# On Convergence of Fixed Points in G-Complete Fuzzy Metric Spaces

Authors: Dong Qiu; Shuai Deng
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We provide some results on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in G-complete fuzzy metric spaces with the H-typet-norms. We improve the
corresponding conclusions in the literature.


---

## Body

## 1. Introduction

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?” This problem posed by Ulam has stimulated a long lasting interest in a particular stability of various types of equations and inequalities. That issue is sometimes called Ulam's type stability. In the recent past, several Ulam stability results concerning the various functional equations were determined in [2–4], respectively, in the fuzzy and intuitionistic fuzzy normed spaces.The concept of fuzzy sets was introduced initially by Zadeh [5] in 1965. After that, to use this concept in topology and analysis many authors have expansively developed the theory of fuzzy sets and applications. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of a fuzzy metric space. This problem was investigated by many authors from different points of view [6–16]. Kramosil and Michalek [7] gave a notion of fuzzy metric space which could be considered as a reformulation, in the fuzzy context, of the notion of probabilistic metric space due to Menger [17]. Later, George and Veeramani [18] introduced and studied a notion of fuzzy metric space which constitutes a modification of the one due to Kramosil and Michalek.These fuzzy metric spaces have been widely accepted as an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. Further, these fuzzy metric spaces have very important applications in studying fixed point theorems [17, 19–28]. In [29], Shen et al. have shown that the convergence of the sequence of fixed points to some sequences of contraction mappings or fuzzy metrics satisfies certain conditions in fuzzy metric spaces. However, they have given the most of results in [29] only for the two specific t-norms, minimum t-norm and product t-norm.In this paper, we will provide some improved results on convergence of fixed points in G-complete fuzzy metric spaces. Specifically, after redefining some basic concepts of [29] in Section 2, we will generalize the results in [29] to the G-complete fuzzy metric spaces with an H-type t-norm in Section 3. Since the H-type t-norms are very important and widely used in fuzzy fixed point theory [17, 20, 24, 30, 31], our results improve the corresponding conclusions in the literature.


## 2. Preliminaries

For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper. LetN denote the set of all positive integers.Definition 1 (see [32]).
A triangular norm (t-norm for short) is a binary operation on the unit interval [
0,1
], that is, a function *
:
[
0,1
]
2
→
[
0,1
], such that for all a
,
b
,
c
,
d
∈
[
0,1
] the following four axioms are satisfied:(T-1)
a
*
1
=
a (boundary condition).
(T-2)
a
*
b
≤
c
*
d whenever a
≤
c and b
≤
d (monotonicity).
(T-3)
a
*
b
=
b
*
a (commutativity).
(T-4)
a
*
(
b
*
c
)
=
(
a
*
b
)
*
c (associativity).At-norm * is said to be continuous if it is a continuous function in [
0,1
]
2. Examples of t-norms are a
*
P
b
=
a
b and a
*
M
b
=
min
⁡
{
a
,
b
}.Definition 2 (see [30]).
At-norm * is of H-type if the family (
x
(
n
)
)
n
∈
N is equicontinuous at the point x
=
1, where x
(
n
) is defined by
(1)
x
(
1
)
=
x
,
x
(
n
)
=
x
(
n
-
1
)
*
x
,
n
≥
2
,
x
∈
[
0,1
]
.
It is obvious that *
M is an H-type t-norm. In fact, there are innumerable H-type t-norms [30].Lemma 3 (see [31]).
Let* be a t-norm. If * is continuous and of H-type, then there exists a strictly increasing sequence (
b
n
)
n
∈
N from the interval [
0,1
) such that lim
⁡
n
→
∞
b
n
=
1 and b
n
*
b
n
=
b
n.In [29], Shen et al. redefined the notion of fuzzy metric space by appending the following condition (FM-6) based on the one in the sense of George and Veeramani [18].Definition 4 (see [33]).
