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Fixed point theorems for fuzzy mappings in metric spaces with an application
Journal of Inequalities and Applications volume 2015, Article number: 78 (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.
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.
Let \((X,d)\) be a metric space. A fuzzy set A in X is a function with domain X and values in \(I=[0,1]\). If A is a fuzzy set and \(x\in X\), then the function value \(A(x)\) is called the grade of membership of x in X. The αlevel set of A, denoted by \(A_{\alpha}\), is defined as

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

\(A_{0}=\overline{ \{x:\mu(x)>0 \}}\),
where \(\overline{B}\) is the closure of the nonfuzzy set B.
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 \(x\in X\).
For \(A, B\in W(X)\), \(\alpha\in(0,1]\), define
where H denotes the Hausdorff distance. Also
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 \(t=0\).
Let Ω be the set of all functions ψ such that \(\psi :[0,+\infty)\rightarrow[0,+\infty)\) is a nondecreasing function with \(\lim_{n\rightarrow\infty}\psi^{n}(t)=0\) for all \(t\in(0,+\infty)\). If \(\psi\in\Omega\), then ψ is called a Ωmap. It is an easy matter to show that if \(\psi\in\Omega\), then

(1)
\(\psi(t)< t\) for all \(t\in(0,+\infty)\),

(2)
\(\psi(0)=0\).
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
A function g is said to be a Gdistance function if \(g : \left.[0,\infty )\right.^{5}\rightarrow[0,\infty)\) is a continuous function and the following properties hold:

(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
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x,y\in X\), where \(L\geq0\), \(\phi\in\Phi\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
Let \(x_{0}\in X\), by Lemma 1.4 there exists \(x_{1}\in X\) such that \(\{x_{1}\} \in T_{1}x_{0}\), which implies that
which is possible only if \(x_{1}\in(T_{1}x_{0})_{\alpha}\). Since \((T_{2}x_{1})_{\alpha}\) is a nonempty compact subset of X, there exists \(x_{2}\in(T_{2}x_{1})_{\alpha}\) such that
Continuing this process, one obtains a sequence \(\{x_{n}\}\) in X such that \(x_{2n+1}\in(T_{1}x_{2n})_{\alpha}\) and \(x_{2n+2}\in (T_{2}x_{2n+1})_{\alpha}\) for all \(n\geq0\) and \(d(x_{2n+1}, x_{2n+2})\leq D_{\alpha}(T_{1}x_{2n}, T_{2}x_{2n+1})\). By the nondecreasing character of ϕ, (2.1) and (2.2), we have
where
and
According to (2.3), (2.4), (2.5), the nondecreasing character of ϕ and g, and the definition of \(P_{\alpha}\), we have
By the rectangle inequality of d and the above inequality, we have
From (2.6) and the nondecreasing character of ϕ, we get
Similarly, it can be shown that
Therefore, for all n, we get
From (ii) of Definition 2.1, we can conclude that \(d(x_{n},x_{n+1})\leq hd(x_{n1},x_{n})\), where \(0< h<1\). By iteration, we have \(d(x_{n},x_{n+1})\leq hd(x_{n1},x_{n})\leq\cdots \leq h^{n} d(x_{0},x_{1})\). Furthermore, for \(m>n\),
It follows that \(\{x_{n}\}\) is a Cauchy sequence in X. Since X is complete, there exists \(x^{*}\) such that \(\lim_{n\rightarrow\infty }x_{n}=x^{*}\). Next, we show that \(\{x^{*}\}\subset T_{2}x^{*}\), \(i=1,2\). Now, by Lemma 1.2,
From
and (2.1), we have
where
and
In (2.7), (2.8) and (2.9), let \(n\rightarrow\infty\), we get
Again, by the nondecreasing character of ϕ, we have
Using (ii) in Definition 2.1, we can get \(p_{\alpha}(x^{*},T_{2}x^{*})=0\). Therefore, we have \(\{x^{*}\}\subset T_{2}x^{*}\). Similarly, \(\{x^{*}\}\subset T_{1}x^{*}\). □
Corollary 2.1
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\), where \(L\geq0\), \(\phi\in\Phi\), \(\varphi\in \Psi\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
Since
hence, by using Theorem 2.1, there exists a point \(x^{*}\) in X such that \(\{ x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\). □
If in Corollary 2.1 we chose \(\phi(t)=t\), we can obtain the following corollary.
