Fixed point theorems for fuzzy mappings in metric spaces with an application
 Jianhua Chen^{1} and
 Xianjiu Huang^{1}Email author
https://doi.org/10.1186/s1366001505992
© Chen and Huang; licensee Springer. 2015
Received: 18 September 2014
Accepted: 13 February 2015
Published: 4 March 2015
Abstract
In this paper, we prove some new fixed point theorems for fuzzy mappings under a Gdistance function and a \(G'\)distance function in complete metric spaces. Our results extend, generalize, and improve some existing results. Moreover, an example and an application are given here to illustrate the usability of the obtained results.
Keywords
MSC
1 Introduction and preliminaries
It is well known that the fuzzy set concept plays an important role in many scientific and engineering applications. The fuzziness appears when we need to perform, on manifold, calculations with imprecision variables. The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Since then, Heilpern [2] introduced the concept of fuzzy mapping and proved a fixed point theorem for fuzzy contraction mappings in a metric linear space, which is a fuzzy extension of the Banach contraction principle. Subsequently several other authors [3–26] have studied existence of fixed points of fuzzy mappings satisfying some different contractive type conditions. Recently, Abbas and Turkoglu [27] also proved some useful fixed point results for fuzzy mappings, which is a fuzzy extension of some existing results. However, the aim of this paper is to prove some new fixed point theorems for fuzzy mappings under a Gdistance function and a \(G'\)distance function in complete metric spaces. Our results extend, generalize, and improve the results of [2, 3, 6–10, 13, 14, 19, 27, 28].
Throughout this paper, we shall use the following notions.

\(A_{\alpha}= \{x:\mu(x)\geq\alpha \}\) if \(\alpha\in(0,1]\),

\(A_{0}=\overline{ \{x:\mu(x)>0 \}}\),
A fuzzy set A in X is said to be an approximate quantity if and only if \(A_{\alpha}\) is compact and convex in X for each \(\alpha\in (0,1]\) and \(\sup_{x\in X}A(x)=1\). We denote by \(W(X)\) the family of all approximate quantities in X. Let \(A, B\in W(X)\), then A is said to be more accurate A than B, denoted by \(A\subset B\), if and only if \(A(x)\leq B(x)\) for each \(x\in X\).
Let \(\alpha\in[0,1]\), then the family \(W_{\alpha}(X)\) is given by {\(A\in I^{X} : A_{\alpha}\) is nonempty convex and compact}. Let X be an arbitrary set, Y be a metric linear space. A mapping T is called a fuzzy mapping if T is a mapping from X into \(W(Y)\), that is, \(T(x)\in W(Y)\) for each x in X. Therefore, a fuzzy mapping T is a fuzzy subset on \(X\times Y\) with membership function \(Tx(y)\). A fuzzy point \(x_{\alpha}\) in X is called a fixed fuzzy point of the fuzzy mapping T if \(\{x_{\alpha}\}\subset Tx\). If \(\{x\}\subset Tx\), then x is a fixed point of T.
The following lemmas are needed in the sequel. Let \((X,d)\) be a metric space.
Lemma 1.1
[2]
Let \(x\in X\) and \(A\in W(X)\). Then \(\{x_{\alpha}\}\subset A\) if \(p_{\alpha}(x,A)=0\).
Lemma 1.2
[2]
Let \(A\in W(X)\), then \(p_{\alpha}(x,A)\leq d(x,y)+p_{\alpha}(y,A)\) for \(x, y\in X\).
Lemma 1.3
[2]
If \(\{x_{\alpha}\}\subset A\), then \(p_{\alpha}(x,B)\leq D_{\alpha}(A,B)\) for each \(A,B\in W(X)\).
Lemma 1.4
[4]
Let \((X,d)\) be a complete metric space and T be a fuzzy mapping from X into \(W(X)\) with \(x_{0}\in X\). Then there exists \(x_{1}\in X\) such that \(\{x_{1}\}\subset Tx_{0}\).
Next, we introduce some classes of functions.
Let Φ be the set of all functions ϕ such that \(\phi :[0,+\infty)\rightarrow[0,+\infty)\) is a continuous and nondecreasing function with \(\phi(t)=0\) if and only if \(t=0\).
Let Ψ be the set of all function φ such that \(\varphi :[0,+\infty)\rightarrow[0,+\infty)\) is lower semicontinuous with \(\varphi(t)=0\) if and only if \(t=0\).
 (1)
\(\psi(t)< t\) for all \(t\in(0,+\infty)\),
 (2)
\(\psi(0)=0\).
2 Fixed point theorems under a Gdistance function
In this section, we will show a fixed theorem for fuzzy mappings under a Gdistance function in complete metric spaces. Inspired by Constantin [5], we give the following definition.
Definition 2.1
 (i)
g is nondecreasing in the 2nd, 3rd, 4th, and 5th variable;
 (ii)
if \(u, v\in[0,\infty)\) are such that \(u\leq g(v,v,u,u+v,0)\) or \(u\leq g(v,u,v,0,u+v)\), then \(u\leq hv\), where \(0< h<1\) is a given constant;
 (iii)
if \(u\in[0,\infty)\) is such that \(u\leq g(u,0,0,u,u)\), then \(u=0\).
