@ Appl. Gen. Topol. 22, no. 2 (2021), 311-319doi:10.4995/agt.2021.14449
© AGT, UPV, 2021

The periodic points of ε-contractive maps in

fuzzy metric spaces

Taixiang Sun a, Caihong Han a,∗, Guangwang Su a, Bin Qin b and

Lue Li a

a
College of Information and Statistics, Guangxi University of Finance and Economics, Nanning,

530003, China (stx1963@163.com,h198204c@163.com, s1g6w3@163.com, li1982lue@163.com)
b

Guangxi (ASEAN) Research Center of Finance and Economics Nanning, 530003, China

(q3009b@163.com)

Communicated by H. Dutta

Abstract

In this paper, we introduce the notion of ε-contractive maps in fuzzy

metric space (X, M, ∗) and study the periodicities of ε-contractive
maps. We show that if (X, M, ∗) is compact and f : X −→ X is
ε-contractive, then P(f) = ∩∞

n=1f
n(X) and each connected component

of X contains at most one periodic point of f, where P(f) is the set of
periodic points of f. Furthermore, we present two examples to illustrate

the applicability of the obtained results.

2010 MSC: 54E35; 54H25.

Keywords: fuzzy metric space; ε-contractive map; periodic point.

1. Introduction

The notion of fuzzy metric spaces was introduced by Kramosil and Michalek
[10] and later was modified by George and Veeramani [3] in order to obtain a
Hausdorff topology in a fuzzy metric space. Recently there has been a great
interest in discussing some properties on discrete dynamical systems in fuzzy

Project supported by NNSF of China (11761011) and NSF of Guangxi (2020GXNS-
FAA297010) and PYMRBAP for Guangxi CU(2021KY0651).

∗Corresponding author.

Received 07 October 2020 – Accepted 25 April 2021

http://dx.doi.org/10.4995/agt.2021.14449


T. Sun, C. Han, G. Su, B. Qin and L. Li

metric spaces. Many authors introduced and investigated the different types
of fuzzy contractive maps and obtained a lot of fixed point theorems (see [1,
2, 5, 6, 8, 11, 12, 14, 15, 16, 17, 18, 19]). Until now, there are very little of
works that investigates the periodic points of discrete dynamical systems in
fuzzy metric spaces because it is much more difficult to find various conditions
to obtain the periodic points of discrete dynamical systems.

In the present paper, we introduce the notion of ε-contractive map in fuzzy
metric space (X, M, ∗) and obtain the following results: (1) If (X, M, ∗) is
compact and f : X −→ X is ε-contractive, then P(f) = ∩∞n=1f

n(X), where
P(f) is the set of periodic points of f. (2) If (X, M, ∗) is compact and f :
X −→ X is ε-contractive, then each connected component of X contains at
most one periodic point of f. Furthermore, if (X, M, ∗) is also connected, then
f has at most one fixed point.

2. Preliminaries

Throughout the paper, let N be the set of all positive integers. Firstly, we
recall the basic definitions and the properties about fuzzy metric spaces.

Definition 2.1 (Schweizer and Sklar [13]). A binary operation T : [0, 1]2 −→
[0, 1] is a continuous t-norm if it satisfies the following conditions (i)-(v):

(i) T (a, b) = T (b, a);
(ii) T (a, b) ≤ T (c, d) for a ≤ c and b ≤ d;
(iii) T (T (a, b), c) = T (a, T (b, c));
(iv) T (a, 0) = 0 and T (a, 1) = a;
(v) T is continuous on [0, 1]2,

where a, b, c, d ∈ [0, 1].

For any a, b ∈ [0, 1], we will use the notation a∗b instead of T (a, b). T (a, b) =
min{a, b}, T (a, b) = ab and T (a, b) = max{a+ b− 1, 0} are the most commonly
used t-norms.

In the present paper, we also use the following definition of a fuzzy metric
space.

Definition 2.2 (George and Veeramani [3]). A triple (X, M, ∗) is called a
fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is
a map defined on X2 × (0, +∞) into [0, 1] satisfying the following conditions
(i)-(v) for any x, y, z ∈ X and s, t ∈ (0, +∞):

(i) M(x, y, t) > 0;
(ii) M(x, y, t) = 1 (for any t > 0) ⇐⇒ x = y;
(iii) M(x, y, t) = M(y, x, t);
(iv) M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s);
(v) Mxy : (0, +∞) −→ [0, 1] is a continuous ( where Mxy(t) = M(x, y, t)).