Corollary 2.2
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\), where \(L\geq0\), \(\varphi\in\Psi\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Remark 2.1
By Theorem 2.1 and the nondecreasing character of ϕ, we have
Therefore,
Again, by the nondecreasing character of ϕ, g and the above inequalities, we have
Remark 2.2
In Remark 2.1, let \(\phi(t)=t\), we have
From Remark 2.1 and Remark 2.2 , we can get the following corollary.
Corollary 2.3
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\), where \(L\geq0\). Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
If in Corollary 2.3 we chose \(\phi(t)=t\), we can obtain the following corollary.
Corollary 2.4
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\), where \(L\geq0\). Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
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)
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\). Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Corollary 2.6
(Park and Jeong [7], Theorem 3.1)
Let \((X,d)\) be a complete metric space and g be a Gdistance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W(X)\) are two fuzzy mappings on X and the following inequality holds:
for all \(x,y\in X\) and \(\beta\in[0,1)\). Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
We can consider the function \(g : \left.[0,\infty)\right.^{5}\rightarrow[0,\infty)\) defined by
Since g is a Gdistance function, hence, by Corollary 2.5, we can obtain Corollary 2.6. □
Remark 2.3
(1) From Corollary 2.5, we can get Theorems 3.2 and 3.4 in Park and Jeong [7]; Theorem 3.2 in Arora and Sharma [8]; Corollaries 3.5 and 3.6 in Rashwan and Ahmed [6].
(2) From Corollary 2.6, we can obtain Theorem 3.1 in Estruch and Vidal [3] and Theorem 3.1 in Heilpern [2].
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
A function g is said to be a \(G'\)distance function if \(g : \left.[0,\infty )\right.^{5}\rightarrow[0,\infty)\) is a continuous function and the following properties hold:

(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
Let \((X,d)\) be a complete metric space and g be a \(G'\)distance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x,y\in X\), where \(L\geq0\), \(\psi\in\Omega\), \(\varphi\in \Psi\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
Let \(x_{0}\in X\), by Lemma 1.4 there exists \(x_{1}\in X\) such that \(\{x_{1}\} \in T_{1}x_{0}\), which implies that
which is possible only if \(x_{1}\in(T_{1}x_{0})_{\alpha}\). Since \((T_{2}x_{1})_{\alpha}\) is a nonempty compact subset of X, there exists \(x_{2}\in(T_{2}x_{1})_{\alpha}\) such that
Continuing this process, for all \(n\geq0\), one obtains a sequence \(\{ x_{n}\}\) in X such that \(x_{2n+1}\in(T_{1}x_{2n})_{\alpha}\) and \(x_{2n+2}\in(T_{2}x_{2n+1})_{\alpha}\). Hence, we have \(d(x_{2n+1}, x_{2n+2})\leq D_{\alpha}(T_{1}x_{2n}, T_{2}x_{2n+1})\). By (3.1) and (3.2), we have
where
and
Now, we prove that \(d(x_{2n+1},x_{2n+2})\leq d(x_{2n},x_{2n+1})\). If \(d(x_{2n+1},x_{2n+2})>d(x_{2n},x_{2n+1})\), then
Since g is a \(G'\)distance function, by (ii) of Definition 3.1 we can conclude that
which is a contraction. Hence, we have \(d(x_{2n+1},x_{2n+2})\leq d(x_{2n},x_{2n+1})\). By (3.3) and the nondecreasing character of ψ, we have \(d(x_{2n+1},x_{2n+2})\leq\psi(d(x_{2n+1},x_{2n+2}))\leq \psi(d(x_{2n},x_{2n+1}))\). Similarly, it can be shown that
Therefore, for all n, we can conclude that \(d(x_{n},x_{n+1})\leq\psi(d(x_{n1},x_{n}))\). Therefore, for positive integers m, n (\(n> m\)), we get
By virtue of the condition \(\sum_{k=1}^{\infty}\psi^{n}(t)<\infty\) for each \(t>0\), from (3.4) we can conclude that \(\{x_{n}\}\) is a Cauchy sequence. Since X is complete, there exists \(x^{*}\) such that \(x_{n}\rightarrow x^{*}\in X\). Next, we show that \(\{x^{*}\}\subset T_{2}x^{*}\). Now, by Lemma 1.2 we have
where
From (3.5) and (3.6) and the properties of ψ, let \(n\rightarrow \infty\), we can get
which is a contraction. Hence, we can get \(p_{\alpha}(x^{*},T_{2}x^{*})=0\). So, by Lemma 1.1 we have \(\{x^{*}\}\subset T_{2}x^{*}\). Similarly, \(\{x^{*}\} \subset T_{1}x^{*}\). □
Next, we give an example to support our results.