Next, we introduce and prove the following results which generalize the results of [2, 3, 6–8].
Theorem 2.1
Proof
Corollary 2.1
Proof
If in Corollary 2.1 we chose \(\phi(t)=t\), we can obtain the following corollary.
Corollary 2.2
Remark 2.1
Remark 2.2
From Remark 2.1 and Remark 2.2 , we can get the following corollary.
Corollary 2.3
If in Corollary 2.3 we chose \(\phi(t)=t\), we can obtain the following corollary.
Corollary 2.4
If in Corollary 2.4 we chose \(L=0\), then we can obtain the following corollary.
Corollary 2.5
(Rashwan and Ahmed [6], Theorem 3.1)
Corollary 2.6
(Park and Jeong [7], Theorem 3.1)
Proof
Since g is a Gdistance function, hence, by Corollary 2.5, we can obtain Corollary 2.6. □
3 Fixed point theorems under a \(G'\)distance function
In this section, we will show some fixed theorems for fuzzy mappings under a \(G'\)distance function in complete metric spaces. However, our results extend and improve some existing results. Inspired by Sedghi et al. [12], we give the following definition.
Definition 3.1
 (i)
g is increasing in each coordinate variable;
 (ii)
\(g(t,t,t,at,bt)\leq t\) for every \(t\in[0,\infty)\), where \(a+b=2\).
Firstly, we prove a fixed point theorem about ‘\(D_{\alpha}(A,B)\)’ under a \(G'\)distance function. The results extend and improve many wellknown results obtained by [14, 27]. Now, we establish and prove the following fixed point theorem.
Theorem 3.1
Proof
Next, we give an example to support our results.
Example 3.1
Corollary 3.1
(Abbas et al. [14], Theorem 3)
Proof
Therefore, g is a \(G'\)distance function. Hence, by Theorem 3.1 we can obtain Corollary 3.1. □
Let \(T_{1}=T_{2}=T\), then we can get the following corollary.
Corollary 3.2
(Abbas and Turkoglu [27], Theorem 2.1)
In Corollary 3.2, let \(\psi(t)=\theta t\), we can get the following corollary.
Corollary 3.3
(Abbas and Turkoglu [27], Theorem 2.2)
If in Corollary 3.3 we let \(L=0\), we can obtain the following corollary.
Corollary 3.4
(Abbas and Turkoglu [27], Theorem 2.3)
Secondly, we prove some fixed point theorems about ‘\(H(A,B)\)’ under a \(G'\)distance function. The results extend and improve some wellknown results obtained by [10, 13]. Now, we establish and prove the following fixed point theorems. But first, we must recall some basic notions in [10] as follows.
Definition 3.2
[10]
Definition 3.3
[10]
A point \(x^{*}\in X\) is called a fixed point of a fuzzy mapping \(F : X\rightarrow K(X)\) if \(F_{x^{*}}(x^{*})\geq F_{x^{*}}(x)\) for any \(x\in X\).
Lemma 3.1
[10]
A point \(x^{*}\in X\) is a fixed point of a fuzzy mapping \(F : X\rightarrow K(X)\) iff \(x^{*}\) is a fixed point of the induced mapping \(\hat{F} : X\rightarrow \operatorname{CB}(X)\).
Lemma 3.3
[11]
For any points \(A,B \in \operatorname{CB}(X)\) with \(H(A,B)<\epsilon\), for each \(a\in A\), there exists an element \(b\in B\) such that \(d(a,b)<\epsilon\).
Now, we present and prove our results.
Theorem 3.2
Then F and G have a common fixed point.
Proof
From (3.15) and (ii) of Definition 3.1, we have \(D(u,\hat{G}u)\leq\psi (D(u,\hat{G}u))< D(u,\hat{G}u)\), which is a contraction and it follows from closedness of \(\hat{G}\) that \(u\in\hat{G}u\). Now, by Lemma 3.1, we insure that u is a common fixed point of F and G. □
Corollary 3.5
(Kamran [13], Theorem 2.5)
Proof
From Corollary 3.5 and the nondecreasing character of ψ, we can get the following corollary.
Corollary 3.6
Next, we give a more general result as follows.
Theorem 3.3
Then \(\{F_{n}\}\) has a common fixed point.
Proof
From (3.23) and (ii) of Definition 3.1, we have \(D(u,\hat{F}_{n}u)\leq \psi(D(u,\hat{F}_{n}u))< D(u,\hat{F}_{n}u)\), which is a contraction and it follows from the closedness of \(\hat{F}_{n}\) that \(u\in\hat{F}_{n}u\) for all \(n=1,2,\ldots \) . Now, by Lemma 3.1, we insure that u is a common fixed point of \(\{F_{n}\}\). □
Corollary 3.7
(Kamran [13], Theorem 2.6)
Proof
From Corollary 3.7 and the nondecreasing character of ψ, we can get the following corollary.
Corollary 3.8
4 Application
In this section, we mainly want to give an application using Theorem 2.1.
Theorem 4.1
Proof
Declarations
Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions. This research has been supported by the National Natural Science Foundation of China (11461043, 11361042 and 11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003 and 20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11346).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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