Remark 2.3 (Grabiec [4]). Mxy is non-decreasing for all x, y ∈ X.

If (X, M, ∗) is a fuzzy metric space, then (M, ∗), or simply M, is called a
fuzzy metric on X. In [3], George and Veeramani showed that every fuzzy

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 312



The periodic points of contractive maps

metric M on X generates a topology τM on X which has as a base the fam-
ily of open sets of the form {BM(x, ε, t) : x ∈ X, ε ∈ (0, 1), t > 0} , where
BM(x, ε, t) = {y ∈ X : M(x, y, t) > 1 − ε} for any x ∈ X, ε ∈ (0, 1) and t > 0.

Definition 2.4 (Gregori and Sapena [7]). Let (X, M, ∗) be a fuzzy metric
space. A sequence of points xn ∈ X is called to converge to x ( denoted
by xn −→ x) ⇐⇒ limn−→+∞ M(xn, x, t) = 1 (for any t > 0), i.e. for each
δ ∈ (0, 1) and t > 0, there exists N ∈ N such that M(xn, x, t) > 1 − δ for all
n ≥ N.

Definition 2.5. (1) A fuzzy metric space (X, M, ∗) is said to be compact if
each sequence of points in X has a convergent subsequence. A subset A of X
is said to be compact if A as a fuzzy metric subspace is compact.

(2) A fuzzy metric space (X, M, ∗) is said to be connected if there exist two
nonempty closed sets U, V of X with U ∩V = ∅ such that X = U ∪V . A subset
A of X is said to be connected if A as a fuzzy metric subspace is connected.

Definition 2.6. Let (X, M, ∗) be a fuzzy metric space and ε ∈ (0, 1). A map
f : X −→ X is said to be ε-contractive if M(x, y, t) > 1 − ε for any x, y ∈ X
with x 6= y and t > 0, then M(f(x), f(y), t) > M(x, y, t).

Denote by C(X, ε) the set of all ε-contractive maps in X.

Remark 2.7.

(1) Let (X, M, ∗) be compact and A be a subset of X. Then A is compact
⇐⇒ A is closed (see [9]).

(2) If f ∈ C(X, ε), then fn ∈ C(X, ε) for any n ∈ N.
(3) If A is a connected component of X, then A is closed.
(4) If f ∈ C(X, ε) and A is a closed subset of X, then f−1(A) is also closed.

Indeed, let a sequence of points xn ∈ f
−1(A) with xn −→ x. Then

there exists N ∈ N such that 1 ≥ M(f(xn), f(x), t) ≥ M(xn, x, t) >
1 − ε for any n ≥ N, which implies 1 ≥ limn−→∞ M(f(xn), f(x), t) ≥
limn−→∞ M(xn, x, t) = 1 and f(xn) −→ f(x). Since f(xn) ∈ A and
A is closed, we see f(x) ∈ A. Thus x ∈ f−1(A) , which implies that
f−1(A) is closed.

(5) By (4) we see that if f ∈ C(X, ε) and A is a connected subset of X,
then f(A) is also connected.

Let (X, M, ∗) be a fuzzy metric space and f : X −→ X. Write f0(x) = x
and fn = f ◦ fn−1 for any x ∈ X and n ∈ N. We write

P(f) = {x : there exists some n ∈ N such that fn(x) = x}.

For any x ∈ X, we write

ω(x, f) = {y : there exists a sequence of positive integers n1 < n2 <

· · · such that lim
k−→∞

M(fnk(x), y, t) = 1 for any t > 0}.

P(f) is called the set of periodic points of f. ω(x, f) is called the set of ω-limit
points of x under f. If f(x) = x, then x is called the fixed point of f.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 313



T. Sun, C. Han, G. Su, B. Qin and L. Li

3. Main results

In this section, we study the set of periodic points of ε-contractive maps in
fuzzy metric spaces.

Lemma 3.1. Let (X, M, ∗) be a compact fuzzy metric space and f ∈ C(X, ε).
Then f(ω(x, f)) = ω(x, f) for any x ∈ X and ω(x, f) = ω(y, f) for any y ∈
ω(x, f).

Proof. Let u ∈ ω(x, f). Then there exists a sequence of positive integers n1 <
n2 < · · · < nk < · · · such that limk−→∞ M(f

nk(x), u, t) = 1 for any t > 0. Thus
for any 0 < δ < ε and t > 0, there exists N ∈ N such that M(fnk(x), u, t) >
1 − δ for all n ≥ N. From which it follows that

M(fnk+1(x), f(u), t) ≥ M(fnk(x), u, t) > 1 − δ.

