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 [2–10] 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 $(X,d)$, denote by $CB(X)$ the family of all nonempty closed bounded subsets of $(X,d)$. As usual, for $\zeta \in X$ and $A\in CB(X)$, we define

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

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

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

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

A fuzzy set in $(X,d)$ is a function with domain X and values in $I=[0,1]$. $I^X$ denotes the collection of all fuzzy sets in X. If A is a fuzzy set and $\zeta \in X$, then the function value $A(\zeta )$ 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:

$$\begin{array}{c}{A}_{\alpha }=\{\zeta :A(\zeta )\ge \alpha \}\text{if }\alpha \in (0,1],\hfill \\ A_0=\text{closure of }\{\zeta :A(\zeta )>0\}.\hfill \end{array}$$

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

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

$$D_{\alpha }(A,B)=H(A_{\alpha },B_{\alpha }),$$

and

$$d_{\mathrm{\infty }}(A,B)=\underset{\alpha \in [0,1]}{sup}{D}_{\alpha }(A_{\alpha },B_{\alpha }).$$

It is well known that $d_{\mathrm{\infty }}$ is a metric on $W(X)$.

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

$$D_{\alpha }(A,B)=H(A_{\alpha },B_{\alpha }),$$

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

In the sequel the letter ℕ 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 $(X,d)$ be a metric space and let $A,B\in CB(X)$ with $H(A,B)<r$, $r>0$. If $a\in A$, then there exists $b\in B$ such that $d(a,b)<r$.

Lemma 2.2 Let $(X,d)$ be a metric space and let ${\{A_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence in $CB(X)$ such that ${lim}_{n\to \mathrm{\infty }}H(A_n,A)=0$, for some $A\in CB(X)$. If ${\xi }_n\in A_n$, for all $n\in \mathbb{N}$, and $d({\xi }_n,\xi )\to 0$, 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 $X\times Y$ with membership function $T(\zeta )$. The value $T(\zeta )(\xi )$ is the grade of membership of ξ in $T(\zeta )$.

If $(X,d)$ 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 $\xi \in T{(\xi )}_1$.

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

Definition 2.3 (compare [12])

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

Clearly, if $\psi :[0,\epsilon )\to [0,1)$ 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 $\epsilon \in (0,\mathrm{\infty }]$, and $\lambda \in (0,1)$. A metric space $(X,d)$ 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 $d({\zeta }_{j-1},{\zeta }_j)<\epsilon$, for all $j=1,2,\dots ,m$). A mapping $T:X\to X$ is called an $(\epsilon ,\lambda )$ uniformly locally contractive mapping if $\zeta ,\zeta \in X$ and $0<d(\zeta ,\zeta )<\epsilon$, implies $d(T\zeta ,T\xi )\le \lambda d(\zeta ,\xi )$. A mapping $T:X\to W(X)$ is called an $(\epsilon ,\lambda )$ uniformly locally contractive fuzzy mapping if $\zeta ,\xi \in X$ and $0<d(\zeta ,\xi )<\epsilon$, imply $d_{\mathrm{\infty }}(T(\zeta ),T(\xi ))\le \lambda d(\zeta ,\xi )$. We remark that a globally contractive mapping can be regarded as an $(\mathrm{\infty },\lambda )$ uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.

Theorem 3.1 Let $\epsilon \in (0,\mathrm{\infty }]$, $(X,d)$ a complete ε-chainable metric space and ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into $I^X$ such that, for each $\zeta \in X$ and $i\in \mathbb{N}$, $T_i{(\zeta )}_1\in CB(X)$. If

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}{D}_1(T_i(\zeta ),T_j(\xi ))\le \psi (d(\zeta ,\xi ))d(\zeta ,\xi ),$$

(1)

for all $i,j\in \mathbb{N}$, where $\psi :[0,\epsilon )\to [0,1)$ is a MT-function, then the sequence ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ has a common fixed point, i.e., there is ${\xi }^{\ast }\in X$ such that ${\xi }^{\ast }\in T_i{({\xi }^{\ast })}_1$, 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 ${\xi }_1\in T_1{({\xi }_0)}_1$. Let

$${\xi }_0={\zeta }_{(1,0)},{\zeta }_{(1,1)},{\zeta }_{(1,2)},\dots ,{\zeta }_{(1,m)}={\xi }_1\in T_1{({\xi }_0)}_1$$

be an arbitrary ε-chain from ${\xi }_0$ to ${\xi }_1$. (We suppose, without loss of generality, that ${\zeta }_{(1,i)}\ne {\zeta }_{(1,j)}$, for each $i,j\in \{0,1,2,\dots ,m\}$ with $i\ne j$.)