The 3-tuple(
X
,
M
,
*
) is said to be a fuzzy metric space if X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set on X
2
×
(
0
,
∞
) satisfying the following conditions, for all x
,
y
,
z
∈
X
,
t
,
s
>
0:(FM-1)
M
(
x
,
y
,
t
)
>
0,
(FM-2)
M
(
x
,
y
,
t
)
=
1 if and only if x
=
y,
(FM-3)
M
(
x
,
y
,
t
)
=
M
(
y
,
x
,
t
),
(FM-4)
M
(
x
,
z
,
t
+
s
)
≥
M
(
x
,
y
,
t
)
*
M
(
y
,
z
,
s
),
(FM-5)
M
(
x
,
y
,
·
)
:
(
0
,
∞
)
→
(
0,1
] is continuous,
(FM-6)
lim
⁡
t
→
∞
⁡
M
(
x
,
y
,
t
)
=
1.Definition 5 (see [7, 29]).
Let(
X
,
M
,
*
) be a fuzzy metric space. Then(a)
a sequence(
x
n
) in X is said to converge to x in X if and only if lim
⁡
n
→
∞
M
(
x
n
,
x
,
t
)
=
1 for all t
>
0; that is, for each r
∈
(
0,1
) and t
>
0, there exists an n
0
∈
N such that M
(
x
n
,
x
,
t
)
>
1
-
r for all n
≥
n
0;
(b)
a sequence(
x
n
) in X is a G-Cauchy sequence if and only if lim
⁡
n
→
∞
M
(
x
n
,
x
n
+
p
,
t
)
=
1 for any p
∈
N and t
>
0;
(c)
the fuzzy metric space(
X
,
M
,
*
) is called G-complete if every G-Cauchy sequence is convergent.Definition 6 (see [7]).
Let(
X
,
M
,
*
) be a fuzzy metric space. A mapping T
:
X
→
X is called a contraction mapping if there exists k
∈
(
0,1
) such that
(2)
M
(
T
x
,
T
y
,
k
t
)
≥
M
(
x
,
y
,
t
)
for every x
,
y
∈
X and t
>
0.Lemma 7.
Let(
X
,
M
,
*
) be a G-complete fuzzy metric space. If T
:
X
→
X is a contraction mapping, then T has a unique fixed point.Definition 8.
Let(
X
,
M
,
*
) be a fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X. T
0
:
X
→
X is a given mapping. The sequence (
T
n
) is said to converge pointwise to T
0 if T
n
x converge to T
0
x for each x
∈
X; that is, for any x
∈
X, r
∈
(
0,1
), and t
>
0, there exists an n
0
∈
N such that
(3)
M
(
T
n
x
,
T
0
x
,
t
)
>
1
-
r
for all n
≥
n
0.Definition 9 (see [29]).
Let(
X
,
M
,
*
) be a fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X. T
0
:
X
→
X is a given mapping. The sequence (
T
n
) is said to converge uniformly to T
0 if for each r
∈
(
0,1
) and t
>
0, there exists an n
0
∈
N such that
(4)
M
(
T
n
x
,
T
0
x
,
t
)
>
1
-
r
for all n
≥
n
0 and x
∈
X.Definition 10.
Let(
X
,
M
,
*
) be a fuzzy metric space. A sequence of self-mappings (
T
n
) is uniformly equicontinuous if for each r
∈
(
0,1
) and t
>
0, there exists an ε
∈
(
0,1
) such that M
(
T
n
x
,
T
n
y
,
t
)
>
1
-
r whenever M
(
x
,
y
,
t
)
>
1
-
ε for all x
,
y
∈
X and n
∈
N.Definition 11 (see [18]).
Let(
X
,
M
,
*
) be a fuzzy metric space. The open ball B
M
(
x
,
r
,
t
) and the closed ball B
M
[
x
,
r
,
t
] with center x
∈
X and radius r
∈
(
0,1
) and t
>
0, respectively, are defined as follows:
(5)
B
M
(
x
,
r
,
t
)
=
{
y
∈
X
:
M
(
x
,
y
,
t
)
>
1
-
r
}
,
B
M
[
x
,
r
,
t
]
=
{
y
∈
X
:
M
(
x
,
y
,
t
)
≥
1
-
r
}
.Lemma 12 (see [18]).
Every open (closed) ball is an open (closed) set.Definition 13.
