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# Common fuzzy fixed points for fuzzy mappings

- Akbar Azam
^{1}and - Ismat Beg
^{2}Email author

**2013**:14

https://doi.org/10.1186/1687-1812-2013-14

© Azam and Beg; licensee Springer. 2013

**Received: **17 August 2012

**Accepted: **9 January 2013

**Published: **22 January 2013

## Abstract

Let $(X,d)$ be a metric space and *S*, *T* be mappings from *X* to a set of all fuzzy subsets of *X*. We obtained sufficient conditions for the existence of a common *α*-fuzzy fixed point of *S* and *T*.

## Keywords

- analysis
- fuzzy set
- fuzzy mapping
*α*-fuzzy fixed point

## 1 Introduction

Fixed point theorems play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in mathematics, physics, engineering, medicine and social sciences. The study of fixed point theorems in fuzzy mathematics was instigated by Weiss [1], Butnariu [2], Singh and Talwar [3], Estruch and Vidal [4], Wang *et al.* [5], Mihet [6], Qiu *et al.* [7] and Beg and Abbas [8]. Heilpern [9] introduced the concept of fuzzy contraction mappings and established the fuzzy Banach contraction principle on a complete metric linear spaces with the ${d}_{\mathrm{\infty}}$-metric for fuzzy sets. Azam and Beg [10] proved common fixed point theorems for a pair of fuzzy mappings satisfying Edelstein, Alber and Guerr-Delabriere type contractive conditions in a metric linear space. Azam *et al.* [11] presented some fixed point theorems for fuzzy mappings under Edelstein locally contractive conditions on a compact metric space with the ${d}_{\mathrm{\infty}}$-metric for fuzzy sets. Frigon and Regan [12] generalized the Heilpern theorem under a contractive condition for 1-level sets (*i.e.*, ${[Tx]}_{1}$) of a fuzzy contraction *T* on a complete metric space, where 1-level sets are not assumed to be convex and compact. Amemiya and Takahashi [13] studied some mathematical properties of contractive type set-valued and fuzzy mappings to obtain fixed points of fuzzy mappings by using the concept of *w*-distance (see [13]) in complete metric spaces. Recently, Zhang *et al.* [14] proved some common fixed point theorems for contraction mappings in a new fuzzy metric space.

The aim of this paper is to obtain a common *α*-fuzzy fixed point of a pair of fuzzy mappings *S* and *T* on a complete metric space under a generalized contractive condition for *α*-level sets (*i.e.*, ${[Sx]}_{\alpha}$, ${[Tx]}_{\alpha}$) of *S* and *T* in connection with Hausdorff metric for fuzzy sets. Our result (Theorem 5) generalizes the results proved by Azam and Arshad [[15], Theorem 4], Bose and Sahani [16] and Vijayaraju and Marudai [[17], Theorem 3.1] among others.

## 2 Preliminaries

*X*. For $A,B\in \mathit{CB}(X)$, define

*X*is a function with domain

*X*and values in $[0,1]$. 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

*A*. The

*α*-

*level set*of

*A*is denoted by ${[A]}_{\alpha}$ and is defined as follows:

*B*. Let $\mathcal{F}(X)$ be the collection of all fuzzy sets in a metric space

*X*. For $A,B\in \mathcal{F}(X)$, $A\subset B$ means $A(x)\le B(x)$ for each $x\in X$. We denote the fuzzy set ${\chi}_{\{x\}}$ by $\{x\}$ unless and until it is stated, where ${\chi}_{A}$ is the characteristic function of the crisp set

*A*. If there exists an $\alpha \in [0,1]$ such that ${[A]}_{\alpha},{[B]}_{\alpha}\in \mathit{CB}(X)$, then define

*A*in a metric linear space

*V*is said to be an

*approximate quantity*if and only if ${[A]}_{\alpha}$ is compact and convex in

*V*for each $\alpha \in [0,1]$ and

We denote the collection of all approximate quantities in a metric linear space *X* by $W(X)$. Let *X* be an arbitrary set, *Y* be a metric space. A mapping *T* is called a *fuzzy mapping* if *T* is a mapping from *X* into $\mathcal{F}(Y)$. A fuzzy mapping *T* is a fuzzy subset on $X\times Y$ with a membership function $T(x)(y)$. The function $T(x)(y)$ is the grade of membership of *y* in $T(x)$.