Since $0<d({\zeta }_{(1,0)},{\zeta }_{(1,1)})<\epsilon$, we deduce that

$$\begin{array}{rcl}{D}_1(T_1({\zeta }_{(1,0)}),T_2({\zeta }_{(1,1)}))& \le & \psi (d({\zeta }_{(1,0)},{\zeta }_{(1,1)}))d({\zeta }_{(1,0)},{\zeta }_{(1,1)})\\ <& \sqrt{\psi (d({\zeta }_{(1,0)},{\zeta }_{(1,1)}))}d({\zeta }_{(1,0)},{\zeta }_{(1,1)})\\ <& d({\zeta }_{(1,0)},{\zeta }_{(1,1)})<\epsilon .\end{array}$$

Rename ${\xi }_1$ as ${\zeta }_{(2,0)}$. Since ${\zeta }_{(2,0)}\in T_1{({\zeta }_{(1,0)})}_1$, using Lemma 2.1 we find ${\zeta }_{(2,1)}\in T_2{({\zeta }_{(1,1)})}_1$ such that

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

Similarly we may choose an element ${\zeta }_{(2,2)}\in T_2{({\zeta }_{(1,2)})}_1$ such that

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

Thus we obtain a set $\{{\zeta }_{(2,0)},{\zeta }_{(2,1)},{\zeta }_{(2,2)},\dots ,{\zeta }_{(2,m)}\}$ of $m+1$ points of X such that ${\zeta }_{(2,0)}\in T_1{({\zeta }_{(1,0)})}_1$ and ${\zeta }_{(2,j)}\in T_2{({\zeta }_{(1,j)})}_1$, for $j=1,2,\dots ,m$, with

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

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

Let ${\zeta }_{(2,m)}={\xi }_2$. Thus the set of points ${\xi }_1={\zeta }_{(2,0)},{\zeta }_{(2,1)},{\zeta }_{(2,2)},\dots ,{\zeta }_{(2,m)}={\xi }_2\in T_2{({\xi }_1)}_1$ is an ε-chain from ${\xi }_0$ to ${\xi }_1$. Rename ${\xi }_2$ as ${\zeta }_{(3,0)}$. Then by the same procedure we obtain an ε-chain

$${\xi }_2={\zeta }_{(3,0)},{\zeta }_{(3,1)},{\zeta }_{(3,2)},\dots ,{\zeta }_{(3,m)}={\xi }_3\in T_3{({\xi }_2)}_1$$

from ${\xi }_2$ to ${\xi }_3$. Inductively, we obtain

$${\xi }_n={\zeta }_{(n+1,0)},{\zeta }_{(n+1,1)},{\zeta }_{(n+1,2)},\dots ,{\zeta }_{(n+1,m)}={\xi }_{n+1}\in T_{n+1}{({\xi }_n)}_1$$

with

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

(2)

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

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

$$\begin{array}{c}{\xi }_1={\zeta }_{(1,m)}={\zeta }_{(2,0)}\in T_1{({\xi }_0)}_1,\hfill \\ {\xi }_2={\zeta }_{(2,m)}={\zeta }_{(3,0)}\in T_2{({\xi }_1)}_1,\hfill \\ {\xi }_3={\zeta }_{(3,m)}={\zeta }_{(4,0)}\in T_3{({\xi }_2)}_1,\hfill \\ ⋮\hfill \\ {\xi }_{n+1}={\zeta }_{(n+1,m)}={\zeta }_{(n+2,0)}\in T_{n+1}{({\xi }_n)}_1,\hfill \end{array}$$

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

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

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

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

Now put

$$M_j=max\{\underset{i=1,\dots ,n_j}{max}\sqrt{\psi (d({\zeta }_{(i,j)},{\zeta }_{(i,j+1)}))},\sqrt{s(l_j)}\}.$$

Then, for every $n>n_j$, we obtain

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

Putting $N=max\{n_j:j=0,1,2,\dots ,m-1\}$, we have

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

for all $n>N+1$. Hence

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

whenever $p>n>N+1$.

