# On locally contractive fuzzy set-valued mappings

## Abstract

We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.

MSC:46S40, 47H10, 54H25.

## 1 Introduction

The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in [1] the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [210] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.

## 2 Preliminaries

Given a metric space $\left(X,d\right)$, denote by $CB\left(X\right)$ the family of all nonempty closed bounded subsets of $\left(X,d\right)$. As usual, for $\zeta \in X$ and $A\in CB\left(X\right)$, we define

$d\left(\zeta ,A\right)=\underset{a\in A}{inf}d\left(\zeta ,a\right).$

Then the Hausdorff metric H on $CB\left(X\right)$ induced by d is defined as

$H\left(A,B\right)=max\left\{\underset{a\in A}{sup}d\left(a,B\right),\underset{b\in B}{sup}d\left(A,b\right)\right\},$

for all $A,B\in CB\left(X\right)$.

A fuzzy set in $\left(X,d\right)$ is a function with domain X and values in $I=\left[0,1\right]$. ${I}^{X}$ denotes the collection of all fuzzy sets in X. If A is a fuzzy set and $\zeta \in X$, then the function value $A\left(\zeta \right)$ is called the grade of membership of ζ in A. The α-level set of a fuzzy set A is denoted by ${A}_{\alpha }$, and it is defined as follows:

According to Heilpern [1], a fuzzy set A in a metric linear space $\left(X,d\right)$ is said to be an approximate quantity if ${A}_{\alpha }$ is compact and convex in X, for each $\alpha \in \left(0,1\right]$, and ${sup}_{\zeta \in X}A\left(\zeta \right)=1$. The family of all approximate quantities of the metric linear space $\left(X,d\right)$ is denoted by $W\left(X\right)$.

Now, for $A,B\in W\left(X\right)$ and $\alpha \in \left[0,1\right]$, define

${D}_{\alpha }\left(A,B\right)=H\left({A}_{\alpha },{B}_{\alpha }\right),$

and

${d}_{\mathrm{\infty }}\left(A,B\right)=\underset{\alpha \in \left[0,1\right]}{sup}{D}_{\alpha }\left({A}_{\alpha },{B}_{\alpha }\right).$

It is well known that ${d}_{\mathrm{\infty }}$ is a metric on $W\left(X\right)$.

In case that $\left(X,d\right)$ is a (non-necessarily linear) metric space, we also define

${D}_{\alpha }\left(A,B\right)=H\left({A}_{\alpha },{B}_{\alpha }\right),$

whenever $A,B\in {I}^{X}$ and ${A}_{\alpha },{B}_{\alpha }\in CB\left(X\right)$, $\alpha \in \left[0,1\right]$.

In the sequel the letter will denote the set of positive integer numbers.

The following well-known properties on the Hausdorff metric (see e.g. [11]) will be useful in the next section.

Lemma 2.1 Let $\left(X,d\right)$ be a metric space and let $A,B\in CB\left(X\right)$ with $H\left(A,B\right), $r>0$. If $a\in A$, then there exists $b\in B$ such that $d\left(a,b\right).

Lemma 2.2 Let $\left(X,d\right)$ be a metric space and let ${\left\{{A}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ be a sequence in $CB\left(X\right)$ such that ${lim}_{n\to \mathrm{\infty }}H\left({A}_{n},A\right)=0$, for some $A\in CB\left(X\right)$. If ${\xi }_{n}\in {A}_{n}$, for all $n\in \mathbb{N}$, and $d\left({\xi }_{n},\xi \right)\to 0$, then $\xi \in A$.

Now, let X be an arbitrary set and let Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into ${I}^{Y}$. In fact, a fuzzy mapping T is a fuzzy subset on $X×Y$ with membership function $T\left(\zeta \right)$. The value $T\left(\zeta \right)\left(\xi \right)$ is the grade of membership of ξ in $T\left(\zeta \right)$.

If $\left(X,d\right)$ is a metric space and T is a (fuzzy) mapping from X into ${I}^{X}$, we say that $\xi \in X$ is a fixed point of T if $\xi \in T{\left(\xi \right)}_{1}$.

We conclude this section with the notion of contractiveness that will be used in our main result.