A fuzzy metric space(
X
,
M
,
*
) is a compact space if (
X
,
τ
M
) is a compact topological space, where τ
M is a topology induced by the fuzzy metric M.Lemma 14 (see [33]).
Every closed subsetA of a compact fuzzy metric space (
X
,
M
,
*
) is compact.Definition 15 (see [33]).
A fuzzy metric space(
X
,
M
,
*
) in which every point has a compact neighborhood is called locally compact.Definition 16 (see [34, 35]).
Let* be a t-norm and a
,
b
∈
[
0,1
]. Define the pseudodifference a
⊖
b by
(6)
a
⊖
b
=
sup
⁡
{
c
:
b
*
c
≤
a
}
.Definition 17.
Let(
X
,
M
0
,
*
) be a fuzzy metric space and let (
M
n
) be a sequence of fuzzy metrics with respect to the same t-norm * on X. The sequence (
M
n
) is said to upper semiconverge uniformly to M
0 if for each r
∈
(
0,1
) and t
>
0, there exists an n
0
∈
N such that
(7)
M
n
(
x
,
y
,
t
)
≥
M
0
(
x
,
y
,
t
)
,
M
0
(
x
,
y
,
t
)
⊖
M
n
(
x
,
y
,
t
)
>
1
-
r
for all n
≥
n
0
,
x
,
y
∈
X.


## 3. Main Results

Theorem 18.
Let(
X
,
M
,
*
) be a fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X. T
0
:
X
→
X is a contraction mapping; that is, there exists a k
∈
(
0,1
) such that M
(
T
0
x
,
T
0
y
,
k
t
)
≥
M
(
x
,
y
,
t
) for every x
,
y
∈
X and t
>
0. A is a compact subset of X. If (
T
n
) converges pointwise to T
0 in A and it is a uniformly equicontinuous sequence, then the sequence (
T
n
) converges uniformly to T
0 in A.Proof.
For eachr
-
∈
(
0,1
), we may choose an appropriate r
∈
(
0,1
) such that
(8)
(
1
-
r
)
*
(
1
-
r
)
*
(
1
-
r
)
>
1
-
r
-
.
At the same time, for any fixed t
>
0, since (
T
n
) is uniformly equicontinuous, there exists ε
∈
(
0
,
r
) such that
(9)
M
(
T
n
x
,
T
n
y
,
t
2
+
k
)
>
1
-
r
,
whenever M
(
x
,
y
,
t
/
(
2
+
k
)
)
>
1
-
ε, for all x
,
y
∈
X and n
∈
N. Now define the open covering C of A as
(10)
C
=
{
B
M
(
x
,
ε
,
t
2
+
k
)
:
x
∈
A
}
.
Since A is compact, there exist x
i
∈
A, i
=
1,2
,
…
,
m such that
(11)
A
⊆
⋃
i
=
1
m
‍
B
M
(
x
i
,
ε
,
t
2
+
k
)
.
Since T
n converges pointwise to T
0 in A, for each x
i, there exists n
i
∈
N such that
(12)
M
(
T
n
x
i
,
T
0
x
i
,
t
2
+
k
)
>
1
-
r
for all n
≥
n
i. Set l
=
max
⁡
{
n
i
,
i
=
1,2
,
…
,
m
}. For any x
∈
A, there exists an x
i
0 such that x
∈
B
(
x
i
0
,
ε
,
t
/
(
2
+
k
)
). Now for all n
≥
l, we have that
(13)
M
(
T
n
x
,
T
0
x
,
t
)
≥
M
(
T
n
x
,
T
n
x
i
0
,
t
2
+
k
)
*
M
(
T
n
x
i
0
,
T
0
x
i
0
,
t
2
+
k
)
*
M
(
T
0
x
i
0
,
T
0
x
,
k
t
2
+
k
)
≥
(
1
-
r
)
*
(
1
-
r
)
*
M
(
x
i
0
,
x
,
t
2
+
k
)
≥
(
1
-
r
)
*
(
1
-
r
)
*
(
1
-
ε
)
≥
(
1
-
r
)
*
(
1
-
r
)
*
(
1
-
r
)
≥
1
-
r
-
.