Since $M_j<1$, for all $j\in \{0,1,2,\dots ,m-1\}$, it follows that ${\{{\xi }_n\}}_{n=1}^{\mathrm{\infty }}$ is a Cauchy sequence. Since $(X,d)$ is complete, there is ${\xi }^{\ast }\in X$ such that ${\xi }_n\to {\xi }^{\ast }$. So for each $\delta \in (0,\epsilon ]$ there is $M_{\delta }\in \mathbb{N}$ such that $n>M_{\delta }$ implies $d({\xi }_n,{\xi }^{\ast })<\delta$. This in view of inequality (1) implies $D_1(T_{n+1}({\xi }_n),T_i({\xi }^{\ast }))<\delta$, for all $i\in \mathbb{N}$. Consequently, $H(T_{n+1}{({\xi }_n)}_1,T_i{({\xi }^{\ast })}_1)\to 0$. Since ${\xi }_{n+1}\in T_{n+1}{({\xi }_n)}_1$ with $d({\xi }_{n+1},{\xi }^{\ast })\to 0$, we deduce from Lemma 2.2 that ${\xi }^{\ast }\in T_i{({\xi }^{\ast })}_1$, for all $i\in \mathbb{N}$. This completes the proof. □

Corollary 3.2 Let $\epsilon \in (0,\mathrm{\infty }]$, $(X,d)$ a complete ε-chainable metric space and ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy mappings from X into $I^X$ such that, for each $\zeta \in X$ and $i\in \mathbb{N}$, $T_i{(\zeta )}_1\in CB(X)$. If

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}{D}_1(T_i(\zeta ),T_j(\xi ))\le \lambda d(\zeta ,\xi ),$$

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

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

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

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}{d}_{\mathrm{\infty }}(T_i(\zeta ),T_j(\xi ))\le \psi (d(\zeta ,\xi ))d(\zeta ,\xi ),$$

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

Proof Since $W(X)\subseteq CB(X)$ and $D_1(T_i(\zeta ),T_j(\xi ))\le d_{\mathrm{\infty }}(T_i(\zeta ),T_j(\xi ))$, for all $i,j\in \mathbb{N}$, the result follows immediately from Theorem 3.1. □

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

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}{d}_{\mathrm{\infty }}(T_i(\zeta ),T_j(\xi ))\le \lambda d(\zeta ,\xi ),$$

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

Corollary 3.5 [4]

Let $\epsilon \in (0,\mathrm{\infty }]$, $(X,d)$ a complete ε-chainable metric linear space and $T_1$, $T_2$, two fuzzy mappings from X into $W(X)$ satisfying the following condition:

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}{d}_{\mathrm{\infty }}(T_i(\zeta ),T_j(\xi ))\le \psi (d(\zeta ,\xi ))d(\zeta ,\xi ),$$

for $i,j=1,2$, where $\psi :[0,\epsilon )\to [0,1)$ is a MT-function. Then $T_1$ and $T_2$ have a common fixed point.

Corollary 3.6 [4, 11]

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

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

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}H(S(\zeta ),S(\xi ))\le \psi (d(\zeta ,\xi ))d(\zeta ,\xi ),$$

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

Proof Define a fuzzy mapping T from X into $I^X$ as $T(\xi )(t)=1$ if $t\in S(\xi )$ and $T(\xi )(t)=0$, otherwise. Then $T{(\xi )}_1=S(\xi )$, for all $\xi \in X$, so $T{(\xi )}_1\in CB(X)$, for all $\xi \in X$. Since

$$D_1(T(\zeta ),T(\xi ))=H(T{(\zeta )}_1,T{(\xi )}_1)=H(S(\zeta ),S(\xi )),$$

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., ${\xi }^{\ast }\in T{({\xi }^{\ast })}_1$. We conclude that ${\xi }^{\ast }\in S({\xi }^{\ast })$. The proof is complete. □

Corollary 3.8 [13]

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

$$\zeta ,\xi \in X,0<d(\zeta ,\xi )<\epsilon \mathit{\text{implies}}H(S(\zeta ),S(\xi ))\le \lambda d(\zeta ,\xi ),$$