Therefore f(u) ∈ ω(x, f) and f(ω(x, f)) ⊂ ω(x, f).
On the other hand, by taking subsequence we may assume that fnk−1(x) −→

v for some v ∈ ω(x, f) since (X, M, ∗) is a compact. Then for any 0 < δ < ε
and t > 0, there exist 0 < δ1 < δ and N ∈ N such that (1−δ1)∗(1−δ1) > 1−δ
and M(fnk(x), u, t/2) > 1 − δ1 and M(f

nk−1(x), v, t/2) > 1 − δ1 for all n ≥ N.
Note that f ∈ C(X, ε), we see that when n ≥ N, one has

M(u, f(v), t) ≥ M(u, fnk(x),
t

2
) ∗ M(fnk(x), f(v),

t

2
)

≥ M(u, fnk(x),
t

2
) ∗ M(fnk−1(x), v,

t

2
)

≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ.

Thus we obtain M(u, f(v), t) = 1 for any t > 0 and u = f(v) ∈ f(ω(x, f)),
which implies ω(x, f) ⊂ f(ω(x, f)).

Now we show that ω(y, f) ⊂ ω(x, f) for any y ∈ ω(x, f). Let z ∈ ω(y, f).
Then there exist two sequences of positive integers k1 < k2 < · · · and r1 <
r2 < · · · such that f

kn(x) −→ y and frn(y) −→ z. Thus for any 0 < δ < ε and
t > 0, there exist 0 < δ1 < δ and N ∈ N such that (1 − δ1) ∗ (1 − δ1) > 1 − δ
and M(y, fkn(x), t/2) > 1 − δ1 and M(z, f

rn(y), t/2) > 1 − δ1 for any n > N.
Note that f ∈ C(X, ε), we see that when n > N, one has

M(z, fkn+rn(x), t) ≥ M(z, frn(y),
t

2
) ∗ M(frn(y), fkn+rn(x),

t

2
)

≥ M(z, frn(y),
t

2
) ∗ M(y, fkn(x),

t

2
)

≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ.

Therefore we obtain z ∈ ω(x, f), which implies ω(y, f) ⊂ ω(x, f).
Finally we show that ω(y, f) = ω(x, f) for any y ∈ ω(x, f). Let z ∈ ω(x, f).

Then there exist two sequences of positive integers k1 < k2 < · · · and r1 <
r2 < · · · such that f

kn(x) −→ y and frn(x) −→ z with rn − kn > n. Thus
for any 0 < δ < ε and t > 0, there exist 0 < δ1 < δ and N ∈ N such that
(1−δ1)∗(1−δ1) > 1−δ and M(y, f

kn(x), t/2) > 1−δ1 and M(z, f
rn(x), t/2) >

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 314



The periodic points of contractive maps

1 − δ1 for any n > N. Note that f ∈ C(X, ε), we see that when n > N, one
has

M(z, frn−kn(y), t) ≥ M(z, frn(x),
t

2
) ∗ M(frn(x), frn−kn(y),

t

2
)

≥ M(z, frn(x),
t

2
) ∗ M(y, fkn(x),

t

2
)

≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ.

Therefore we obtain z ∈ ω(y, f), which implies ω(y, f) = ω(x, f). The proof is
completed. �

Theorem 3.2. Let (X, M, ∗) be a compact fuzzy metric space and f ∈ C(X, ε).
Then P(f) = ∩∞n=1f

n(X).

Proof. Let Y = ∩∞n=1f
n(X). Then f(Y ) = Y . It is easy to see that P(f) ⊂ Y .

In the following we show that Y ⊂ P(f).
Let x0 ∈ Y . Then there exists a sequence of points xn ∈ Y such that

f(xn) = xn−1 for every n ∈ N. Since X is compact, there exist a subsequence
{xnk}

∞
k=1 of {xn}

∞
n=1 and a point u ∈ Y satisfying xnk −→ u . Thus, for any

0 < δ < ε and t > 0, there exits N ∈ N such that M(xnk , u, t) ≥ 1 − δ for all
k ≥ N. Note that f ∈ C(X, ε), we have

M(x0, f
nk(u), t) = M(fnk(xnk ), f

nk (u), t) > · · · > M(xnk , u, t) ≥ 1 − δ

for all k ≥ N. Therefore x0 ∈ ω(u, f), which with Lemma 3.1 implies x0 ∈
ω(u, f) = ω(x0, f).