Definition 2.3 (compare [12])

Let $\epsilon \in \left(0,\mathrm{\infty }\right]$. A function $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition (i.e., $lim{sup}_{r\to {t}^{+}}\psi \left(r\right)<1$, for all $t\in \left[0,\epsilon \right)$).

Clearly, if $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is a nondecreasing function or a nonincreasing function, then it is a MT-function. So the set of MT-functions is a rich class.

## 3 Fixed points of fuzzy mappings

Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein [13], Beg and Azam [14], Holmes [15], Hu [11], Hu and Rosen [16], Ko and Tasi [17], Kuhfitting [18] and Nadler [19].

Heilpern [1] established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam et al. [4, 5], and Lee and Cho [10] further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem 3.1 below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature.

Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, and $\lambda \in \left(0,1\right)$. A metric space $\left(X,d\right)$ is said to be ε-chainable if given $\zeta ,\xi \in X$, there exists an ε-chain from ζ to ξ (i.e., a finite set of points $\zeta ={\zeta }_{0}$, ${\zeta }_{1},{\zeta }_{2},\dots ,{\zeta }_{m}=\xi$ such that $d\left({\zeta }_{j-1},{\zeta }_{j}\right)<\epsilon$, for all $j=1,2,\dots ,m$). A mapping $T:X\to X$ is called an $\left(\epsilon ,\lambda \right)$ uniformly locally contractive mapping if $\zeta ,\zeta \in X$ and $0, implies $d\left(T\zeta ,T\xi \right)\le \lambda d\left(\zeta ,\xi \right)$. A mapping $T:X\to W\left(X\right)$ is called an $\left(\epsilon ,\lambda \right)$ uniformly locally contractive fuzzy mapping if $\zeta ,\xi \in X$ and $0, imply ${d}_{\mathrm{\infty }}\left(T\left(\zeta \right),T\left(\xi \right)\right)\le \lambda d\left(\zeta ,\xi \right)$. We remark that a globally contractive mapping can be regarded as an $\left(\mathrm{\infty },\lambda \right)$ uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.

Theorem 3.1 Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into ${I}^{X}$ such that, for each $\zeta \in X$ and $i\in \mathbb{N}$, ${T}_{i}{\left(\zeta \right)}_{1}\in CB\left(X\right)$. If

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0
(1)

for all $i,j\in \mathbb{N}$, where $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is a MT-function, then the sequence ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ has a common fixed point, i.e., there is ${\xi }^{\ast }\in X$ such that ${\xi }^{\ast }\in {T}_{i}{\left({\xi }^{\ast }\right)}_{1}$, for all $i\in \mathbb{N}$.

Proof Let ${\xi }_{0}$ be an arbitrary, but fixed element of X. Find ${\xi }_{1}\in X$ such that ${\xi }_{1}\in {T}_{1}{\left({\xi }_{0}\right)}_{1}$. Let

${\xi }_{0}={\zeta }_{\left(1,0\right)},\phantom{\rule{2em}{0ex}}{\zeta }_{\left(1,1\right)},{\zeta }_{\left(1,2\right)},\dots ,{\zeta }_{\left(1,m\right)}={\xi }_{1}\in {T}_{1}{\left({\xi }_{0}\right)}_{1}$

be an arbitrary ε-chain from ${\xi }_{0}$ to ${\xi }_{1}$. (We suppose, without loss of generality, that ${\zeta }_{\left(1,i\right)}\ne {\zeta }_{\left(1,j\right)}$, for each $i,j\in \left\{0,1,2,\dots ,m\right\}$ with $i\ne j$.)