By the arbitrariness of r
- and t, we get that the sequence (
T
n
) converges uniformly to T
0 in A.Theorem 19.
Let(
X
,
M
,
*
) be a G-complete fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X, where the t-norm * is of H-type. T
0
:
X
→
X is a contraction mapping; that is, there exists a k
∈
(
0,1
) such that M
(
T
0
x
,
T
0
y
,
k
t
)
≥
M
(
x
,
y
,
t
) for every x
,
y
∈
X and t
>
0, and then T
0 has a unique fixed point x
0
∈
X. If there exists at least one fixed point x
n for each T
n and the sequence (
T
n
) converges uniformly to T
0, then x
n converges to x
0.Proof.
Since* is of H-type, by Lemma 3 there exists a strictly increasing sequence (
r
i
) from the interval [
0,1
) such that lim
⁡
i
→
∞
r
i
=
1 and r
i
*
r
i
=
r
i.
Suppose thatx
n does not converge to x
0. Thus, without loss of generality, we can suppose that there exists t
0
>
0 and 1
-
r
′
∈
(
r
i
) such that
(14)
1
-
r
′
>
M
(
x
n
,
x
0
,
t
0
)
for all n
∈
N. Let α be a given number in (
k
,
1
). According to the condition (FM-6) of Definition 4, for any x
n, we can find a p
(
n
)
∈
N such that
(15)
M
(
x
n
,
x
0
,
t
0
(
a
k
)
p
(
n
)
)
>
1
-
r
′
.
In addition, since the sequence (
T
n
) converges uniformly to T
0, there exists an n
0
∈
N such that
(16)
M
(
T
n
x
,
T
0
x
,
(
1
-
a
)
t
0
)
>
1
-
r
′
for all x
∈
X and n
≥
n
0. Now for n
≥
n
0, we get that
(17)
1
-
r
′
>
M
(
x
n
,
x
0
,
t
0
)
=
M
(
T
n
x
n
,
T
0
x
0
,
t
0
)
≥
M
(
T
n
x
n
,
T
0
x
n
,
(
1
-
a
)
t
0
)
*
M
(
T
0
x
n
,
T
0
x
0
,
a
t
0
)
≥
M
(
T
n
x
n
,
T
0
x
n
,
(
1
-
a
)
t
0
)
*
M
(
x
n
,
x
0
,
a
k
t
0
)
≥
M
(
T
n
x
n
,
T
0
x
n
,
(
1
-
a
)
t
0
)
*
M
(
T
n
x
n
,
T
0
x
n
,
a
k
(
1
-
a
)
t
0
)
*
M
(
x
n
,
x
0
,
(
a
k
)
2
t
0
)
≥
M
(
T
n
x
n
,
T
0
x
n
,
(
1
-
a
)
t
0
)
*
M
(
T
n
x
n
,
T
0
x
n
,
a
k
(
1
-
a
)
t
0
)
*
⋯
*
M
(
T
n
x
n
,
T
0
x
n
,
(
a
k
)
p
(
n
)
-
1
(
1
-
a
)
t
0
)
*
M
(
x
n
,
x
0
,
(
a
k
)
p
(
n
)
t
0
)
.
Since
(18)
(
a
k
)
i
(
1
-
a
)
t
0
≥
(
1
-
a
)
t
0
,
i
=
1,2
,
…
,
p
(
n
)
-
1
,
by the monotonicity of fuzzy metric M with respect to t, we have that
(19)
M
(
T
n
x
n
,
T
0
x
n
,
(
a
k
)
i
(
1
-
a
)
t
0
)
≥
M
(
T
n
x
n
,
T
0
x
n
,
(
1
-
a
)
t
0
)
>
1
-
r
′
,
i
=
1,2
,
…
,
p
(
n
)
-
1
.
Thus from inequality (15) and inequality (17), we get
(20)
1
-
r
′
>
(
1
-
r
′
)
*
(
1
-
r
′
)
*
⋯
*
(
1
-
r
′
)
.
But since * is of H-type and 1
-
r
′
∈
(
r
n
), inequality (20) is a contradiction. Thence, we get that x
n converges to x
0.Theorem 20.