**Definition 1** Let *S*, *T* be fuzzy mappings from *X* into $\mathcal{F}(X)$. A point *z* in *X* is called an *α*-*fuzzy fixed point* of *T* if $z\in {[Tz]}_{\alpha}$. The point *z* is called a common *α*-fuzzy fixed point of *S* and *T* if $z\in {[Sz]}_{\alpha}\cap {[Tz]}_{\alpha}$. When $\alpha =1$, it is called a common fixed point of fuzzy maps.

For the sake of convenience, we first state some known results for subsequent use in the next section.

**Lemma 2** [18]

*Let*$(X,d)$

*be a metric space and*$A,B\in \mathit{CB}(X)$,

*then for each*$a\in A$,

**Lemma 3** [18]

*Let* $(X,d)$ *be a metric space and* $A,B\in \mathit{CB}(X)$, *then for each* $a\in A$, $\beta >0$, *there exists an element* $b\in B$ *such that* $d(a,b)\u2a7dH(A,B)+\beta $.

**Lemma 4** [19]

*Let* *V* *be a metric linear space*, $T:V\to W(V)$ *be a fuzzy mapping and* ${x}_{0}\in V$. *Then there exists* ${x}_{1}\in V$ *such that* $\{{x}_{1}\}\subset T({x}_{0})$.

## 3 Common fuzzy fixed points

In this section, we establish Theorem 5 on the existence of an *α*-fuzzy fixed point of a fuzzy mapping and also obtain a fixed point of fuzzy mappings (see Corollaries 8 and 9) and multivalued mappings (see Corollary 7).

**Theorem 5**

*Let*$(X,d)$

*be a complete metric space and let*

*S*,

*T*

*be fuzzy mappings from*

*X*

*to*$\mathcal{F}(X)$

*satisfying the following conditions*:

*for all* $x,y\in X$, *where* ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$, ${a}_{5}$ *are nonnegative real numbers and* ${\sum}_{i=1}^{5}{a}_{i}<1$ *and either* ${a}_{1}={a}_{2}$ *or* ${a}_{3}={a}_{4}$. *Then there exists* $z\in X$ *such that* $z\in {[Sz]}_{\alpha (z)}\cap {[Tz]}_{\alpha (z)}$.

*Proof*We consider the following three possible cases:

- (i)
${a}_{1}+{a}_{3}+{a}_{5}=0$;

- (ii)
${a}_{2}+{a}_{4}+{a}_{5}=0$;

- (iii)
${a}_{1}+{a}_{3}+{a}_{5}\ne 0$, ${a}_{2}+{a}_{4}+{a}_{5}\ne 0$.

*Case*(i): For $x\in X$, there exists $\alpha (x)\in (0,1]$ such that is a nonempty closed bounded subset of

*X*. Take and similarly . Then by Lemma 2, we obtain

*Case*(iii): Let

*X*. For convenience, we denote $\alpha ({x}_{0})$ by ${\alpha}_{1}$. Let ${x}_{1}\in {[S{x}_{0}]}_{{\alpha}_{1}}$; for this ${x}_{1}$, there exists ${\alpha}_{2}\in (0,1]$ such that ${[T{x}_{1}]}_{{\alpha}_{2}}$ is a nonempty closed bounded subset of

*X*. Since ${a}_{1}+{a}_{3}+{a}_{5}>0$, by Lemma 3, there exists ${x}_{2}\in {[T{x}_{1}]}_{{\alpha}_{2}}$ such that

*X*is complete, there exists $u\in X$ such that ${x}_{n}\to u$. Now,

we can show that $u\in {[Tu]}_{\alpha (u)}$, which implies that $u\in {[Su]}_{\alpha (u)}\cap {[Tu]}_{\alpha (u)}$.

and $\{x\}$ is a crisp set. □

In the following, we furnish an illustrative example to highlight the utility of Theorem 5.

**Example 6**Let $X=\{1,2,3\},\{1\},\{2\},\{3\}$ be crisp sets. Define $d:X\times X\to \mathbb{R}$ as follows:

*X*is not linear, therefore [[12], Theorems 2.1, 2.2] and main results in [9–11, 16] are not applicable to find $1\in {[T1]}_{\frac{3}{4}}$. Now, ${[Sx]}_{\alpha (x)}=\{t:S(x)(t)=\frac{3}{4}\}=\{1\}$ for all $x\in X$ and

if $y=2$. Hence, for ${a}_{1}={a}_{2}={a}_{3}={a}_{5}=0$, ${a}_{4}=\frac{5}{17}$, the conditions of Theorem 5 are satisfied to obtain $1\in {[S1]}_{\frac{3}{4}}\cap {[T1]}_{\frac{3}{4}}$. Since ${a}_{3}\ne {a}_{4}$, the results proved by Vijayaraju and Marudai [[17], Theorem 3.1] and Azam and Arshad [[15], Theorem 4] are also not applicable.