Since $0, we deduce that

$\begin{array}{rcl}{D}_{1}\left({T}_{1}\left({\zeta }_{\left(1,0\right)}\right),{T}_{2}\left({\zeta }_{\left(1,1\right)}\right)\right)& \le & \psi \left(d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\right)d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\\ <& \sqrt{\psi \left(d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\right)}d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\\ <& d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)<\epsilon .\end{array}$

Rename ${\xi }_{1}$ as ${\zeta }_{\left(2,0\right)}$. Since ${\zeta }_{\left(2,0\right)}\in {T}_{1}{\left({\zeta }_{\left(1,0\right)}\right)}_{1}$, using Lemma 2.1 we find ${\zeta }_{\left(2,1\right)}\in {T}_{2}{\left({\zeta }_{\left(1,1\right)}\right)}_{1}$ such that

$\begin{array}{rcl}d\left({\zeta }_{\left(2,0\right)},{\zeta }_{\left(2,1\right)}\right)& <& \sqrt{\psi \left(d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\right)}d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)\\ <& d\left({\zeta }_{\left(1,0\right)},{\zeta }_{\left(1,1\right)}\right)<\epsilon .\end{array}$

Similarly we may choose an element ${\zeta }_{\left(2,2\right)}\in {T}_{2}{\left({\zeta }_{\left(1,2\right)}\right)}_{1}$ such that

$\begin{array}{rcl}d\left({\zeta }_{\left(2,1\right)},{\zeta }_{\left(2,2\right)}\right)& <& \sqrt{\psi \left(d\left({\zeta }_{\left(1,1\right)},{\zeta }_{\left(1,2\right)}\right)\right)}d\left({\zeta }_{\left(1,1\right)},{\zeta }_{\left(1,2\right)}\right)\\ <& d\left({\zeta }_{\left(1,1\right)},{\zeta }_{\left(1,2\right)}\right)<\epsilon .\end{array}$

Thus we obtain a set $\left\{{\zeta }_{\left(2,0\right)},{\zeta }_{\left(2,1\right)},{\zeta }_{\left(2,2\right)},\dots ,{\zeta }_{\left(2,m\right)}\right\}$ of $m+1$ points of X such that ${\zeta }_{\left(2,0\right)}\in {T}_{1}{\left({\zeta }_{\left(1,0\right)}\right)}_{1}$ and ${\zeta }_{\left(2,j\right)}\in {T}_{2}{\left({\zeta }_{\left(1,j\right)}\right)}_{1}$, for $j=1,2,\dots ,m$, with

$\begin{array}{rcl}d\left({\zeta }_{\left(2,j\right)},{\zeta }_{\left(2,j+1\right)}\right)& <& \sqrt{\psi \left(d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right)\right)}d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right)\\ <& d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right)<\epsilon ,\end{array}$

for $j=0,1,2,\dots ,m-1$.

Let ${\zeta }_{\left(2,m\right)}={\xi }_{2}$. Thus the set of points ${\xi }_{1}={\zeta }_{\left(2,0\right)},{\zeta }_{\left(2,1\right)},{\zeta }_{\left(2,2\right)},\dots ,{\zeta }_{\left(2,m\right)}={\xi }_{2}\in {T}_{2}{\left({\xi }_{1}\right)}_{1}$ is an ε-chain from ${\xi }_{0}$ to ${\xi }_{1}$. Rename ${\xi }_{2}$ as ${\zeta }_{\left(3,0\right)}$. Then by the same procedure we obtain an ε-chain

${\xi }_{2}={\zeta }_{\left(3,0\right)},\phantom{\rule{2em}{0ex}}{\zeta }_{\left(3,1\right)},{\zeta }_{\left(3,2\right)},\dots ,{\zeta }_{\left(3,m\right)}={\xi }_{3}\in {T}_{3}{\left({\xi }_{2}\right)}_{1}$

from ${\xi }_{2}$ to ${\xi }_{3}$. Inductively, we obtain

${\xi }_{n}={\zeta }_{\left(n+1,0\right)},\phantom{\rule{2em}{0ex}}{\zeta }_{\left(n+1,1\right)},{\zeta }_{\left(n+1,2\right)},\dots ,{\zeta }_{\left(n+1,m\right)}={\xi }_{n+1}\in {T}_{n+1}{\left({\xi }_{n}\right)}_{1}$

with

$\begin{array}{rcl}d\left({\zeta }_{\left(n+1,j\right)},{\zeta }_{\left(n+1,j+1\right)}\right)& <& \sqrt{\psi \left(d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)\right)}d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)\\ <& d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)<\epsilon ,\end{array}$
(2)

for $j=0,1,2,\dots ,m-1$.