Let(
X
,
M
,
*
) be a G-complete fuzzy metric space. If T
:
X
→
X is a self-mapping of X such that the iteration mapping T
m is a contraction mapping for a certain positive integer m, then T has a unique fixed point.Proof.
Without loss of generality, we supposem
>
1. Since T
m is a contraction mapping, by Lemma 7, T
m has a unique fixed point x. Thus T
m
x
=
x and T
(
T
m
x
)
=
T
x which implies T
m
(
T
x
)
=
T
x. It follows that T
x
=
x; that is, x is a fixed point of T. If T
y
=
y for some y
∈
X, then we can get that T
m
y
=
y which implies x
=
y. Thus T has a unique fixed point.Theorem 21.
Let(
X
,
M
,
*
) be a G-complete fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X, where the t-norm * is of H-type. Suppose that T
0
:
X
→
X is a self-mapping such that T
0
m is a contraction mapping for a certain positive integer m. If there exists at least one fixed point x
n for each T
n and the sequence (
T
n
) converges uniformly to T
0, then x
n converges to x
0.Proof.
It is a corollary of Theorems19 and 20.Theorem 22.
Let(
X
,
M
,
*
) be a locally compact fuzzy metric space and let (
T
n
) be a sequence of self-mappings of X, where the t-norm * is of H-type. T
0
:
X
→
X is a contraction mapping; that is, there exists a k
∈
(
0,1
) such that M
(
T
0
x
,
T
0
y
,
k
t
)
≥
M
(
x
,
y
,
t
) for every x
,
y
∈
X and t
>
0. If the following conditions are satisfied:(i)
T
n
m is a contraction mapping for a certain number m
=
m
(
n
),
(ii)
(
T
n
) converges pointwise to T
0 and it is a uniformly equicontinuous sequence,
(iii)
T
n
x
n
=
x
n
,
n
=
0,1
,
2
,
…,then x
n converges to x
0.Proof.
Since* is of H-type, by Lemma 3 there exists a strictly increasing sequence (
r
i
) from the interval [
0,1
) such that lim
⁡
i
→
∞
r
i
=
1 and r
i
*
r
i
=
r
i. Since X is a locally compact space, for the given point x
0, we can choose a t
0
>
0 and an r
′ such that 1
-
r
′
∈
(
r
i
) and the closed ball B
M
[
x
0
,
r
′
,
t
0
] is a compact set of X. Since T
n is uniformly equicontinuous and pointwise convergent on B
M
[
x
0
,
r
′
,
t
0
], by Theorem 18, T
n converges uniformly to T
0 on B
M
[
x
0
,
r
′
,
t
0
]. Thus there exists n
0
∈
N such that
(21)
M
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
>
1
-
r
′
for all x
∈
B
M
[
x
0
,
r
′
,
t
0
] and n
≥
n
0. Consequently, for all x
∈
B
M
[
x
0
,
r
′
,
t
0
] and n
≥
n
0, we have that
(22)
M
(
T
n
x
,
x
0
,
t
0
)
=
M
(
T
n
x
,
T
0
x
0
,
t
0
)
≥
M
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
*
M
(
T
0
x
,
T
0
x
0
,
k
t
0
)
≥
M
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
*
M
(
x
,
x
0
,
t
0
)
≥
1
-
r
′
*
1
-
r
′
=
1
-
r
′
.
Then we have T
n
x
∈
B
M
[
x
0
,
r
′
,
t
0
], and then by inductive reasoning T
n
i
x
∈
B
M
[
x
0
,
r
′
,
t
0
] for all i
∈
N whenever x
∈
B
M
[
x
0
,
r
′
,
t
0
] and n
≥
n
0. Especially, B
M
[
x
0
,
r
′
,
t
0
] is an invariant set for T
n
m
(
n
) whenever n
≥
n
0. It follows that the fixed point x
n of T
n is contained in the set B
M
[
x
0
,
r
′
,
t
0
] whenever n
≥
n
0. Now by Theorem 19, we get that x
n converges to x
0.Theorem 23.
