Some fixed point results for fuzzy homotopic mappings

Abstract

The first purpose of this paper is to define a homotopy for fuzzy spaces. We continue our work by showing that the property of having a fixed point is invariant by this homotopy. These theorems generalize and improve well-known results.

1 Introduction and preliminaries

The study of fuzzy metric spaces has been developing since 1971. The well-known fixed point theorem of Banach was extended by Grabiec [1]. On the other hand, a number of authors have studied the conditions under which the property of having a fixed point is invariant in metric spaces. For example, see [2, 3].

To seek completeness, we briefly recall some basic concepts used in the following.

Definition 1.1

[4]

A binary operation $*:[0,1] \times[0,1] \rightarrow [0,1]$ is called a continuous t-norm if $([0,1],\ast)$ is an abelian topological monoid with unit 1 such that $a \ast b \leq c\ast d$ whenever $a \leq c$ and $b \leq d$ for all $a,b,c,d \in[0,1]$.

Definition 1.2

[5]

The 3-tuple $(X,M,*)$ is called a fuzzy metric space if X is an arbitrary non-empty set, ∗ is a continuous t-norm, M is a fuzzy set on $X^{2}\times[0,\infty)$ satisfying the following conditions, for each $x,y,z\in X$ and $t,s >0$,

• (1) $M(x,y,t)>0$,

• (2) $M(x,y,t)=1$ if and only if $x=y$,

• (3) $M(x,y,t)=M(y,x,t)$,

• (4) $M(x,y,t)\ast M(y,z,s)\leq M(x,z,t+s)$,

• (5) $M(x,y,t):(0,\infty)\rightarrow[0,1]$ is continuous.

Example 1.3

[6]

Let $X=\Bbb{N}$, define $a\ast b=ab$ for all $a,b\in [0,1]$, let M be a fuzzy set on $X^{2}\times[0,\infty)$ as follows:

$$M(x,y,t) = \textstyle\begin{cases} \frac{x+t}{y+t}, & x \leq y, \\ \frac{y+t}{x+t}, & y >x. \end{cases} $$

Then $(X,M,\ast)$ is a fuzzy metric space.

Example 1.4

[7]

Let $(X,d)$ be a metric space. Define $a\ast b=ab$ for all $x,y\in X$ and $t>0$,

$$M(x,y,t)=\frac{t}{t+d(x,y)}. $$

Then $(X,M,\ast)$ is a fuzzy metric space. We call this fuzzy metric M induced by the metric d the standard fuzzy metric. If $(X,d)$ is a complete metric space, then also $(X,M,\ast)$ is complete.

Lemma 1.5

[1]

$M(x,y,\cdot)$ is non-decreasing for all $x,y \in X$.

Remark 1.6

[5]

• (a) In a fuzzy metric space $(X,M,\ast)$, whenever $M(x,y,t)>1-r$ for x, y in $X,t>0$, $0< r<1$, we can find $0< t_{0}< t$ such that $M(x,y,t_{0})>1-r$.

• (b) For any $r_{1}>r_{2}$, we can find $r_{3}$ such that $r_{1} \ast r_{3}\geq r_{2}$, and for any $r_{4}$, we can find $r_{5}$ such that $r_{5} \ast r_{5}\geq r_{4}$ ($r_{1},r_{2},r_{3},r_{4},r_{5}\in(0,1)$).

George and Veeramani introduced Hausdorff topology in fuzzy metric spaces. They showed that this topology is first countable.

Definition 1.7

[5]

Let $(X,M,\ast)$ be a fuzzy metric space. For $t>0$ and $0< r<1$, the open ball $B(x,r,t)$ with center $x \in X$ is defined by $B(x,r,t)=\lbrace y \in X:M(x,y,t)>1-r \rbrace$.

A subset $A \subseteq X$ is called open if for each $x \in A$ there exist $t>0$ and $0< r<1$ such that $B(x,r,t)\subseteq A$. Let τ denote the family of all open subsets of X. Then τ is a topology on X induced by the fuzzy metric $(X,M,\ast)$. This topology also is metrizable (see [7]).

Definition 1.8

Let $(X,M,\ast)$ be a fuzzy metric space.

• (1) A sequence $\lbrace x _{n}\rbrace$ is said to be convergent to a point $x \in X$ if $\lim_{n\rightarrow\infty} M(x_{n},x,t)=1$ for all $t>0$.

• (2) A sequence $\lbrace x_{n}\rbrace$ is said to be Cauchy sequence if

  $$\lim_{n,m\rightarrow\infty} M(x_{n},x_{m},t)=1$$

  for all $t>0$.