By x0 ∈ ω(x0, f) we see that there exist 1 < n1 < n2 < · · · < nk < · · · such
that limk−→∞ M(f

nk(x0), x0, t) = 1. Choose nr with M(f
nr(x0), x0, t) > 1−ε.

In the following we show fnr(x0) = x0, which implies x0 ∈ P(f). In-
deed, if fnr(x0) 6= x0, then M(f

nr(x0), x0, t0) < 1 for some t0 > 0. Since
M(fnr(x0), x0, t) is continuous and non-decreasing in (0, +∞) and f ∈ C(X, ε),
we see that there exists γ > 0 such that M(fnr(x0), x0, t0) ≤ M(f

nr (x0), x0, t0+
γ) < M(fnr+1(x0), f(x0), t0). Thus

M(fnr(x0), x0, t0 + γ)

≥ M(fnr(x0), f
nk+nr (x0),

γ

2
) ∗ M(fnk+nr (x0), f

nk(x0), t0) ∗ M(f
nk(x0), x0,

γ

2
)

≥ M(x0, f
nk(x0),

γ

2
) ∗ M(f1+nr(x0), f(x0), t0) ∗ M(f

nk(x0), x0,
γ

2
).

Taking the limit on both sides in the above as k −→ ∞ we obtain

M(fnr(x0), x0, t0+γ) ≥ 1∗M(f
1+nr(x0), f(x0), t)∗1 = M(f

1+nr(x0), f(x0), t0).

This leads a contradiction. The proof is completed. �

Remark 3.3. It is easy to see that if (X, M, ∗) is a compact fuzzy metric space
and f ∈ C(X, ε), then ∪x∈Xω(x, f) = P(f) since f(ω(x, f)) = ω(x, f) and
P(f) ⊂ ∪x∈Xω(x, f) ⊂ ∩

∞
n=1f

n(X) = P(f).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 315



T. Sun, C. Han, G. Su, B. Qin and L. Li

Lemma 3.4. Let (X, M, ∗) be a fuzzy metric space, u ∈ X and f ∈ C(X, ε).
Write J(u, f) = {x ∈ X : u ∈ ω(x, f)}. Then J(u, f) is a closed subset of X.
Furthermore, if (X, M, ∗) is compact, then J(u, f) is a open subset of X.

Proof. Let x1, x2, · · · , xn, · · · ∈ J(u, f) and x ∈ X such that limn−→∞ M(xn, x, t) =
1 for any t > 0. For any n ∈ N, there exist 1 < k1n < k2n < · · · < krn < · · ·
such that limr−→∞ M(f

krn(xn), u, t) = 1 for any t > 0. Then for any 0 < δ < ε
and t > 0, we can choose 0 < δ1 < δ with (1 − δ1) ∗ (1 − δ1) > 1 − δ and
N ∈ N such that (by taking subsequence ) M(fknn(xn), u, t/2) ≥ 1 − δ1 and
M(xn, x, t/2) > 1 − δ1 for any n ≥ N. Then

M(fknn(x), u, t) ≥ M(fknn(x), fknn(xn),
t

2
) ∗ M(fknn(xn), u,

t

2
)

≥ M(x, xn,
t

2
) ∗ M(fknn(xn), u,

t

2
)

≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ.

Thus u ∈ ω(x, f) and J(u, f) is closed.
Now we prove the second part of the lemma. Assume that X is compact.

Then for any 0 < δ < ε, there exists 0 < δ1 < δ < ε such that (1−δ1)∗(1−δ1) >
1 − δ.

Let x ∈ J(u, f). We prove that B(x, δ1, t/2) ⊂ J(u, f). Let y ∈ B(x, δ1, t/2).
Since x ∈ J(u, f), there exist n1 < n2 < · · · < nk < · · · such that
limk−→∞ M(f

nk(x), u, t/2) = 1 for any t > 0. Thus for any t > 0, there
exists N ∈ N such that M(fnk(x), u, t/2) ≥ 1 − δ1 for any k > N. Therefore
we have that for k > N and t > 0,

M(fnk(y), u, t) ≥ M(fnk(y), fnk (x),
t

2
) ∗ M(fnk(x), u,

t

2
)

≥ M(y, x,
t

2
) ∗ M(fnk(x), u,

t

2
)

≥ (1 − δ1) ∗ (1 − δ1) > 1 − δ.