Consequently, we construct a sequence ${\left\{{\xi }_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ of points of X with

$\begin{array}{c}{\xi }_{1}={\zeta }_{\left(1,m\right)}={\zeta }_{\left(2,0\right)}\in {T}_{1}{\left({\xi }_{0}\right)}_{1},\hfill \\ {\xi }_{2}={\zeta }_{\left(2,m\right)}={\zeta }_{\left(3,0\right)}\in {T}_{2}{\left({\xi }_{1}\right)}_{1},\hfill \\ {\xi }_{3}={\zeta }_{\left(3,m\right)}={\zeta }_{\left(4,0\right)}\in {T}_{3}{\left({\xi }_{2}\right)}_{1},\hfill \\ ⋮\hfill \\ {\xi }_{n+1}={\zeta }_{\left(n+1,m\right)}={\zeta }_{\left(n+2,0\right)}\in {T}_{n+1}{\left({\xi }_{n}\right)}_{1},\hfill \end{array}$

for all $n\in \mathbb{N}$.

For each $j\in \left\{0,1,2,\dots ,m-1\right\}$, we deduce from (2) that ${\left\{d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)\right\}}_{n=1}^{\mathrm{\infty }}$ is a decreasing sequence of non-negative real numbers and therefore there exists ${l}_{j}\ge 0$ such that

$\underset{n\to \mathrm{\infty }}{lim}d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)={l}_{j}.$

By assumption, $lim{sup}_{t\to {l}_{j}^{+}}\psi \left(t\right)<1$, so there exists ${n}_{j}\in \mathbb{N}$ such that $\psi \left(d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)\right), for all $n\ge {n}_{j}$ where $lim{sup}_{t\to {l}_{j}^{+}}\psi \left(t\right).

Now put

${M}_{j}=max\left\{\underset{i=1,\dots ,{n}_{j}}{max}\sqrt{\psi \left(d\left({\zeta }_{\left(i,j\right)},{\zeta }_{\left(i,j+1\right)}\right)\right)},\sqrt{s\left({l}_{j}\right)}\right\}.$

Then, for every $n>{n}_{j}$, we obtain

$\begin{array}{rcl}d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)& <& \sqrt{\psi \left(d\left({\zeta }_{\left(n-1,j\right)},{\zeta }_{\left(n-1,j+1\right)}\right)\right)}d\left({\zeta }_{\left(n-1,j\right)},{\zeta }_{\left(n-1,j+1\right)}\right)\\ <& \sqrt{s\left({l}_{j}\right)}d\left({\zeta }_{\left(n-1,j\right)},{\zeta }_{\left(n-1,j+1\right)}\right)\\ \le & {M}_{j}d\left({\zeta }_{\left(n-1,j\right)},{\zeta }_{\left(n-1,j+1\right)}\right)\\ \le & {\left({M}_{j}\right)}^{2}d\left({\zeta }_{\left(n-2,j\right)},{\zeta }_{\left(n-2,j+1\right)}\right)\\ \le & \cdots \\ \le & {\left({M}_{j}\right)}^{n-1}d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right).\end{array}$

Putting $N=max\left\{{n}_{j}:j=0,1,2,\dots ,m-1\right\}$, we have

$\begin{array}{rcl}d\left({\xi }_{n-1},{\xi }_{n}\right)& =& d\left({\zeta }_{\left(n,0\right)},{\zeta }_{\left(n,m\right)}\right)\le \sum _{j=0}^{m-1}d\left({\zeta }_{\left(n,j\right)},{\zeta }_{\left(n,j+1\right)}\right)\\ <& \sum _{j=0}^{m-1}{\left({M}_{j}\right)}^{n-1}d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right),\end{array}$

for all $n>N+1$. Hence

$\begin{array}{rcl}d\left({\xi }_{n},{\xi }_{p}\right)& \le & d\left({\xi }_{n},{\xi }_{n+1}\right)+d\left({\xi }_{n+1},{\xi }_{n+2}\right)+\cdots +d\left({\xi }_{p-1},{\xi }_{p}\right)\\ <& \sum _{j=0}^{m-1}{\left({M}_{j}\right)}^{n}d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right)+\cdots +\sum _{j=0}^{m-1}{\left({M}_{j}\right)}^{p-1}d\left({\zeta }_{\left(1,j\right)},{\zeta }_{\left(1,j+1\right)}\right),\end{array}$

whenever $p>n>N+1$.

