzzy metric space (R, M, !). Let G0 be a the complete
graph and define a mapping T : R → R as Tx = x

2
for all x ∈ R. Then T

is a G0-quasi-contraction with respect to η ∈ H with respect to η ∈ H with
a quasi-contractive constant λ = 1

2
. Meanwhile, T nx = x

2n
and δt(O(x; n)) =

|x|
t
(1 − 1

2n
) for all x ∈ R and all n ∈ N ∪ {0}. Now, let x0 be a positive real

numbers. Take a mapping η ∈ H of the form η(t) = 1
t
− 1 for t ∈ (0, 1] and put

n = 2, i = 0 and j = 1 in Lemma 2.14. Hence, we have

η
#
M(x, Tx, t)

$
=

|x0 − Tx0|
t

=
x0
2t

>
x0
2t

*
1 −

1

22

+
= λδt

#
O(x0; 2)

$
.

Lemma 2.17. Let (X, M, !) be a fuzzy metric space endowed with a graph G
and T : X → X be a G-fuzzy H-quasi-contraction with respect to η ∈ H. Then
for all x ∈ CT and each n ∈ N, there exists a positive integer k no more than
n such that

δt
#
O(x; n)

$
= η

#
M(x, T kx, t)

$

for all t > 0.

Proof. Suppose that x ∈ CT and n ∈ N be given. On the one hand, if
δt
#
O(x; n)

$
= 0, then O(x; n) is singleton. In particular, x is a fixed point

for T and η
#
M(T ix, T jx, t)

$
= 0 for all i, j = 0, 1, . . . , n and each t > 0.

Hence, the assertion holds trivially for any positive integer k no more than n.
On the other hand, if since O(x; n) is a finite set, it follows that there exist

distinct nonnegative integers i and j no more that n such that δt
#
O(x; n)

$
=

η
#
M(T ix, T jx, t)

$
for all t > 0. If both the integers i and j are assumed to be

positive, then from Lemma 2.15, we have

δt
#
O(x; n)

$
= η

#
M(T ix, T jx, t)

$
≤ λδt

#
O(x; n)

$

for all t > 0, where λ ∈ (0, 1) is a quasi-contraction constant of T , which is a
contradiction. Therefore, either i or j must be zero. □

Remark 2.18. Combining Lemmas 2.15 and 2.17, one can easily obtain that if
(X, M, !) is a fuzzy metric space endowed with a graph G and T : X → X be
a G-fuzzy H-quasi-contraction with respect to η ∈ H with a quasi-contraction

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 187



M. Dinarvand

constant λ, then for all x ∈ CT and each n ∈ N, there exists a positive integer
k no more than n such that

η
#
M(T ix, T jx, t)

$
≤ λδt

#
O(x; n)

$
= λη

#
M(x, T kx, t)

$
i, j = 1, . . . , n

for all t > 0.

Lemma 2.19. Let (X, M, !) be a fuzzy metric space endowed with a graph G
and T : X → X be a G-fuzzy H-quasi-contraction with respect to η ∈ H such
that

(i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈
!
M(T ix, T jx, t) :

x ∈ X, t > 0, i, j ∈ N
"
;

(ii)
!
η(M(x, Tx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then the sequence {T nx} is Cauchy for all x ∈ CT .

Proof. Suppose that x ∈ CT and n ∈ N ∪ {0} be given. If n = 0, then there
remains nothing to prove since δt

#
O(x; 0)

$
= 0. Otherwise, from Lemma 2.17,

there exists a positive integer k no more than n such that

δt
#
O(x; n)

$
= η

#
M(x, T kx, t)

$
for all t > 0.(2.3)

Now, we can choose a strictly decreasing sequence of positive numbers {ai}
with

,∞
i=1 ai = 1 and so thanks to (i) and (2.3), we obtain

δt
#
O(x; n)

$
= η

#
M(x, T kx, t)

$

= η
#
M(x, T kx,

∞-

i=1

ait)
$

≤ η
#
M(x, Tx,

∞-

i=j+1

ait)
$
+ η

#
M(Tx, T kx,

j-

i=1

ait)
$

for all t > 0 and any j. Hence, by substituting i = 1 and j = k in Lemma 2.15,
we get

δt
#
O(x; n)

$
≤ lim sup

j→∞
η
#
M(x, Tx,

∞-

i=j+1

ait)
$
+ η

#
M(Tx, T kx, t)

$

≤ lim sup
j→∞

η
#
M(x, Tx,

∞-

i=j+1

ait)
$
+ λδt

#
O(x; n)

$
,

from which it follows that

δt
#
O(x; n)