By taking subsequence we may assume that fnk(y) −→ v. Taking the limit on
both sides in the above as k −→ ∞ we obtain

M(v, u, t) ≥ 1 − δ.

If J(u, f) = ∅, then J(u, f) is open.
If J(u, f) 6= ∅, then by Remark 3.3 we see that v, u ∈ P(f). Let m and n

be the periods of u and v, respectively. Note that f ∈ C(X, ε), we have

M(u, v, t) = M(fmn(u), fmn(v), t) > M(u, v, t),

which is impossible unless u = v. Hence u ∈ ω(y, f) and B(x, δ1, t/2) ⊂ J(u, f).
Since x is an arbitrarily chosen point of J(u, f), we see that J(u, f) is an open
subset of X. The proof is completed. �

Theorem 3.5. If (X, M, ∗) is a compact fuzzy metric space and f ∈ C(X, ε),
then each connected component of X contains at most one periodic point of f.
Furthermore, if (X, M, ∗) is also connected, then f has at most one fixed point.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 316



The periodic points of contractive maps

Proof. Let p ∈ P(f) and Y (p) be the connected component of X containing p
and r be the period of p. Write h = fr. By Remark 2.7 we see that h ∈ C(X, ε)
and h(Y (p)) is connected and Y (p) is a compact. Since p ∈ Y (p) ∩ h(Y (p)),
we have h(Y (p)) ⊂ Y (p). Replace X and f of Lemma 3.4 by Y (p) and h,
respectively, and write J(p, h) ⊂ Y (p) to be as in Lemma 3.4. By Lemma 3.4
we see that J(p, h) is both closed and open in Y (p). Also, since p ∈ ω(p, h)
and Y (p) is connected, we have that J(p, h) = Y (p)

In the following, we show P(f) ∩ Y (p) = {p}. Indeed, if P(f) ∩ Y (p) 6= {p}
and q ∈ P(f) ∩ Y (p) − {p}, then q ∈ P(h) ∩ Y (p) = P(h) ∩ J(p, h) and
p ∈ ω(q, h). Let n1 < n2 < · · · < nk < · · · such that h

nk(q) −→ p. Then for
any 0 < δ < ε and t > 0, there exists N ∈ N such that M(hnk (q), p, t) > 1 − δ
for any k ≥ N. Let s be the period of q. we have

M(p, hnk(q), t) = M(hsnk (p), h(s+1)nk (q), t) > M(p, hnk(q), t).

This will lead a contradiction. The proof is completed. �

In the following we present two examples to illustrate the applicability of
the obtained results.

Example 3.6. Let X = [0, 1/3] ∪ [2/3, 1] ⊂ (−∞, +∞). Define s ∗ t = st for
any s, t ∈ [0, 1], and let M : X × X × (0, ∞) −→ [0, 1] by, for any x, y ∈ X and
t > 0,

M(x, y, t) =

{

1
1+|x−y|

, if t ≥ 1,
t

t+|x−y|
, if 0 < t < 1.

Then (X, M, ∗) is a compact fuzzy metric space. Take k ∈ (0, 1) and define
f : X −→ X by, for any x ∈ X,

f(x) =

{

kx + 2
3
, if x ∈ [0, 1

3
],

k(1 − x), if x ∈ [2
3
, 1].

We claim that f ∈ C(X, 1/4). Indeed, for any x, y ∈ X and t > 0 with
M(x, y, t) > 1−1/4, we have |x−y| < 1/3. Then x, y ∈ [0, 1/3] or x, y ∈ [2/3, 1],
from which it follows that

|f(x) − f(x)| = k|x − y| < |x − y|.

Thus M(f(x), f(y), t) > M(x, y, t) and f ∈ C(X, 1/4). By Theorem 3.2 and
Theorem 3.5 we see that P(f) = ∩∞n=1f

n(X) contains at most 2 points, and
[0, 1/3] ∩ P(f) contains at most one point, and [2/3, 1] ∩ P(f) contains at most
one point. In fact, we have P(f) = {k/3(k2 + 1), (3k2 + 2)/3(k2 + 1)} with
f(k/3(k2 + 1)) = (3k2 + 2)/3(k2 + 1) and f((3k2 + 2)/3(k2 + 1)) = k/3(k2 + 1).