Since ${M}_{j}<1$, for all $j\in \left\{0,1,2,\dots ,m-1\right\}$, it follows that ${\left\{{\xi }_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ is a Cauchy sequence. Since $\left(X,d\right)$ is complete, there is ${\xi }^{\ast }\in X$ such that ${\xi }_{n}\to {\xi }^{\ast }$. So for each $\delta \in \left(0,\epsilon \right]$ there is ${M}_{\delta }\in \mathbb{N}$ such that $n>{M}_{\delta }$ implies $d\left({\xi }_{n},{\xi }^{\ast }\right)<\delta$. This in view of inequality (1) implies ${D}_{1}\left({T}_{n+1}\left({\xi }_{n}\right),{T}_{i}\left({\xi }^{\ast }\right)\right)<\delta$, for all $i\in \mathbb{N}$. Consequently, $H\left({T}_{n+1}{\left({\xi }_{n}\right)}_{1},{T}_{i}{\left({\xi }^{\ast }\right)}_{1}\right)\to 0$. Since ${\xi }_{n+1}\in {T}_{n+1}{\left({\xi }_{n}\right)}_{1}$ with $d\left({\xi }_{n+1},{\xi }^{\ast }\right)\to 0$, we deduce from Lemma 2.2 that ${\xi }^{\ast }\in {T}_{i}{\left({\xi }^{\ast }\right)}_{1}$, for all $i\in \mathbb{N}$. This completes the proof. □

Corollary 3.2 Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into ${I}^{X}$ such that, for each $\zeta \in X$ and $i\in \mathbb{N}$, ${T}_{i}{\left(\zeta \right)}_{1}\in CB\left(X\right)$. If

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

for all $i,j\in \mathbb{N}$, where $\lambda \in \left(0,1\right)$, then the sequence ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ has a common fixed point.

Proof Apply Theorem 3.1 when ψ is the MT-function defined as $\psi \left(t\right)=\lambda$, for all $t\in \left[0,\epsilon \right)$. □

Corollary 3.3 Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric linear space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into $W\left(X\right)$ satisfying the following condition:

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

for all $i,j\in \mathbb{N}$, where $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is a MT-function. Then the sequence ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ has a common fixed point.

Proof Since $W\left(X\right)\subseteq CB\left(X\right)$ and ${D}_{1}\left({T}_{i}\left(\zeta \right),{T}_{j}\left(\xi \right)\right)\le {d}_{\mathrm{\infty }}\left({T}_{i}\left(\zeta \right),{T}_{j}\left(\xi \right)\right)$, for all $i,j\in \mathbb{N}$, the result follows immediately from Theorem 3.1. □

Corollary 3.4 Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric linear space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into $W\left(X\right)$ satisfying the following condition:

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

for all $i,j\in \mathbb{N}$, where $\lambda \in \left(0,1\right)$. Then the sequence ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ has a common fixed point.

Corollary 3.5 [4]

Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric linear space and ${T}_{1}$, ${T}_{2}$, two fuzzy mappings from X into $W\left(X\right)$ satisfying the following condition:

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

for $i,j=1,2$, where $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is a MT-function. Then ${T}_{1}$ and ${T}_{2}$ have a common fixed point.

Corollary 3.6 [4, 11]

Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric linear space and T: $X\to W\left(X\right)$ an $\left(\epsilon ,\lambda \right)$ uniformly locally contractive fuzzy mapping. Then T has a fixed point.

Corollary 3.7 Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric space and S be a multivalued mapping from X into $CB\left(X\right)$ satisfying the following condition:

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

where $\psi :\left[0,\epsilon \right)\to \left[0,1\right)$ is a MT-function. Then S has a fixed point.