$
≤

1

1 − λ
lim sup
j→∞

η
#
M(x, Tx,

∞-

i=j+1

ait)
$
.(2.4)

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 188



Quasi-contractions in fuzzy metric spaces with a graph

If m, n ∈ N with m ≥ n ≥ 2, since T n−1x ∈ CT , then by putting i = m − n + 1
and j = 1 in Lemma 2.15, we obtain

η
#
M(T mx, T nx, t)

$
= η

#
M(T m−n+1T n−1x, TT n−1x, t)

$

≤ λδt
#
O(T n−1x; m − n + 1)

$
,(2.5)

for all t > 0, where λ ∈ (0, 1) is a quasi-contractive constant of T . Moreover,
due to (2.3), there exists a positive integer k no more than m − n + 1 such that

δt
#
O(T n−1x; m − n + 1)

$
= η

#
M(T n−1x, T k+n−1x, t)

$
(2.6)

for all t > 0. Because n ≥ 2, it follows that T n−2x ∈ CT and so putting i = 1
and j = k + 1 in Lemma 2.15, we obtain

η
#
M(T n−1x, T k+n−1x, t)

$
= η

#
M(TT n−2x, T k+1T n−2x, t)

$

≤ λδt
#
O(T n−2x; m − n + 2)

$
(2.7)

for all t > 0. Combining (2.5), (2.6) and (2.7) together with (2.4) and using
induction, we get

η
#
M(T mx, T nx, t)

$
≤ λδt

#
O(T n−1x; m − n + 1)

$

= λη
#
M(T n−1x, T k+n−1x, t)

$

≤ λ2δt
#
O(T n−2x; m − n + 2)

$

...

≤ λnδt
#
O(x; m)

$

≤
λn

1 − λ
lim sup
j→∞

η
#
M(x, Tx,

∞-

i=j+1

ait)
$
,

which implies from (ii) that

lim
m,n→∞

η
#
M(T mx, T nx, t)

$
= 0.

Hence, limm,n→∞ M(T
nx, T mx, t) = 1. This means that {T nx} is a Cauchy

sequence. □

Following Petruşel and Rus [16], we introduce the concept of a Picard and
weakly Picard operator in fuzzy metric spaces as follows.

Definition 2.20. Let (X, M, !) be a fuzzy metric space and T : X → X be a
mapping.

(i) T is called a Picard operator if T has a unique fixed point x∗ ∈ X and
limn→∞ M(T

nx, x∗, t) = 1 for all x ∈ X and each t > 0.
(ii) T is called a weakly Picard operator if {T nx} is a convergent sequence

and its limit (which depends on x) is a fixed point of T for all x ∈ X.

It is clear that each Picard operator is weakly Picard operator but the iden-
tity mapping of any fuzzy metric space with more than one point shows that
the converse is not generally true. In fact, the set of fixed points of a weakly

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 189



M. Dinarvand

Picard operator can have any arbitrary cardinality. Nevertheless, one can eas-
ily see that a weakly Picard operator is Picard one if and only if it has a unique
fixed point.

Motivated by Jachymski [13], we define a weaker type of continuity of self-
maps in fuzzy metric spaces endowed with a graph as follows.

Definition 2.21. Let (X, M, !) be a fuzzy metric space endowed with a graph
G and T : X → X be a mapping. We say that T is orbitally G-continuous on
X if limn→∞ M

#
T anx, y, t

$
= 1 implies limn→∞ M

#
T(T anx), Ty, t

$
= 1 for all

x, y ∈ X and each t > 0 and all sequences {an} of positive integers such that
(T anx, T an+1x) ∈ E(G) for all n ∈ N.

It is clear that a continuous mapping on a fuzzy metric space is orbitally
G-continuous for all graphs G but the converse is not true in general as the
following example shows.

Example 2.22. Let (R+, d) be the usual (Euclidean) metric space of all non-
negative real numbers and (R+, Md, !) be the standard fuzzy metric space
induced by d. Consider a mapping T : R+ → R+ defined by the rule

Tx =

)
x
3
, x ∕= 0,

1, x = 0.