Let(
X
,
M
0
,
*
) be a G-complete fuzzy metric space and let A be a compact subset of X. If (
M
n
) is a sequence of fuzzy metrics of X with respect to * and (
T
n
) is a sequence of self-mappings of X satisfying the following conditions:(i)
(
M
n
) upper semiconverges uniformly to M
0,
(ii)
T
n is a contraction mapping with respect to the fuzzy metric M
n for n
=
0,1
,
2
,
…,
(iii)
(
T
n
) converges pointwise to T
0 with respect to M
0,then T
n converges uniformly to T
0 in A with respect to the fuzzy metric M
0.Proof.
For eachr
∈
(
0,1
), we can choose an ε
∈
(
0,1
) such that
(23)
(
1
-
ε
)
*
(
1
-
ε
)
>
1
-
r
.
Since M
n upper semiconverges uniformly to M
0, for ε
∈
(
0,1
) and t
>
0, there exists n
(
ε
,
t
) such that
(24)
M
n
(
x
,
y
,
t
)
≥
M
0
(
x
,
y
,
t
)
,
M
0
(
x
,
y
,
t
)
⊖
M
n
(
x
,
y
,
t
)
>
1
-
ε
for all x
,
y
∈
X and n
≥
n
(
ε
,
t
). It should be noted that
(25)
M
0
(
x
,
y
,
t
)
≥
(
M
0
(
x
,
y
,
t
)
⊖
M
n
(
x
,
y
,
t
)
)
*
M
n
(
x
,
y
,
t
)
because of the continuity of *.
Now for anyx
,
y
∈
X satisfying M
0
(
x
,
y
,
t
)
>
1
-
ε and n
≥
n
(
ε
,
t
), we get that
(26)
M
0
(
T
n
x
,
T
n
y
,
t
)
≥
(
M
0
(
T
n
x
,
T
n
y
,
t
)
⊖
M
n
(
T
n
x
,
T
n
y
,
t
)
)
*
M
n
(
T
n
x
,
T
n
y
,
t
)
≥
(
1
-
ε
)
*
M
n
(
T
n
x
,
T
n
y
,
t
)
≥
(
1
-
ε
)
*
M
n
(
x
,
y
,
t
k
n
)
,
where k
n
∈
(
0,1
) is the constant in Definition 6 for T
n. By the monotonicity of fuzzy metric M with respect to t, we have that
(27)
M
n
(
x
,
y
,
t
k
n
)
≥
M
n
(
x
,
y
,
t
)
>
M
0
(
x
,
y
,
t
)
>
1
-
ε
.
Now from inequality (26), for n
≥
n
(
ε
,
t
), we get that
(28)
M
0
(
T
n
x
,
T
n
y
,
t
)
≥
(
1
-
ε
)
*
(
1
-
ε
)
>
1
-
r
,
whenever M
0
(
x
,
y
,
t
)
>
1
-
ε. Thus T
n is uniformly equicontinuous in X with respect to the fuzzy metric M
0. Since A is a compact subset of X and T
n converges pointwise to T
0 with respect to M
0, it follows from Theorem 18 that T
n converges uniformly to T
0 in A with respect to the fuzzy metric M
0.Theorem 24.
Let(
X
,
M
0
,
*
) be a locally compact fuzzy metric space where the t-norm * is of H-type. T
0
:
X
→
X is a contraction mapping; that is, there exists a k
∈
(
0,1
) such that M
0
(
T
0
x
,
T
0
y
,
k
t
)
≥
M
0
(
x
,
y
,
t
) for every x
,
y
∈
X and t
>
0. If (
M
n
) is a sequence of fuzzy metrics of X with respect to * and (
T
n
) is a sequence of self-mappings of X satisfying the following conditions:(i)
(
M
n
) upper semiconverges uniformly to M
0;
(ii)
T
n is a contraction mapping with respect to the fuzzy metric M
n and T
n
x
n
=
x
n for n
=
0,1
,
2
,
…;
(iii)
(
T
n
) converges pointwise to T
0 with respect to M
0.Then the sequence of fixed points(
x
n
) converges to x
0.Proof.