Proof Define a fuzzy mapping T from X into ${I}^{X}$ as $T\left(\xi \right)\left(t\right)=1$ if $t\in S\left(\xi \right)$ and $T\left(\xi \right)\left(t\right)=0$, otherwise. Then $T{\left(\xi \right)}_{1}=S\left(\xi \right)$, for all $\xi \in X$, so $T{\left(\xi \right)}_{1}\in CB\left(X\right)$, for all $\xi \in X$. Since

${D}_{1}\left(T\left(\zeta \right),T\left(\xi \right)\right)=H\left(T{\left(\zeta \right)}_{1},T{\left(\xi \right)}_{1}\right)=H\left(S\left(\zeta \right),S\left(\xi \right)\right),$

for all $\zeta ,\xi \in X$, we deduce that condition (1) of Theorem 3.1 is satisfied for T. Hence T has a fixed point ${\xi }^{\ast }$, i.e., ${\xi }^{\ast }\in T{\left({\xi }^{\ast }\right)}_{1}$. We conclude that ${\xi }^{\ast }\in S\left({\xi }^{\ast }\right)$. The proof is complete. □

Corollary 3.8 [13]

Let $\epsilon \in \left(0,\mathrm{\infty }\right]$, $\left(X,d\right)$ a complete ε-chainable metric space and S be a multivalued mapping from X into $CB\left(X\right)$ satisfying the following condition:

$\zeta ,\xi \in X,\phantom{\rule{1em}{0ex}}0

where $\lambda \in \left(0,1\right)$. Then S has a fixed point.

Corollary 3.9 ([20, 21], see also [9, 13])

Let $\left(X,d\right)$ be a complete metric space, S a multivalued mapping from X into $CB\left(X\right)$ and $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ a MT-function such that

$H\left(S\zeta ,S\xi \right)\le \psi \left(d\left(\zeta ,\xi \right)\right)d\left(\zeta ,\xi \right),$

for all $\zeta ,\xi \in X$. Then S has a fixed point in X.

Proof Apply Corollary 3.8 with $\epsilon =\mathrm{\infty }$. □

We conclude the paper with two examples to support Theorem 3.1 and Corollary 3.2.

Example 3.10 Let $\left(X,d\right)$ be the compact, and thus complete, metric space such that $X=\left[0,1\right]$, and $d\left(x,y\right)=|x-y|$, for all $x,y\in X$. Let λ be a constant such that $\lambda \in \left[1/14,1\right)$ and let ${\left\{{T}_{k}\right\}}_{k=1}^{\mathrm{\infty }}$ be the sequence of fuzzy mappings defined from X into ${I}^{X}$ as follows:

For each $x,y\in X$ with $x\ne y$, and $i,j\in \mathbb{N}$ we have

${D}_{1}\left({T}_{i}\left(x\right),{T}_{j}\left(y\right)\right)=H\left({T}_{i}{\left(x\right)}_{1},{T}_{j}{\left(y\right)}_{1}\right)=H\left(\left[0,x/14\right],\left[0,y/14\right]\right)=\frac{1}{14}|x-y|.$

Hence, for $\psi \left(t\right)=\lambda$, the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any $\epsilon \in \left(0,\mathrm{\infty }\right]$, whereas X is not linear. Therefore all previous relevant fixed point results Corollaries 3.3-3.6 on metric linear spaces are not applicable.

Example 3.11 Let $\left(X,d\right)$ be the complete metric space such that $X=\left[0,\mathrm{\infty }\right)$, $d\left(x,x\right)=0$, for all $x\in X$, and $d\left(x,y\right)=max\left\{x,y\right\}$ whenever $x\ne y$ (in the sequel we shall write $x\vee y$ instead of $max\left\{x,y\right\}$).

Note that a sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ is a Cauchy sequence in $\left(X,d\right)$ if and only if $d\left({x}_{n},0\right)\to 0$. Moreover, $x=0$ is the only non-isolated point of X for the topology induced by d.

Let $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ be the MT-function defined as

and let ${\left\{{T}_{k}\right\}}_{k=1}^{\mathrm{\infty }}$ be the sequence of fuzzy mappings defined from X into ${I}^{X}$ as follows:

Observe that, for $0\le x\le 1$,

${T}_{k}{\left(x\right)}_{1}=\left[\frac{x}{4k},\frac{x}{2k}\right],$

and, for $x>1$,

${T}_{k}{\left(x\right)}_{1}=\left[\frac{x}{2k},\frac{{x}^{2}}{k\left(1+x\right)}\right).$

Therefore ${T}_{k}{\left(x\right)}_{1}\in CB\left(X\right)$, for all $x\in X$ and $k\in \mathbb{N}$ (recall that each $x\ne 0$ is an isolated point for the induced topology, so every bounded interval belongs to $CB\left(X\right)$).