Then it is clear that T is not continuous at x = 0 and in particular, T is not
continuous on the whole R+. Now, suppose that R+ is endowed with a graph
G = (V (G), E(G)), where V (G) = R+ and E(G) = {(x, x) : x ∈ R+}, i.e.
E(G) contains all loops. If x, y ∈ R+ and {an} is a sequence of positive integers
with (T anx, T an+1x) ∈ E(G) for all n ∈ N such that limn→∞ M

#
T anx, y, t

$
= 1

for any t > 0, then {T anx} is necessarily a constant sequence. Hence, T anx = y
for all n ∈ N and so limn→∞ M

#
T(T anx), Ty, t

$
= 1 for each t > 0. Therefore,

T is orbitally G-continuous on R+.

Now, we are ready to prove our main theorem on the existence of a fixed point
for a G-fuzzy H-quasi-contraction in the setup of M-complete fuzzy metric
spaces endowed with a graph.

Theorem 2.23. Let (X, M, !) be an M-complete fuzzy metric space endowed
with a graph G and T : X → X be a G-fuzzy H-quasi-contraction with respect
to η ∈ H such that

(i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈
!
M(T ix, T jx, t) :

x ∈ X, t > 0, i, j ∈ N
"
;

(ii)
!
η(M(x, Tx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then the restriction of T to CT is a weakly Picard operator if either T is

orbitally &G-continuous on X or G is a (&C)-graph.
In particular, whenever either T is orbitally &G-continuous on X or G is a

(&C)-graph, T has a fixed point in X if and only if CT ∕= ∅.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 190



Quasi-contractions in fuzzy metric spaces with a graph

Proof. If CT = ∅, then there remains nothing to prove. So assume that CT is
nonempty. Note that since T preserves the edges of &G, it follows immediately
that CT is T-invariant, i.e. T maps CT into itself.

Now, suppose that x ∈ CT is given. By virtue of Lemma 2.19, {T nx} is a
Cauchy sequence. As (X, M, !) is M-complete, there exists x∗ ∈ X (depending
on x) such that limn→∞ M(T

nx, x∗, t) = 1 for all t > 0. We shall show that x∗

is a fixed point for T .

To this end, note that from x ∈ CT , we have (T nx, T n+1x) ∈ E( &G) for all
n ∈ N ∪ {0}. On the one hand, if T is orbitally &G-continuous on X, then
limn→∞ M(T

nx, x∗, t) = 1 for all t > 0 implies limn→∞ M(T
n+1x, Tx∗, t) =

limn→∞ M(T(T
nx), Tx∗, t) = 1 for all t > 0. By the uniqueness of the limit,

we get M(x∗, Tx∗, t) = 1 for all t > 0, i.e. Tx∗ = x∗.

On the other hand, if G is a (&C)-graph, then there exists a strictly increasing
sequence {nk} of positive integers such that (T nkx, x∗) ∈ E( &G) for all k ∈ N.
Due to Proposition 2.14, if λ ∈ (0, 1) is a quasi-contractive constant of T ,
then T is a &G-fuzzy H-quasi-contraction with respect to η ∈ H with a quasi-
contractive constant λ. Let t > 0 is given. Hence, for all " > 0 and k ∈ N, we
have

M(Tx∗, x∗, t + ") ≥ M(Tx∗, T nk+1x, ") ! M(T nk+1x, x∗, t)

which together with (i) and (FQ2) yields

η
#
M(x∗, Tx∗, t + ")

$
≤ η

#
M(x∗, T nk+1x, ")

$
+ η

#
M(Tx∗, T nk+1x, t)

$

≤ η
#
M(Tx∗, T nk+1x, t)

$
+ λ max

!
η
#
M(x∗, T nkx, t)

$
,

η
#
M(x∗, Tx∗, t)

$
, η
#
M(T nkx∗, T nk+1x∗, t)

$
,

η
#
M(x∗, T nk+1x∗, t)

$
, η
#
M(T nkx, Tx∗, t)

$"
.

On taking the limit as k → ∞ in the above inequality, we obtain

η
#
M(x∗, Tx∗, t + ")

$
≤ λη

#
M(x∗, Tx∗, t)

$
,

which implies that

η
#
M(x∗, Tx∗, t)

$
= lim

!→0+
η
#
M(x∗, Tx∗, t + ")

$
≤ λη

#
M(x∗, Tx∗, t)

$
.