Example 3.1
Let \(X=[0,1]\) and let \(d : X\times X\rightarrow[0,+\infty)\) be the Euclidean metric. Let \(\alpha\in(0,\frac{1}{2})\) and suppose that \(T_{1},T_{2} : X\rightarrow I^{X}\) defined by
and for any \(z\in(0,1)\),
Also they are defined by
and for any \(z\in(0,1)\),
Then
and
Consequently,
Let \(g(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}\) and let \(k : [0,\infty)\rightarrow [0,\infty)\), \(\psi(t)=qt\), \(q\in(0,1)\). Thus \(\psi(t)< t\) and \(\sum^{\infty} _{n=1}(qt)^{n}<\infty\). \(D_{\alpha}(T_{1}(x),T_{2}(y))=0\leq\psi (d(x,y))\) for each \(x,y\in X\). Hence, all the conditions of Theorem 3.1 are satisfied. There exists a fuzzy point of fuzzy mappings \(T_{1}\) and \(T_{2}\). We can see by the definition of \(T_{1}\) and \(T_{2}\) that \(\{0\}\) is a fixed point.
Corollary 3.1
(Abbas et al. [14], Theorem 3)
Let \((X,d)\) be a complete metric space and g be a \(G'\)distance function. Suppose that \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x,y\in X\), where \(L\geq0\), \(\psi\in\Omega\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
We can consider the function \(g : \left.[0,\infty)\right.^{5}\rightarrow[0,\infty)\) defined by
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)
Let \((X,d)\) be a complete metric space and g be a \(G'\)distance function. Suppose that \(T : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x\in X\), where \(L\geq0\), \(\psi\in\Omega\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset Tx^{*}\).
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)
Let \((X,d)\) be a complete metric space and g be a \(G'\)distance function. Suppose that \(T : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x\in X\), where \(L\geq0\), \(\theta\in(0,1)\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset Tx^{*}\).
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)
Let \((X,d)\) be a complete metric space and g be a \(G'\)distance function. Suppose that \(T : X\rightarrow W_{\alpha}(X)\) are two fuzzy mappings on X satisfying the following conditions:
for all \(x\in X\), where \(L\geq0\), \(\theta\in(0,1)\), and
Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset Tx^{*}\).
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.
Throughout the rest of the paper, we shall use the following notations for a metric space \((X,d)\). \(\operatorname{CB}(X)\) is the set of all nonempty closed bounded subsets of X:
It is clear that H is a metric on \(\operatorname{CB}(X)\) called the Hausdorff metric induced by d [2].
For simplicity we write \(D(A,x)\) instead of \(D(A,\{x\})\):
Definition 3.2
[10]
A fuzzy mapping F is a fuzzy setvalued mapping of X into \(K(X)\). For a fuzzy mapping \(F : X \rightarrow K(X)\) and a mapping \(\wedge: K(X)\rightarrow \operatorname{CB}(X)\), the composition \(\wedge \circ F\) will be denoted by \(\hat{F}\). In other words,
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 3.2
[10]
For any points \(w,a \in X\), \(D(a,B)\leq d(a,w) + D(w,B)\).
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
Let \((X,d)\) be a complete metric space. Let \(F, G : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}, \hat{G} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by F, G according to (3.7). Let \(\psi\in\Omega \) and g be a \(G'\)distance function. Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
where
Then F and G have a common fixed point.
