Research Article | Open Access

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

**Academic Editor:**Liviu Cădariu

#### 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-type *t*-norms. We improve the
corresponding conclusions in the literature.

#### 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 -norms, minimum -norm and product -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 -norm in Section 3. Since the H-type -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. Let denote the set of all positive integers.

*Definition 1 (see [32]). *A triangular norm (-norm for short) is a binary operation on the unit interval , that is, a function , such that for all the following four axioms are satisfied:(T-1) (boundary condition).(T-2) whenever and (monotonicity).(T-3) (commutativity).(T-4) (associativity).

A -norm is said to be continuous if it is a continuous function in . Examples of -norms are and .

*Definition 2 (see [30]). *A -norm is of H-type if the family is equicontinuous at the point , where is defined by
It is obvious that is an H-type -norm. In fact, there are innumerable H-type -norms [30].

Lemma 3 (see [31]). *Let be a -norm. If is continuous and of H-type, then there exists a strictly increasing sequence from the interval such that and .*

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 is said to be a fuzzy metric space if is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions, for all :(FM-1),(FM-2) if and only if ,(FM-3),(FM-4),(FM-5) is continuous,(FM-6).

*Definition 5 (see [7, 29]). *Let be a fuzzy metric space. Then(a)a sequence in is said to converge to in if and only if for all ; that is, for each and , there exists an such that for all ;(b)a sequence in is a G-Cauchy sequence if and only if for any and ;(c)the fuzzy metric space is called G-complete if every G-Cauchy sequence is convergent.

*Definition 6 (see [7]). *Let be a fuzzy metric space. A mapping is called a contraction mapping if there exists such that
for every and .

Lemma 7. *Let be a G-complete fuzzy metric space. If is a contraction mapping, then has a unique fixed point.*

*Definition 8. *Let be a fuzzy metric space and let be a sequence of self-mappings of . is a given mapping. The sequence is said to converge pointwise to if converge to for each ; that is, for any , , and , there exists an such that
for all .

*Definition 9 (see [29]). *Let be a fuzzy metric space and let be a sequence of self-mappings of . is a given mapping. The sequence is said to converge uniformly to if for each and , there exists an such that
for all and .

*Definition 10. *Let be a fuzzy metric space. A sequence of self-mappings is uniformly equicontinuous if for each and , there exists an such that whenever for all and .

*Definition 11 (see [18]). *Let be a fuzzy metric space. The open ball and the closed ball with center and radius and , respectively, are defined as follows:

Lemma 12 (see [18]). *Every open (closed) ball is an open (closed) set.*

*Definition 13. *A fuzzy metric space is a compact space if is a compact topological space, where is a topology induced by the fuzzy metric .

Lemma 14 (see [33]). *Every closed subset of a compact fuzzy metric space is compact.*

*Definition 15 (see [33]). *A fuzzy metric space in which every point has a compact neighborhood is called locally compact.

*Definition 16 (see [34, 35]). *Let be a -norm and . Define the pseudodifference by

*Definition 17. *Let be a fuzzy metric space and let be a sequence of fuzzy metrics with respect to the same -norm on . The sequence is said to upper semiconverge uniformly to if for each and , there exists an such that
for all .

#### 3. Main Results

Theorem 18. *Let be a fuzzy metric space and let be a sequence of self-mappings of . is a contraction mapping; that is, there exists a such that for every and . is a compact subset of . If converges pointwise to in and it is a uniformly equicontinuous sequence, then the sequence converges uniformly to in .*

* Proof. *For each , we may choose an appropriate such that
At the same time, for any fixed , since is uniformly equicontinuous, there exists such that
whenever , for all and . Now define the open covering of as
Since is compact, there exist , such that
Since converges pointwise to in , for each , there exists such that
for all . Set . For any , there exists an such that . Now for all , we have that
By the arbitrariness of and , we get that the sequence converges uniformly to in .

Theorem 19. *Let be a G-complete fuzzy metric space and let be a sequence of self-mappings of , where the -norm is of H-type. is a contraction mapping; that is, there exists a such that for every and , and then has a unique fixed point . If there exists at least one fixed point for each and the sequence converges uniformly to , then converges to .*

* Proof. *Since is of H-type, by Lemma 3 there exists a strictly increasing sequence from the interval such that and .

Suppose that does not converge to . Thus, without loss of generality, we can suppose that there exists and such that
for all . Let be a given number in . According to the condition (FM-6) of Definition 4, for any , we can find a such that
In addition, since the sequence converges uniformly to , there exists an such that
for all and . Now for , we get that
Since
by the monotonicity of fuzzy metric with respect to , we have that
Thus from inequality (15) and inequality (17), we get
But since is of H-type and , inequality (20) is a contradiction. Thence, we get that converges to .

Theorem 20. *Let be a G-complete fuzzy metric space. If is a self-mapping of such that the iteration mapping is a contraction mapping for a certain positive integer , then has a unique fixed point.*

* Proof. *Without loss of generality, we suppose . Since is a contraction mapping, by Lemma 7, has a unique fixed point . Thus and which implies . It follows that ; that is, is a fixed point of . If for some , then we can get that which implies . Thus has a unique fixed point.