As λ ∈ (0, 1), it then follows that η
#
M(x∗, Tx∗, t)

$
= 0 for all t > 0. Thus

M(x∗, Tx∗, t) = 1 for all t > 0 or equivalently, Tx∗ = x∗.
Finally, since CT contains all fixed points of T , it follows that x

∗ ∈ CT .
Consequently, the restriction of T

..
CT

: CT → CT is a weakly Picard operator.
□

By putting G = G0 in Theorem 2.23, we obtain the following generalization
of Ćirić’s fixed point theorem [7] on M-complete fuzzy metric spaces in the
sense of George and Veeramani.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 191



M. Dinarvand

Corollary 2.24. Let (X, M, !) be an M-complete fuzzy metric space and T :
X → X be a fuzzy H-quasi-contraction with respect to η ∈ H such that

(i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈
!
M(fix, fjx, t) :

x ∈ X, t > 0, i, j ∈ N
"
;

(ii)
!
η(M(x, fx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then T is a Picard operator.

Proof. The set CT is nonempty because CT = X. Therefore, due to Theorem
2.23, the mapping T = T

..
CT

is a weakly Picard operator. In particular, T has

a fixed point in X. To see that T is a Picard operator, it sufficies to show that
T has a unique fixed point in X. To this end, suppose that x∗ and y∗ are two
fixed points for T in X. Thus, from (2.1), we have

η
#
M(x∗, y∗, t)

$
= η

#
M(Tx∗, Ty∗, t)

$

≤ λ max
!
η
#
M(x∗, y∗, t)

$
, η
#
M(x∗, Tx∗, t)

$
, η
#
M(y∗, Ty∗, t)

$
,

η
#
M(x∗, Ty∗, t)

$
, η
#
M(y∗, Tx∗, t)

$"

= λη
#
M(x∗, y∗, t)

$
,

for all t > 0, where λ ∈ (0, 1) is the quasi-contractive constant. Therefore,
η
#
M(x∗, y∗, t)

$
= 0 which by our assumptions about η ∈ H implies that

M(x∗, y∗, t) = 1 for all t > 0. Hence, x∗ = y∗. □

Remark 2.25. By a subtle look at the proof of Corollary 2.24 and use an ar-
gument similar to that appeared there, we see that both the ends of any link
of G can not be fixed points for a G-fuzzy H-quasi-contraction with respect
to η ∈ H, i.e. if x ∕= y, Tx = x and Ty = y, then (x, y) ∕∈ E(G). Roughly
speaking, no G-fuzzy H-quasi-contraction with respect to η ∈ H can keep both
the ends of a link of G fixed. In particular,

• if G = G0, then no fuzzy H-quasi-contraction with respect to η ∈ H
can have two distinct fixed points;

• if ≼ is a partial order on X, then neither a G1-fuzzy H-quasi-contraction
with respect to η ∈ H nor a G2-fuzzy H-quasi-contraction with respect
to η ∈ H can have two distinct fixed points which are comparable
elements of (X, ≼).

By taking G = G1 or G = G2 in Theorem 2.23, we obtain the ordered
version of Ćirić’s fixed point theorem on ordered fuzzy H-quasi-contractions
with respect to η ∈ H in M-complete fuzzy metric spaces equipped with a
partial order as follows.

Corollary 2.26. Let (X, ≼) be a partially ordered set and (X, M, !) be an M-
complete fuzzy metric space. Suppose that T : X → X be a mapping which
maps comparable elements of (X, ≼) onto comparable elements and satisfies
(2.2) such that for η ∈ H,

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 192



Quasi-contractions in fuzzy metric spaces with a graph

(i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈
!
M(fix, fjx, t) :

x ∈ X, t > 0, i, j ∈ N
"
;

(ii)
!
η(M(x, fx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then the restriction of T to the set of all points x ∈ X whose every two iterates
under T are comparable elements of (X, ≼) is a weakly Picard operator if either
T is orbitally G2-continuous on X or the quadruple (X, M, !, ≼) satisfies (∗).

In particular, whenever either T is orbitally G2-continuous on X or the
quadruple (X, M, !, ≼) satisfies (∗), T has a fixed point in X if and only if
there exists x ∈ X such that T mx and T nx are comparable elements of (X, ≼)
for all m, n ∈ N ∪ {0}.

Because G-fuzzy H-Banach contractions with respect to η ∈ H, G-fuzzy H-
Kannan contractions with respect to η ∈ H, G-fuzzy H-Chatterjea contractions
with respect to η ∈ H, G-fuzzy H-Ćirić-Reich-Rus contractions with respect to
η ∈ H and G-fuzzy H-Hardy-Rogers contractions with respect to η ∈ H are all
a G-fuzzy H-quasi-contraction with respect to η ∈ H, we have also the following
fixed point theorem for these contractions as a consequence of Theorem 2.23.