Proof
Let \(x_{0}\) be a point in X. Since \(\hat{F}x_{0}\) is nonempty, there exists a point \(x_{1}\in\hat{F}x_{0}\). Let
Then, by inequalities (3.8) and (3.9), we have \(H(\hat{F}x_{0},\hat{G}x_{1})<\epsilon\). Now, using Lemma 3.3 and the properties of ψ, we obtain \(x_{2}\in X\) such that \(x_{2}\in\hat {G}x_{1}\) and
where
Now, we prove that \(d(x_{1},x_{2})\leq d(x_{0},x_{1})\). If \(d(x_{1},x_{2})> d(x_{0},x_{1})\), by (3.10) we get
Hence,
which is a contraction. Therefore,
Thus by inequalities (3.10), (3.11) and (3.12), we have \(d(x_{1},x_{2})<\psi (d(x_{0},x_{1}))\). Continuing in the fashion, we produce a sequence \(\{x_{n}\} \) of points of X such that for all \(n\geq1\),
and
Consequently,
for all \(n\geq1\). Therefore, for positive integers m, n (\(n> m\)), we get
By virtue of the condition \(\sum_{k=1}^{\infty}\psi^{n}(t)<\infty\) for each \(t>0\), from (3.13), we can conclude that \(\{x_{n}\}\) is a Cauchy sequence. Let \(x_{n}\rightarrow u\in X\). Using Lemma 3.2, we have
In (3.14), let \(n\rightarrow\infty\) and using the properties of ψ, we have
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)
Let \((X,d)\) be a complete metric space. Let \(F, G : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}, \hat{G} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by F, G according to (3.7). Let \(\psi\in\Omega \). Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
then F and G have a common fixed point.
Proof
We can consider the function \(g : \left.[0,\infty)\right.^{5}\rightarrow[0,\infty)\) defined by
Let \(x_{1}=t\), \(x_{2}=t\), \(x_{3}=t\), \(x_{4}=at\), \(x_{5}=bt\) and \(a+b=2\). Therefore,
Hence, g is a \(G'\)distance function. By Theorem 3.2, we can get Corollary 3.5. □
From Corollary 3.5 and the nondecreasing character of ψ, we can get the following corollary.
Corollary 3.6
Let \((X,d)\) be a complete metric space. Let \(F, G : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}, \hat{G} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by F, G according to (3.7). Let \(\psi\in\Omega \). Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
then F and G have a common fixed point.
Next, we give a more general result as follows.
Theorem 3.3
Let \((X,d)\) be a complete metric space. Let \(F_{i}, F_{j} : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}_{i}, \hat{F}_{j} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by \(F_{i}\), \(F_{j}\) according to (3.7). Let \(\psi\in \Omega\) and g be a \(G'\)distance function. Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
where
Then \(\{F_{n}\}\) has a common fixed point.
Proof
Let \(x_{0}\) be a point in X. Since \(\hat{F}_{1}x_{0}\) is nonempty, there exists a point \(x_{1}\in\hat{F}_{1}x_{0}\). Let
Then by inequality (3.16), we have \(H(\hat{F}_{1}x_{0},\hat{F}_{2}x_{1})<\epsilon\). Now, using Lemma 3.3 and the properties of ψ, we obtain \(x_{2}\in X\) such that \(x_{2}\in\hat{F}_{2}x_{1}\) and
where
Now, we prove that \(d(x_{1},x_{2})\leq d(x_{0},x_{1})\). If \(d(x_{1},x_{2})> d(x_{0},x_{1})\), by (3.18) we get
Hence
which is a contraction. Therefore,
Thus, by inequalities (3.18), (3.19) and (3.20), we have \(d(x_{1},x_{2})<\psi(d(x_{0},x_{1}))\). Continuing in the fashion, we produce a sequence \(\{x_{n}\}\) of points of X such that for all \(n\geq1\),
and
Consequently,
for all \(n\geq1\). Therefore, for positive integers m, n (\(n> m\)), we get
By virtue of the condition \(\sum_{k=1}^{\infty}\psi^{n}(t)<\infty\) for each \(t>0\), from (3.21), we can conclude that \(\{x_{n}\}\) is a Cauchy sequence. Let \(x_{n}\rightarrow u\in X\). Using Lemma 3.2 we have
In (3.22), let \(n\rightarrow\infty\) and using the properties of ψ, we have
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)
Let \((X,d)\) be a complete metric space. Let \(F_{i}, F_{j} : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}_{i}, \hat{F}_{j} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by \(F_{i}\), \(F_{j}\) according to (3.7). Let \(\psi\in \Omega\) and g be a \(G'\)distance function. Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
then \(\{F_{n}\}\) has a common fixed point.