**Corollary 7**

*Let*$(X,d)$

*be a complete metric space and*$F,G:X\to \mathit{CB}(X)$

*be multivalued mappings*.

*Suppose that for all*$x,y\in X$,

*where* ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$, ${a}_{5}$ *are nonnegative real numbers and* ${\sum}_{i=1}^{5}{a}_{i}<1$ *and either* ${a}_{1}={a}_{2}$ *or* ${a}_{3}={a}_{4}$. *Then there exists* $u\in X$ *such that* $u\in Fu\cap Gu$.

*Proof*Consider a mapping $\alpha :X\to (0,1]$ and a pair of fuzzy mappings $S,T:X\to \mathcal{F}(X)$ defined by

□

**Corollary 8** [16]

*Let*$(X,d)$

*be a complete metric linear space and*$S,T:X\to W(X)$

*be fuzzy mappings*,

*and for all*$x,y\in X$,

*where* ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$, ${a}_{5}$ *are nonnegative real numbers and* ${\sum}_{i=1}^{5}{a}_{i}<1$ *and either* ${a}_{1}={a}_{2}$ *or* ${a}_{3}={a}_{4}$. *Then there exists* $u\in X$ *such that* $\{u\}\subset T(u)$, $\{u\}\subset S(u)$.

*Proof*Let $x\in X$, then by Lemma 4 there exists $y\in X$ such that $y\in {[Sx]}_{1}$. Similarly, we can find $z\in X$ such that $z\in {[Tx]}_{1}$. It follows that for each $x\in X$, ${[Sx]}_{\alpha (x)}$, ${[Tx]}_{\alpha (x)}$ are nonempty closed bounded subsets of

*X*. As $\alpha (x)=\alpha (y)=1$, by the definition of a ${d}_{\mathrm{\infty}}$-metric for fuzzy sets, we have

Now, by Theorem 5, we obtain $u\in X$ such that $u\in {[Su]}_{1}\cap {[Tu]}_{1}$, *i.e.*, $\{u\}\subset T(u)$, $\{u\}\subset S(u)$. □

*i.e.*,

**Corollary 9**

*Let*$(X,d)$

*be a complete metric space and*$S,T:X\to \mathcal{F}(X)$

*be fuzzy mappings such that for all*$x\in X$, $\stackrel{\u02c6}{S}(x)$, $\stackrel{\u02c6}{T}(x)$

*nonempty closed bounded subsets of*

*X*

*and*

*for all* $x,y\in X$, *where* ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$, ${a}_{5}$ *are nonnegative real numbers and* ${\sum}_{i=1}^{5}{a}_{i}<1$ *and either* ${a}_{1}={a}_{2}$ *or* ${a}_{3}={a}_{4}$. *Then there exists a point* ${x}^{\ast}\in X$ *such that* $T({x}^{\ast})({x}^{\ast})\u2a7eT({x}^{\ast})(x)$ *and* $S({x}^{\ast})({x}^{\ast})\u2a7eS({x}^{\ast})(x)$ *for all* $x\in X$.

*Proof*By Corollary 7, there exists ${x}^{\ast}\in X$ such that ${x}^{\ast}\in \stackrel{\u02c6}{S}{x}^{\ast}\cap \stackrel{\u02c6}{T}{x}^{\ast}$. Then by [[21], Lemma 1], we obtain

for all $x\in X$. □

**Remark 10**The result proved by Vijayaraju and Marudai [[17], Theorem 3.1] and Azam and Arshad [[15], Theorem 4] is the case when ${a}_{1}\ne {a}_{2}$ and ${a}_{3}={a}_{4}$ in Theorem 5. Corollary 9 also generalizes the results proved by Heilpern [9] and Frigon and Regan [[12], Theorems 2.1, 2.2] for

## 4 Conclusion

In this paper, we obtained fixed point results for fuzzy set-valued mappings under a generalized contractive condition related to the ${d}_{\mathrm{\infty}}$-metric which is useful for computing Hausdorff dimensions. These dimensions help us to understand ${e}^{\mathrm{\infty}}$-space which is used in high energy physics. Our results are also useful in geometric problems arising in high energy physics. This is because events in this case are mostly fuzzy sets.

## Declarations

### Acknowledgements

Authors are grateful to the editor and referees for their valuable suggestions and critical remarks for improving this paper.

## Authors’ Affiliations

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## Copyright

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