Theorem 21. *Let be a G-complete fuzzy metric space and let be a sequence of self-mappings of , where the -norm is of H-type. Suppose that is a self-mapping such that is a contraction mapping for a certain positive integer . If there exists at least one fixed point for each and the sequence converges uniformly to , then converges to .*

* Proof. *It is a corollary of Theorems 19 and 20.

Theorem 22. *Let be a locally compact fuzzy metric space and let be a sequence of self-mappings of , where the -norm is of H-type. is a contraction mapping; that is, there exists a such that for every and . If the following conditions are satisfied:*(i)* is a contraction mapping for a certain number ,*(ii)* converges pointwise to and it is a uniformly equicontinuous sequence,*(iii)*,**then converges to .*

* Proof. *Since is of H-type, by Lemma 3 there exists a strictly increasing sequence from the interval such that and . Since is a locally compact space, for the given point , we can choose a and an such that and the closed ball is a compact set of . Since is uniformly equicontinuous and pointwise convergent on , by Theorem 18, converges uniformly to on . Thus there exists such that
for all and . Consequently, for all and , we have that
Then we have , and then by inductive reasoning for all whenever and . Especially, is an invariant set for whenever . It follows that the fixed point of is contained in the set whenever . Now by Theorem 19, we get that converges to .

Theorem 23. *Let be a G-complete fuzzy metric space and let be a compact subset of . If is a sequence of fuzzy metrics of with respect to and is a sequence of self-mappings of satisfying the following conditions:*(i)* upper semiconverges uniformly to ,*(ii)* is a contraction mapping with respect to the fuzzy metric for ,*(iii)* converges pointwise to with respect to ,**then converges uniformly to in with respect to the fuzzy metric .*

* Proof. *For each , we can choose an such that
Since upper semiconverges uniformly to , for and , there exists such that
for all and . It should be noted that
because of the continuity of .

Now for any satisfying and , we get that
where is the constant in Definition 6 for . By the monotonicity of fuzzy metric with respect to , we have that
Now from inequality (26), for , we get that
whenever . Thus is uniformly equicontinuous in with respect to the fuzzy metric . Since is a compact subset of and converges pointwise to with respect to , it follows from Theorem 18 that converges uniformly to in with respect to the fuzzy metric .

Theorem 24. *Let be a locally compact fuzzy metric space where the -norm is of H-type. is a contraction mapping; that is, there exists a such that for every and . If is a sequence of fuzzy metrics of with respect to and is a sequence of self-mappings of satisfying the following conditions:*(i)* upper semiconverges uniformly to ;*(ii)* is a contraction mapping with respect to the fuzzy metric and for ;*(iii)* converges pointwise to with respect to .*

Then the sequence of fixed points converges to .

* Proof. *Since is of H-type, by Lemma 3 there exists a strictly increasing sequence from the interval such that and . Since is a locally compact space, for the given point , we can choose a and an such that and the closed ball is a compact set of .

It follows form Theorem 23 that converges uniformly to in with respect to the fuzzy metric . Thus there exists such that
for all and .

Meantime, since upper semiconverges uniformly to , for and , there exists such that
for all and . Let . If and , then we have that and . This implies that because is of H-type and
Thus, we have
for all . But for , it is easy to see that
because
Therefore, we get , for all . Now for and , we have
Then we have , which implies that is an invariant set in with respect to for . It follows that the fixed point of is contained in the set whenever . Now by Theorem 19, we get that converges to .

Theorem 25. *Let be a compact fuzzy metric space where the -norm is of H-type. is a sequence of fuzzy metrics of with respect to and is a sequence of self-mappings of satisfying the following conditions:*(i)* upper semiconverges uniformly to ;*(ii)* is a contraction mapping with respect to the fuzzy metric for ;*(iii)* converges pointwise to with respect to .**If is the fixed point of and there is a subsequence of which converges to , then .*

* Proof. *Since is of H-type, by Lemma 3, there exists a strictly increasing sequence from the interval such that and .

Let be the closure of the set . By Lemma 14, we can easily know that is a compact set. According to Theorem 23, it follows that the subsequence converges uniformly to in with respect to ; that is, for each and , there exists an such that for all and . Now for , we have that
where is the constant in Definition 6 for . Thus converges to . Then .

*Example 26. *Let be a strictly increasing sequence from the interval , and let , for all . Since , by Lemma 3 in [36], the ordinal sum is an H-type -norm. Let . Define a fuzzy set on by
for all and . Then we can get that is a G-complete fuzzy metric space. Let be a mapping defined by for all , and let be a sequence of self-mappings of defined by
for all and . By some simple calculations, we can get that is a contraction mapping and converges uniformly to . Thus by Theorem 19, we obtain that the sequence of fixed point of converges to the fixed point of . In fact, we have that and
for all .

#### 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-type -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.

#### Conflict of Interests

The authors have no conflicts of interests regarding this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11201512), the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001), and the Science and Technology Project of Chongqing Municipal Education Committee of China (Grant no. KJ120520, KJ1400426).

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Copyright © 2014 Dong Qiu and Shuai Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.