Corollary 2.27. Let (X, M, !) be an M-complete fuzzy metric space endowed
with a graph G and T : X → X be a G-fuzzy H-Banach contraction (a G-
fuzzy H-Kannan contraction, a G-fuzzy H-Chatterjea contraction, a G-fuzzy
H-Ćirić-Reich-Rus contraction, or a G-fuzzy H-Hardy-Rogers contraction) with
respect to η ∈ H such that

(i) τ ≥ r ! s implies η(τ) ≤ η(r) + η(s) for all r, s, τ ∈
!
M(fix, fjx, t) :

x ∈ X, t > 0, i, j ∈ N
"
;

(ii)
!
η(M(x, fx, ti)) : i ∈ N

"
is bounded for all x ∈ X and any sequence

{tn} ⊆ (0, ∞), tn ↓ 0.
Then the restriction of T to CT is a weakly Picard operator if either T is

orbitally &G-continuous on X or G is a (&C)-graph.
In particular, whenever either T is orbitally &G-continuous on X or G is a

(&C)-graph, T has a fixed point in X if and only if CT ∕= ∅.

Remark 2.28. By comparing Corollary 2.27 as a version of Theorem 2.23 for
several types of contractions with some recent results in graph metric fixed point
theory, one can see easily that our results can be viewed as the improvement
and generalization of corresponding results in [3, 4, 6, 12, 13, 14, 18, 17] and
several other comparable results.

3. Conclusion

Despite noted improvements in computer skill and its remarkable success in
facilitating many areas of research, computers are not designed to handle situ-
ations wherein uncertainties are involved. Fuzzy set theory has provided many
important tools in mathematics and related disciplines to resolve the issues of

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 193



M. Dinarvand

uncertainty and ambiguity. In the present work, we investigated sufficient con-
ditions which guarantee the existence of a fixed point for a new notion called
G-fuzzy H-quasi-contraction using directed graphs in the setting of fuzzy metric
spaces endowed with a graph. A large number of different types of contractive
mappings formulated using directed graphs satisfy the presented contractive
condition and our main result is a natural generalization of [2, Definition 2.3]
from fuzzy metric spaces to fuzzy metric spaces with a graph and enriches our
knowledge of fixed points in such spaces. As a new work, it will be interesting
to study common fixed point results for two or more than two mappings on
fuzzy metric spaces endowed with a graph G by considering the function η ∈ H.

References

[1] S. M. A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Some fixed point results on a
metric space with a graph, Topology Appl. 159, no. 3 (2012), 659–663.

[2] A. Amini-Harandi and D. Miheţ, Quasi-contractive mappings in fuzzy metric spaces,
Iranian J. Fuzzy Syst. 12, no. 4 (2015), 147–153.

[3] F. Bojor, Fixed points of Kannan mappings in metric spaces endowed with a graph, An.
Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 20, no. 1 (2012), 31–40.

[4] F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a
graph, Nonlinear Anal. 75 (2012), 3895–3901.

[5] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier
Publishing Co., Inc., New York, 1976.

[6] S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727–730.

[7] Lj. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.
45, no. 2 (1974), 267–273.

[8] M. Dinarvand, Fixed point results for (ϕ, ψ)-contractions in metric spaces endowed with
a graph, Mat. Vesn. 69, no. 1 (2017), 23–38.

[9] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1988), 385–389.
[10] V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets

Syst. 125 (2002), 245–252.
[11] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst.

64 (1994), 395–399.
[12] G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich,

Canadian Math. Bull. 16 (1973), 201–206.
[13] J. Jachymski, The contraction principle for mappings on a metric space with a graph,

Proc. Amer. Math. Soc. 136, no. 4 (2008), 1359–1373.
[14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76.
[15] I. Kramosil and J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetica

11, no. 5 (1975), 336–344.
[16] A. Petruşel and I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math.

Soc. 134, no. 2 (2006), 411–418.
[17] S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972), 26–42.
[18] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans.

Amer. Math. Soc. 226 (1977), 257–290.
[19] S. Shukla, Fixed point theorems of G-fuzzy contractions in fuzzy metric spaces endowed

with a graph, Commun. Math. 22 (2014), 1–12.
[20] D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces,

Fuzzy Sets Syst. 222 (2013), 108–114.
[21] L. A. Zadeh, Fuzzy Sets, Inform. Control, 10, no. 1 (1960), 385–389.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 194