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 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:

  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.