We show that condition (1) of Theorem 3.1 is satisfied for $\epsilon =\mathrm{\infty }$ and ψ as defined above. Indeed, let $x,y\in X$ with $x\ne y$ and $j,k\in \mathbb{N}$. Assume without loss of generality that $x>y$.

If $x,y>1$, for each $b\in {T}_{j}{\left(y\right)}_{1}$, we obtain

$d\left({T}_{k}{\left(x\right)}_{1},b\right)=\underset{a\in {T}_{k}{\left(x\right)}_{1}}{inf}\left(a\vee b\right)\le \frac{{x}^{2}}{k\left(1+x\right)}\vee b\le \frac{{x}^{2}}{k\left(1+x\right)}\vee \frac{{y}^{2}}{j\left(1+y\right)}.$

Similarly, for each $a\in {T}_{k}{\left(x\right)}_{1}$, we obtain

$d\left(a,{T}_{j}{\left(y\right)}_{1}\right)\le \frac{{x}^{2}}{k\left(1+x\right)}\vee \frac{{y}^{2}}{j\left(1+y\right)}.$

Consequently

$\begin{array}{rcl}{D}_{1}\left({T}_{k}\left(x\right),{T}_{j}\left(y\right)\right)& =& H\left({T}_{k}{\left(x\right)}_{1},{T}_{j}{\left(y\right)}_{1}\right)\le \frac{{x}^{2}}{k\left(1+x\right)}\vee \frac{{y}^{2}}{j\left(1+y\right)}\\ \le & \frac{{\left(x\vee y\right)}^{2}}{1+\left(x\vee y\right)}=\frac{d\left(x,y\right)}{1+d\left(x,y\right)}d\left(x,y\right)\\ =& \psi \left(d\left(x,y\right)\right)d\left(x,y\right).\end{array}$

If $x>1$ and $y\le 1$, we deduce, in a similar way, that

$\begin{array}{rcl}{D}_{1}\left({T}_{k}\left(x\right),{T}_{j}\left(y\right)\right)& =& H\left({T}_{k}{\left(x\right)}_{1},{T}_{j}{\left(y\right)}_{1}\right)\le \frac{{x}^{2}}{k\left(1+x\right)}\vee \frac{y}{2j}\\ \le & \frac{{x}^{2}}{1+x}\vee \frac{y}{2}\le \frac{{x}^{2}}{1+x}\vee \frac{x}{2}=\frac{{x}^{2}}{1+x}\\ =& \frac{{\left(x\vee y\right)}^{2}}{1+\left(x\vee y\right)}=\frac{d\left(x,y\right)}{1+d\left(x,y\right)}d\left(x,y\right)\\ =& \psi \left(d\left(x,y\right)\right)d\left(x,y\right).\end{array}$

Finally, if $x,y\le 1$, we deduce

$\begin{array}{rcl}{D}_{1}\left({T}_{k}\left(x\right),{T}_{j}\left(y\right)\right)& =& H\left({T}_{k}{\left(x\right)}_{1},{T}_{j}{\left(y\right)}_{1}\right)\le \frac{x}{2k}\vee \frac{y}{2j}\\ \le & \frac{x\vee y}{2}=\psi \left(d\left(x,y\right)\right)d\left(x,y\right).\end{array}$

We have shown that all conditions of Theorem 3.1 are satisfied (in fact $x=0$ is the only fixed point of T).

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## Acknowledgements

The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

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Correspondence to Salvador Romaguera.

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The authors declare that they have no competing interests.

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The three authors contributed equally in writing this article. They read and approved the final manuscript.

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Ahmad, J., Azam, A. & Romaguera, S. On locally contractive fuzzy set-valued mappings. J Inequal Appl 2014, 74 (2014). https://doi.org/10.1186/1029-242X-2014-74