Proof
We can consider the function \(g : \left.[0,\infty)\right.^{5}\rightarrow[0,\infty)\) defined by
Let \(x_{1}=t\), \(x_{2}=t\), \(x_{3}=t\), \(x_{4}=at\), \(x_{5}=bt\) and \(a+b=2\). Therefore,
Hence, g is a \(G'\)distance function. By Theorem 3.3, we can get Corollary 3.7. □
From Corollary 3.7 and the nondecreasing character of ψ, we can get the following corollary.
Corollary 3.8
Let \((X,d)\) be a complete metric space. Let \(F_{i}, F_{j} : X \rightarrow K(X)\) be two fuzzy mappings and \(\hat{F}_{i}, \hat{F}_{j} : X\rightarrow \operatorname{CB}(X)\) be mappings induced by \(F_{i}\), \(F_{j}\) according to (3.7). Let \(\psi\in \Omega\). Suppose that for any \(x, y\ (x\neq y)\in X \), the following holds:
then \(\{F_{n}\}\) has a common fixed point.
Application
In this section, we mainly want to give an application using Theorem 2.1.
Theorem 4.1
Let \((X,d)\) be a complete metric linear space, and let \(T_{1}, T_{2} : X\rightarrow W_{\alpha}(X)\) be two fuzzy mappings on X satisfying the following conditions:
where \(\alpha_{i}>0\) (\(i=1,2,3,4,5\)), \(\alpha_{1}+\alpha_{2}+\alpha_{4}+\alpha _{5}<1\), \(\alpha_{1}+\alpha_{3}<1\). Then there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\).
Proof
We can consider the function \(g : \left.[0,\infty)\right.^{5}\rightarrow[0,\infty)\) defined by
Next we prove that g is a Gdistance function. Firstly, obviously, g is nondecreasing in the 2nd, 3rd, 4th, and 5th variable. Secondly,
If \(u\leq v\), from (4.2) we can get
Hence, there exists \(h=\sqrt{\alpha_{1}+\alpha_{2}+\alpha_{4}+\alpha _{5}}<1\) such that \(u\leq hv\), where \(0< h<1\). If \(u>v\), from (4.2) we can get
which is a contraction. Therefore, (ii) of Definition 2.1 holds. Thirdly, since \(u\leq g(u,0,0, u,u)=[(\alpha_{1}+\alpha_{3})u^{2}]^{\frac {1}{2}}=\sqrt{\alpha_{1}+\alpha_{3}}u< u\), which is a contraction. Hence, \(u=0\). Therefore, Theorem 4.1 satisfies all the conditions of Theorem 2.1. Hence, there exists a point \(x^{*}\) in X such that \(\{x^{*}\}\subset T_{1}x^{*}\) and \(\{x^{*}\}\subset T_{2}x^{*}\). □
Remark 4.1
In Theorem 4.1, we can conclude that for \(\alpha_{i}>0\) (\(i=1,2,3,4,5\)), \(\alpha_{1}+\alpha_{2}+\alpha_{4}+\alpha_{5}<1\), \(\alpha_{1}+\alpha_{3}<1\), Theorem 2.1 improves Theorem 3.1 in [9].
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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).
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Chen, J., Huang, X. Fixed point theorems for fuzzy mappings in metric spaces with an application. J Inequal Appl 2015, 78 (2015). https://doi.org/10.1186/s1366001505992
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DOI: https://doi.org/10.1186/s1366001505992
MSC
 47H10
 54H25
Keywords
 fixed point
 fuzzymapping
 fuzzy set
 Gdistance function
 \(G'\)distance function