Example 3.7. Let X = [0, 1/3] ∪ [2/3, 1] ⊂ (−∞, +∞). Define s ∗ t = st for
any s, t ∈ [0, 1], and let M : X × X × (0, ∞) −→ [0, 1] by, for any x, y ∈ X and
t > 0,

M(x, y, t) =
t

t + |x − y|
.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 317



T. Sun, C. Han, G. Su, B. Qin and L. Li

Then (X, M, ∗) is a compact fuzzy metric space. Take k ∈ (0, 1) and define
f : X −→ X by, for any x ∈ X,

f(x) =

{

kx + 2
3
, if x ∈ [0, 1

3
],

k(1 − x) + 2
3
, if x ∈ [2

3
, 1].

We claim that f ∈ C(X, 1/4). Indeed, if x, y ∈ [0, 1/3] or x, y ∈ [2/3, 1], then
|f(x) − f(y)| = k|x − y| < |x − y|, which follows that

M(f(x), f(y), t) > M(x, y, t).

If x ∈ [0, 1/3] and y ∈ [2/3, 1], then |f(x) − f(y)| = 1/3 < |x − y|, which also
follow that

M(f(x), f(y), t) > M(x, y, t).

By Theorem 3.2 and Theorem 3.5 we see that P(f) = ∩∞n=1f
n(X) contains at

most 2 points, and [0, 1/3]∩P(f) contains at most one point, and [2/3, 1]∩P(f)
contains at most one point. In fact, we have P(f) = {3k+2/(3k+3)} ⊂ [2/3, 1].

4. Conclusion

In this paper, we introduce the notion of ε-contractive maps in a fuzzy metric
space, and study the periodicities of ε-contractive maps, and obtain the follow-
ing result: If f is a ε-contractive map in compact fuzzy metric space (X, M, ∗),
then P(f) = ∩∞n=1f

n(X) and each connected component of X contains at most
one periodic point of f. Furthermore, we present two examples to illustrate
the applicability of the obtained results.

Acknowledgements. The authors thanks the referee for his/her valuable
suggestions which improved the paper.

References

[1] M. Abbas, M. Imdad and D. Gopal, ψ-weak contractions in fuzzy metric spaces, Iranian
J. Fuzzy Syst. 8 (2011), 141–148.

[2] I. Beg, C. Vetro, D, Gopal and M. Imdad, (φ,ψ)-weak contractions in intuitionistic fuzzy
metric spaces, J. Intel. Fuzzy Syst. 26 (2014), 2497–2504.

[3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst.
64 (1994), 395–399.

[4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1989), 385–389.
[5] V. Gregori and J. J. Miñana, Some remarks on fuzzy contractive mappings, Fuzzy Sets

Syst. 251 (2014), 101–103.
[6] V. Gregori and J. J. Miñana, On fuzzy Ψ-contractive sequences and fixed point theorems,

Fuzzy Sets Syst. 300 (2016), 93–101.
[7] V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets

Syst. 125 (2002), 245–252.
[8] J. Harjani, B. López and K. Sadarangani, Fixed point theorems for cyclic weak contrac-

tions in compact metric spaces, J. Nonl. Sci. Appl. 6 (2013), 279–284.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 318



The periodic points of contractive maps

[9] X. Hu, Z. Mo and Y. Zhen, On compactnesses of fuzzy metric spaces (Chinese), J.
Sichuan Norm. Univer. (Natur. Sei.) 32 (2009), 184–187.

[10] I. Kramosil and J. Michàlek, Fuzzy metrics and statistical metric spaces, Kybernetika
11 (1975), 336–344.

[11] D. Mihet, Fuzzy Ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy
Sets Sys. 159 (2008), 739–744.

[12] D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst.
251 (2014), 83–91.

[13] B. Schweizer and A. Sklar, Statistical metrics paces, Pacif. J. Math. 10 (1960), 385–389.
[14] Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math.

Letters 25 (2012), 138–141.
[15] S. Shukla, D. Gopal and A. F. Roldán-López-de-Hierro, Some fixed point theorems in

1-M-complete fuzzy metric-like spaces, Inter. J. General Syst. 45 (2016), 815–829.
[16] S. Shukla, D. Gopal and W. Sintunavarat, A new class of fuzzy contractive mappings

and fixed point theorems, Fuzzy Sets Syst. 359 (2018), 85–94.
[17] D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces,

Fuzzy Sets Syst. 222 (2013), 108–114.
[18] D. Zheng and P. Wang, On probabilistic Ψ-contractions in Menger probabilistic metric

spaces, Fuzzy Sets Syst. 350 (2018), 107–110.

[19] D. Zheng and P. Wang, Meir-Keeler theorems in fuzzy metric spaces, Fuzzy Sets Syst.
370 (2019), 120–128.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 319