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

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

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

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

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

$$\begin{array}{c}\text{if }x=0,T_k(x)(y)=\{\begin{array}{ll}1& \text{if }y=0,\\ 1/3k& \text{if }0<y\le 1/100,\\ 0& \text{if }1/100<y\le 1,\end{array}k\in \mathbb{N},\hfill \\ \text{if }x\ne 0,T_k(x)(y)=\{\begin{array}{ll}1& \text{if }0\le y\le x/14,\\ \lambda /2k& \text{if }x/14<y\le x/12,\\ \lambda /3k& \text{if }x/12<y<x,\\ 0& \text{if }x\le y\le 1,\end{array}k\in \mathbb{N}.\hfill \end{array}$$

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

$$D_1(T_i(x),T_j(y))=H(T_i{(x)}_1,T_j{(y)}_1)=H([0,x/14],[0,y/14])=\frac{1}{14}|x-y|.$$

Hence, for $\psi (t)=\lambda$, the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any $\epsilon \in (0,\mathrm{\infty }]$, 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 $(X,d)$ be the complete metric space such that $X=[0,\mathrm{\infty })$, $d(x,x)=0$, for all $x\in X$, and $d(x,y)=max\{x,y\}$ whenever $x\ne y$ (in the sequel we shall write $x\vee y$ instead of $max\{x,y\}$).

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

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

$$\psi (t)=\{\begin{array}{ll}1/2& \text{if }0\le t\le 1,\\ t/(t+1)& \text{if }t>1,\end{array}$$

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

$$\begin{array}{c}\text{if }0\le x\le 1,T_k(x)(y)=\{\begin{array}{ll}1& \text{if }x/4k\le y\le x/2k,\\ 0& \text{otherwise,}\end{array}k\in \mathbb{N},\hfill \\ \text{if }x>1,T_k(x)(y)=\{\begin{array}{ll}1& \text{if }x/2k\le y<x^2/k(1+x),\\ 0& \text{otherwise},\end{array}k\in \mathbb{N}.\hfill \end{array}$$

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

$$T_k{(x)}_1=[\frac{x}{4k},\frac{x}{2k}],$$

and, for $x>1$,

$$T_k{(x)}_1=[\frac{x}{2k},\frac{x^2}{k(1+x)}).$$

Therefore $T_k{(x)}_1\in CB(X)$, 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 $CB(X)$).

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 $b\in T_j{(y)}_1$, we obtain

$$d(T_k{(x)}_1,b)=\underset{a\in T_k{(x)}_1}{inf}(a\vee b)\le \frac{x^2}{k(1+x)}\vee b\le \frac{x^2}{k(1+x)}\vee \frac{y^2}{j(1+y)}.$$

Similarly, for each $a\in T_k{(x)}_1$, we obtain

$$d(a,T_j{(y)}_1)\le \frac{x^2}{k(1+x)}\vee \frac{y^2}{j(1+y)}.$$

Consequently

$$\begin{array}{rcl}{D}_1(T_k(x),T_j(y))& =& H(T_k{(x)}_1,T_j{(y)}_1)\le \frac{x^2}{k(1+x)}\vee \frac{y^2}{j(1+y)}\\ \le & \frac{{(x\vee y)}^2}{1+(x\vee y)}=\frac{d(x,y)}{1+d(x,y)}d(x,y)\\ =& \psi (d(x,y))d(x,y).\end{array}$$

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

$$\begin{array}{rcl}{D}_1(T_k(x),T_j(y))& =& H(T_k{(x)}_1,T_j{(y)}_1)\le \frac{x^2}{k(1+x)}\vee \frac{y}{2j}\\ \le & \frac{x^2}{1+x}\vee \frac{y}{2}\le \frac{x^2}{1+x}\vee \frac{x}{2}=\frac{x^2}{1+x}\\ =& \frac{{(x\vee y)}^2}{1+(x\vee y)}=\frac{d(x,y)}{1+d(x,y)}d(x,y)\\ =& \psi (d(x,y))d(x,y).\end{array}$$

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

$$\begin{array}{rcl}{D}_1(T_k(x),T_j(y))& =& H(T_k{(x)}_1,T_j{(y)}_1)\le \frac{x}{2k}\vee \frac{y}{2j}\\ \le & \frac{x\vee y}{2}=\psi (d(x,y))d(x,y).\end{array}$$

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