Since* is of H-type, by Lemma 3 there exists a strictly increasing sequence (
r
i
) from the interval [
0,1
) such that lim
⁡
i
→
∞
r
i
=
1 and r
i
*
r
i
=
r
i. Since X is a locally compact space, for the given point x
0, we can choose a t
0
>
0 and an r
′ such that 1
-
r
′
∈
(
r
i
) and the closed ball B
M
0
[
x
0
,
r
′
,
t
0
] is a compact set of X.
It follows form Theorem23 that T
n converges uniformly to T
0 in B
M
[
x
0
,
r
′
,
t
0
] with respect to the fuzzy metric M
0. Thus there exists n
0
∈
N such that
(29)
M
0
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
>
1
-
r
′
for all x
∈
B
M
[
x
0
,
r
′
,
t
0
] and n
≥
n
0.
Meantime, sinceM
n upper semiconverges uniformly to M
0, for r
′
∈
(
0,1
) and t
0
>
0, there exists n
1 such that
(30)
M
n
(
x
,
y
,
t
0
)
≥
M
0
(
x
,
y
,
t
0
)
,
M
0
(
x
,
y
,
t
0
)
⊖
M
n
(
x
,
y
,
t
0
)
>
1
-
r
′
for all x
,
y
∈
X and n
≥
n
1. Let n
′
=
max
⁡
{
n
0
,
n
1
}. If n
≥
n
′ and x
∈
B
M
n
[
x
0
,
r
′
,
t
0
], then we have that M
n
(
x
,
x
0
,
t
0
)
≥
1
-
r
′ and M
0
(
x
,
x
0
,
t
)
⊖
M
n
(
x
,
x
0
,
t
)
>
1
-
r
′. This implies that M
0
(
x
,
x
0
,
t
)
≥
1
-
r
′ because * is of H-type and
(31)
M
0
(
x
,
x
0
,
t
0
)
≥
(
M
0
(
x
,
x
0
,
t
0
)
⊖
M
n
(
x
,
x
0
,
t
0
)
)
*
M
n
(
x
,
x
0
,
t
0
)
.
Thus, we have
(32)
B
M
n
[
x
0
,
r
′
,
t
0
]
⊆
B
M
0
[
x
0
,
r
′
,
t
0
]
,
for all n
≥
n
′. But for n
≥
n
′, it is easy to see that
(33)
B
M
n
[
x
0
,
r
′
,
t
0
]
⊇
B
M
0
[
x
0
,
r
′
,
t
0
]
,
because
(34)
M
n
(
x
,
y
,
t
0
)
≥
M
0
(
x
,
y
,
t
0
)
≥
1
-
r
′
.
Therefore, we get B
M
n
[
x
0
,
r
′
,
t
0
]
=
B
M
0
[
x
0
,
r
′
,
t
0
], for all n
≥
n
′. Now for x
∈
B
M
0
[
x
0
,
r
′
,
t
0
] and n
≥
n
′, we have
(35)
M
n
(
T
n
x
,
x
0
,
t
0
)
≥
M
0
(
T
n
x
,
x
0
,
t
0
)
=
M
0
(
T
n
x
,
T
0
x
0
,
t
0
)
≥
M
0
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
*
M
0
(
T
0
x
,
T
0
x
0
,
k
t
0
)
≥
M
0
(
T
n
x
,
T
0
x
,
(
1
-
k
)
t
0
)
*
M
0
(
x
,
x
0
,
t
0
)
≥
1
-
r
′
*
1
-
r
′
=
1
-
r
′
.
Then we have T
n
x
∈
B
M
n
[
x
0
,
r
′
,
t
0
]
=
B
M
0
[
x
0
,
r
′
,
t
0
], which implies that B
M
0
[
x
0
,
r
′
,
t
0
] is an invariant set in X with respect to M
n for n
≥
n
′. It follows that the fixed point x
n of T
n is contained in the set B
M
0
[
x
0
,
r
′
,
t
0
] whenever n
≥
n
′. Now by Theorem 19, we get that x
n converges to x
0.Theorem 25.