• (3) A fuzzy metric space in which every Cauchy sequence is convergent to a point $x \in X$ is said to be complete.

Definition 1.9

Let $(X,M\ast)$ be a fuzzy metric space and $A\subseteq X$. Closure of the set A is the smallest closed set containing A, denoted by A̅. Interior of the set A is the largest open set contained in A, denoted by $A^{\circ}$. Obviously, having in mind the Hausdorff topology and the definition of converging sequences, we have that the next remark holds.

Remark 1.10

$x\in\overline{A}$ if and only if there exists a sequence $\lbrace x_{n}\rbrace$ in A such that $x_{n}\rightarrow x$.

We also need the following definitions.

Definition 1.11

Let $(X,M,\ast)$ be a fuzzy metric space, $A\subseteq X$ $\overline{A}\setminus A^{\circ}$ is called boundary of A and denoted by ∂A.

Definition 1.12

[6]

Let A be a non-empty subset of fuzzy metric space $(X,M,\ast)$. For each $x\in X$ and $t>0$, define

$$M(x,A,t)=\sup \bigl\lbrace M(x,y,t):y \in A \bigr\rbrace . $$

The following lemma is essential in proving our result.

Lemma 1.13

[8]

Let $(X,M,*)$ be a fuzzy metric space such that for every $x,y\in X$, $t>0$ and $h>1$,

$$ \lim_{n \rightarrow\infty}\ast^{\infty}_{i=n}M\bigl(x,y,th^{i} \bigr)=1. $$

(1.13)

Suppose that $\lbrace x _{n}\rbrace$ is a sequence in X such that, for all $n \in\Bbb{N}$,

$$M(x_{n},x_{n+1},kt)\geq M(x_{n-1},x_{n},t), $$

where $0< K<1$, then $\lbrace x _{n}\rbrace$ is a Cauchy sequence.

Definition 1.14

Let $(X,M,\ast)$ be a fuzzy metric space. A map $F :X\rightarrow X$ is said to be fuzzy contraction if there exists a constant $0 < \alpha< 1$ with

$$M(Fx,Fy,\alpha t)\geq M(x,y,t). $$

Kiany and Amini proved the following improvement of Gregori and Sapena’s fixed point theorem.

Theorem 1.15

[8]

Let $(X,M,\ast)$ be a complete fuzzy metric space. Suppose that $F:X\rightarrow X$ is a fuzzy contractive map. Furthermore, assume that $(X,M,\ast)$ satisfies (1.13) for some $x_{0} \in X$, each $t>0$ and $h>1$. Then F has a fixed point.

2 Main results

Let $(X,M,\ast)$ be a complete fuzzy metric space.

Lemma 2.1

If $0< a<1$, $0< p<1$, $t,N>0$ all are given, then there exists $\epsilon >0$ such that, if we have $\vert\lambda- \lambda_{0} \vert\leq \epsilon$, then

$$\frac{at}{ N\vert\lambda-\lambda_{0}\vert+at} \geq p. $$

Proof

Put $0< \epsilon\leq\frac{at(1-p)}{pN}$. We have

$$\begin{gathered} \epsilon pN \leq at(1-p), \\\epsilon pN \leq at-atp, \\\epsilon pN+atp \leq at, \\p(\epsilon N+at) \leq at, \\ p \leq\frac{at}{\epsilon N+at}. \end{gathered}$$

Since $\vert\lambda-\lambda_{0}\vert\leq\epsilon$, we get

$$p \leq\frac{at}{ N\vert\lambda-\lambda_{0}\vert+at} $$

or

$$\frac{at}{ N\vert\lambda-\lambda_{0}\vert+at} \geq p. $$

 □

Definition 2.2

Let $(X,M,\ast)$ be a fuzzy metric space and A be a closed subset of X and $x_{0}\notin A$, then we say X has a real distance if $\sup \lbrace M(x,A,t): \forall t>0 \rbrace<1$.

Example 2.3

Then $(X,M,\ast)$ is a complete standard fuzzy metric, then X has a real distance, because if A is a closed subset of X and $x_{0}\notin A$, then $\inf \lbrace d(x_{0},A)\rbrace>0$.

Example 2.4

Suppose that $(X,M,\ast)$ is the same as in Example 1.3, $A\in X$ is an arbitrary subset of X and $x_{0}\notin A$, then $M(x_{0},A,t)\leq \frac{1}{2}$.

