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# Some fixed point results for fuzzy homotopic mappings

Fixed Point Theory and Applications20172017:15

https://doi.org/10.1186/s13663-017-0609-9

• Received: 10 February 2017
• Accepted: 1 August 2017
• Published:

## 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.

## Keywords

• fixed point
• fuzzy contraction
• fuzzy homotopic map

• 47H10
• 54H25

## 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. (1)

$$M(x,y,t)>0$$,

2. (2)

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

3. (3)

$$M(x,y,t)=M(y,x,t)$$,

4. (4)

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

5. (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]

1. (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 $$M(x,y,t_{0})>1-r$$.

2. (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. (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. (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. (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 . 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:
1. (a)

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

2. (b)

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

3. (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 $$t>0$$;

4. (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:
1. (a)

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

2. (b)

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

3. (c)

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

4. (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.

## Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

(1)
Department of Mathematics, Islamic Azad University, Ahvaz Branch, Ahvaz, Iran

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## Copyright

© The Author(s) 2017

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