Let(
X
,
M
,
*
) be a compact fuzzy metric space where the t-norm * is of H-type. (
M
n
) is a sequence of fuzzy metrics of X with respect to * and (
T
n
) is a sequence of self-mappings of X satisfying the following conditions:(i)
(
M
n
) upper semiconverges uniformly to M
0;
(ii)
T
n is a contraction mapping with respect to the fuzzy metric M
n for n
=
0,1
,
2
,
…;
(iii)
(
T
n
) converges pointwise to T
0 with respect to M
0.
Ifx
n is the fixed point of T
n and there is a subsequence (
x
n
k
) of (
x
n
) which converges to x
0
∈
X, then T
0
x
0
=
x
0.Proof.
Since* is of H-type, by Lemma 3, there exists a strictly increasing sequence (
r
i
) from the interval [
0,1
) such that lim
⁡
i
→
∞
r
i
=
1 and r
i
*
r
i
=
r
i.
LetK be the closure of the set {
x
n
:
n
∈
N
}. By Lemma 14, we can easily know that K is a compact set. According to Theorem 23, it follows that the subsequence (
T
n
k
) converges uniformly to T
0 in K with respect to M
0; that is, for each r
∈
(
r
i
) and t
>
0, there exists an k
0
∈
N such that M
(
T
n
x
,
T
0
x
,
t
/
2
)
>
r for all x
∈
K and k
≥
k
0. Now for k
≥
k
0, we have that
(36)
M
(
T
n
k
x
n
k
,
T
0
x
0
,
t
)
≥
M
(
T
n
k
x
n
k
,
T
0
x
n
k
,
t
2
)
*
M
(
T
0
x
n
k
,
T
0
x
0
,
t
2
)
≥
r
*
M
(
x
n
k
,
x
0
,
t
2
k
0
)
≥
r
*
r
=
r
,
where k
0
∈
(
0,1
) is the constant in Definition 6 for T
0. Thus T
n
k
x
n
k
=
x
n
k converges to T
0
x
0. Then T
0
x
0
=
x
0.Example 26.
Let(
1
-
1
/
2
n
)
n
∈
N be a strictly increasing sequence from the interval [
0,1
), and let (
α
n
,
β
n
)
=
(
1
-
1
/
2
n
,
1
-
1
/
2
n
+
1
), for all n
∈
N. Since lim
⁡
n
→
∞
1
-
1
/
2
n
=
1, by Lemma 3 in [36], the ordinal sum *
=
(
〈
(
α
n
,
β
n
)
,
*
P
〉
)
n
∈
N is an H-type t-norm. Let X
=
[
-
1
/
2,1
/
2
]. Define a fuzzy set M on X
2
×
(
0
,
∞
) by
(37)
M
(
x
,
y
,
t
)
=
t
t
+
|
x
-
y
|
,
for all x
,
y
∈
X and t
>
0. Then we can get that (
X
,
M
,
*
) is a G-complete fuzzy metric space. Let T
0 be a mapping defined by T
0
(
x
)
=
x
2
/
4 for all x
∈
X, and let (
T
n
) be a sequence of self-mappings of X defined by
(38)
T
n
(
x
)
=
(
1
4
+
1
4
n
)
x
2
-
1
4
n
,
for all x
∈
X and n
∈
N. By some simple calculations, we can get that T
0 is a contraction mapping and (
T
n
) converges uniformly to T
0. Thus by Theorem 19, we obtain that the sequence (
x
n
) of fixed point of (
T
n
) converges to the fixed point x
0 of T
0. In fact, we have that x
0
=
0 and
(39)
x
n
=
1
-
1
+
(
1
/
4
n
)
(
1
+
(
1
/
4
n
-
1
)
)
(
1
/
2
)
+
(
1
/
2
2
n
-
1
)
,
for all n
∈
N.


## 4. Conclusions

In this paper, we provide some results on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in G-complete fuzzy metric spaces with the H-typet-norms. We improve the corresponding conclusions in the literature. The first solution to the Ulam problem was obtained through a classical approach known from the fixed point theory, by iteration of a simple operator. Later it has been shown that some fixed point theorems can be directly applied in investigations of Ulam's type stability. Thus we hope our results would provide a background to ongoing work in the problems of those related fields.



---

*Source: 579454-2014-07-20.xml*