Definition 2.5

Let $F:\overline{U}\rightarrow X$ and $G:\overline {U}\rightarrow X$ be two fuzzy contractions. We say that F and G are fuzzy homotopic • maps if there exists $H:\overline{U}\times[0,1]\rightarrow X$ with the following properties:

• (a) $H(\cdot,0)=G$ and $H(\cdot,1)=F$;

• (b) $x\neq H(x,s)$ for $x\in\partial U$ and $s\in[0,1]$;

• (c) there exists K, $0< K<1$, such that $M(H(x,s),H(y,s),Kt)\geq M(x,y,t)$ for every $x,y \in\overline{U}$, $s\in[0,1]$ and $t>0$;

• (d) there exists N, $N\geq0$, such that $M(H(x,s_{0}),H(y,s_{1}),t)\geq\frac{t}{t+N\vert s_{0}-s_{1} \vert}$ for every $x \in\overline{U}$, $t >0$ and $s_{0},s_{1}\in[0,1]$.

Theorem 2.6

Let $(X,M,\ast)$ be a fuzzy complete metric space and U be an open subset of X. Suppose that $F:\overline{U}\rightarrow X$ and $G:\overline{U}\rightarrow X$ are two homotopic fuzzy maps and G has a fixed point in U. Assume that $(X,M,\ast)$ satisfies (1.13) for some $x_{0} \in X$ and also X has a real distance, then F has a fixed point in U.

Proof

Consider the set

$$A= \bigl\lbrace \lambda\in[0,1]:x=H(x,\lambda) \text{ for some } x\in U\bigr\rbrace , $$

where H is a homotopy between F and G as described in Definition 2.5. Notice that A is non-empty since G has a fixed point, that is, $0\in A$. We will show that A is both open and closed in $[0,1]$ and, by connectedness, we have that $A=[0,1]$. As a result, F has a fixed point in U. We break the argument into two steps.

Step one. A is open in $[0,1]$.

Since A is non-empty, there exists $x_{0} \in U$ with $x_{0}=H(x_{0},\lambda_{0})$. Since X has a real distance, there exists $0< r^{*}<1$ such that

$$M(x_{0},\partial U,t)>1-r^{*}. $$

So we can choose r, $0< r<1$, such that $1-r^{*}>1-r$.

Now if

$$x\in\overline{B(x_{0},r,t)}\quad \Longrightarrow \quad M(x_{0},x,t)>1-r. $$

From Remark 1.6(a) we can find $t_{0}$, $0< t_{0}< t$, such that $M(x_{0},x,t_{0})>1-r$. Let

$$ r_{0}=M(x_{0},x,t )>1-r. $$

(1)

Since $r_{0}>1-r$, we can find s, $0< s<1$, such that

$$ r_{0}>1-s>1-r.$$

(2)

Now, for given $r_{0}$ and s, from Remark 1.6(b) we can find p, $0< p<1$, such that

$$ r_{0} \ast p\geq1-s. $$

(3)

Now consider Lemma 2.1 with $a=(1-K)$, p, N, $t_{0}$, we can find ϵ such that if $\vert\lambda-\lambda_{0} \vert\leq\epsilon$, then we have

$$ \frac{(1-K)t_{0}}{(1-K)t_{0}+N\vert\lambda-\lambda_{0} \vert}\geq p. $$

(4)

Thus, for each fixed $\lambda\in(\lambda_{0}-\epsilon,\lambda _{0}+\epsilon)$ and $x\in\overline{{\bullet}B(x_{0},r,t})$, we have

$$ \begin{aligned}[b] M\bigl(x_{0},H(x,\lambda),t\bigr) &\geq M\bigl(x_{0},H(x, \lambda),t_{0}\bigr) \\&\geq M\bigl(H(x_{0},\lambda_{0}),H(x_{0}, \lambda),(1-K)t_{0}\bigr) \ast M\bigl(H(x_{0},\lambda ),H(x, \lambda),Kt_{0}\bigr). \end{aligned}$$

(5)

By Definition 2.5(d) we know that

$$M\bigl(H(x_{0},\lambda_{0}),H(x_{0}, \lambda),(1-K)t_{0}\bigr) \geq\frac{(1-K)t_{0}}{ (1-K)t_{0}+N\vert\lambda-\lambda_{0}\vert}. $$

Also by Definition 2.5(c) we know that

$$M\bigl(H(x_{0},\lambda),H(x,\lambda),Kt_{0}\bigr)\geq M(x_{0},x,t_{0}). $$

Substitution of these expressions into (5) reveals

$$ M\bigl(x_{0},H(x,\lambda),t\bigr)\geq\frac{(1-K)t_{0}}{(1-K)t_{0}+N\vert\lambda -\lambda_{0}\vert}\ast M(x_{0},x,t_{0}). $$

(6)

Now from substitution of (1) and (4) into the (6) we have

$$M\bigl(x_{0},H(x,\lambda),t\bigr) \geq p \ast r_{0}. $$

From (3) we get

$$M\bigl(x_{0},H(x,\lambda),t\bigr) \geq1-s. $$

Then from (2) we get

$$M\bigl(x_{0},H(x,\lambda),t\bigr)\geq1-r. $$

Thus, for each fixed $\lambda\in(\lambda_{0}-\epsilon,\lambda _{0}+\epsilon)$,

$$H(\cdot,\lambda):\overline{B(x_{0},r,t)} \rightarrow\overline{B(x_{0},r,t)}. $$

We can apply Theorem 1.15 to deduce that $H(\cdot,\lambda)$ has a fixed point in U. Thus $\lambda\in A$ for any $\lambda\in(\lambda _{0}-\epsilon,\lambda_{0}+\epsilon)$ and therefore A is open in $[0,1]$.

Step two. A is closed in $[0,1]$.

To see this, let

$$\lbrace\lambda_{n} \rbrace_{n=1}^{\infty}\subseteq A\quad \text{with } \lambda_{n} \rightarrow\lambda\in[0,1]\text{ as } n\rightarrow \infty. $$

We must show that $\lambda\in A$. Since $\lambda_{n}\in A$ for $n=1,2,\ldots$ , there exists $x_{n}\in U$ with $x_{n} =H(x_{n},\lambda _{n})$. Hence, by Lemma 1.13, we know $\lbrace x_{n}\rbrace$ is a Cauchy sequence. Since $(X,M,\ast)$ is a fuzzy complete metric space, then there exists $x \in X$ such that $\lim_{n\rightarrow\infty }x_{n} =x$, that means

$$ \lim_{n\rightarrow\infty}M(x_{n},x,t) =1 \quad\text{for each } t>0.$$

(7)

On the other hand, from $\lambda_{n}\rightarrow\lambda$, we have

$$\frac{(1-K)t}{ N\vert\lambda_{n}- \lambda\vert+(1-K)t}\rightarrow1. $$

In addition, $x=H(x,\lambda)$ since

$$ \begin{aligned}[b] M\bigl(x_{n},H(x,\lambda),t\bigr)&=M\bigl(H(x_{n}, \lambda_{n}),H(x,\lambda),t\bigr) \\&\geq M\bigl(H(x_{n},\lambda_{n} ),H(x_{n}, \lambda),(1-K)t\bigr) \ast M\bigl(H(x_{n},\lambda),H(x,\lambda),Kt \bigr).\end{aligned}$$

(8)

By Definition 2.5(d) we have

$$ M\bigl(H(x_{n},\lambda_{n} ),H(x_{n}, \lambda),(1-K)t \bigr)\geq\frac {(1-K)t}{(1-K)t+N \vert\lambda_{n}-\lambda\vert}. $$

(9)

Also, by Definition 2.5(c), we have

$$ M\bigl(H(x_{n},\lambda),H(x,\lambda),Kt\bigr)\geq M(x_{n},x,t). $$

(10)

Now from substitution of (9) and (10) in (8) we get

$$M\bigl(x_{n},H(x,\lambda),t\bigr)\geq\frac{(1-K)t}{(1-K)t+N\vert\lambda _{n}-\lambda\vert} \ast M(x_{n},x,t). $$

As seen above, on the left-hand side of this inequality, both limits exist and are equal to one. So, for each $t>0$, we must have

$$\lim _{n\rightarrow\infty}M\bigl(x_{n},H(x, \lambda),t\bigr)=1. $$

From (7) we get $H(x,\lambda)=x$. Thus $\lambda\in A$, and A is closed in $[0,1]$. □

Example 2.7

Let $X=\Bbb{R}$, $M(x,y,t)=\frac{t}{t+ \vert x-y \vert}$, $a \ast b=ab$, then $(X,M,\ast)$ is a complete fuzzy metric space. Also $(X,M,\ast)$ satisfies (1.13). Let $N>0$ be a fixed real number and $f(x):X \rightarrow X$ be given by

$$f(x) = \textstyle\begin{cases} \frac{x}{2}, & 0\leq x \leq2N; \\ N,& \text{else}. \end{cases} $$

Also define

$$g(x)=(1-\beta) f(x) \quad\text{for } 0< \beta< 1. $$

It is easy to show $\vert f(x)-f(y) \vert\leq\frac{1}{2} \vert x-y \vert$ for all $x,y \in X$. Now we have $M(f(x),f(y), \frac{t}{2} )\geq M(x,y,t)$. Because f is a fuzzy contraction ($\alpha=\frac{1}{2}$), g is a fuzzy contraction, too. Let $s,s_{0},s_{1}\in[0,1]$, $t>0$. We define $H:X\times[0,1]\rightarrow X$

$$H(x,s)=sf(x)+(1-s)g(x). $$

It is obvious that H satisfies Definition 2.5(a) and Definition 2.5(b). We need only check that Definition 2.5(c) and Definition 2.5(d) are true. For Definition 2.5(c), we have

$$\begin{aligned} \big\vert H(x,s)-H(y,s)\big\vert &=\big\vert sf(x)+(1-s)g(x)-sf(y)-(1-s)g(y)\big\vert \\&= \big\vert s\bigl(f(x)-f(y)\bigr)+(1-s) (1- \beta) \bigl(f(x)-f(y)\bigr) \big\vert \\&= \big\vert f(x)-f(y) \bigl( s+(1-s) (1-\beta) \bigr) \big\vert \\&\leq\big\vert f(x)-f(y) \bigl( s+(1-s) \bigr) \big\vert \\&= \big\vert f (x)-f(y)\big\vert \leq\bigg\vert \frac{1}{2}( x-y) \bigg\vert . \end{aligned}$$

For $K=\frac{1}{2}$, we have

$$M\bigl(H(x,s),H(y,s),Kt\bigr)\geq M(x,y,t). $$

For Definition 2.5(d), we have

$$\begin{aligned} \big\vert H(x,s_{0})-H(x,s_{1})\big\vert &=\big\vert s_{0}f(x)+(1-s_{0})g(x)-s_{1}f(x)-(1-s_{1})g(x) \big\vert \\&= \big\vert (s_{0}-s_{1}) \bigl(f(x)-g(x)\bigr)\big\vert \\&= \big\vert (s_{0}-s_{1}) \bigl(f(x)-f(x)+\beta f(x)\bigr)\big\vert \\&= \big\vert (s_{0}-s_{1})\beta f(x)\big\vert \\&\leq\big\vert (s_{0}-s_{1})\beta N\big\vert \\&\leq\big\vert (s_{0}-s_{1}) N\big\vert . \end{aligned}$$

So

$$M\bigl(H(x,s_{0}),H(x,s_{1}),t\bigr) =\frac{t}{t+\vert H(x,s_{0})-H(x,s_{1})\vert }\geq \frac{t}{t+N \vert s_{0}-s_{1}\vert}. $$

Now f and g are two fuzzy homotopic contractive maps. Notice that f has a fixed point in zero. We can now apply Theorem 2.6 to deduce that there exists x with $x=g(x)$.

Now, as a result of Theorem 2.6, we can prove the following theorem due to Fournier [3].

Theorem 2.8

Let $(X,d)$ be a complete metric space and U be an open subset of X. Suppose that $F:U\rightarrow X$ and $G:U\rightarrow X$ if there exists $H:\overline{U}\times[0,1]\rightarrow X$ with the following properties:

• (a) $H(\cdot,0)=G$ and $H(\cdot,1)=F$;

• (b) $x\neq H(x,s)$ for $x\in\partial U$ and $s\in[0,1]$;

• (c) there exists K, $0\leq K< 1$, such that $d(H(x,s),H(y,s))\leq Kd(x,y)$ for every $x,y \in\overline{U}$, $s\in[0,1]$;

• (d) there exists N, $N\geq0$, such that $d(H(x,s),H(y,p))\leq N\vert s-p\vert$ for every $x,y \in\overline{U}$ and $s,p \in[0,1]$. Suppose that F and G are two contractive maps and G has a fixed point in U, then F has a fixed point in U.

Proof

Let $(X,M,\ast)$ be a standard fuzzy metric space induced by the metric d with $a*b=\min \lbrace a,b \rbrace$. Notice that F and G are two contractive maps, so they are fuzzy contractive maps in the induced fuzzy metric space. Now we can see that condition (1.13) is satisfied. Also X has a real distance. Since $(X,d)$ is a complete metric space, $(X,M,\ast)$ is a complete fuzzy metric space. It is easy to see that $(X,M,\ast)$ satisfies all the conditions Definition 2.5(a), Definition 2.5(b), Definition 2.5(c) and Definition 2.5(d). We can apply Theorem 2.6 to deduce that F has a fixed point. □

3 Conclusions

Motivated by the results of Frigon, I slightly modified the definition of homotopic contractive maps. I proved that the property of having a fixed point is invariant by homotopy for fuzzy contractive maps. This investigation could be extended to a fuzzy quasi-metric space with possible application to the study of analysis of probabilistic metric spaces.