Some fixed point results for fuzzy homotopic mappings
- Fatemeh Kiany^{1}Email author
https://doi.org/10.1186/s13663-017-0609-9
© The Author(s) 2017
Received: 10 February 2017
Accepted: 1 August 2017
Published: 3 October 2017
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
MSC
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]
- (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]
Example 1.4
[7]
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
- (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 iffor all \(t>0\).$$\lim_{n,m\rightarrow\infty} M(x_{n},x_{m},t)=1$$
- (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]
The following lemma is essential in proving our result.
Lemma 1.13
[8]
Definition 1.14
Kiany and Amini proved the following improvement of Gregori and Sapena’s fixed point theorem.
2 Main results
Let \((X,M,\ast) \) be a complete fuzzy metric space.
Lemma 2.1
Proof
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
- (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
Step one. A is open in \([0,1] \).
Step two. A is closed in \([0,1] \).
Example 2.7
Now, as a result of Theorem 2.6, we can prove the following theorem due to Fournier [3].
Theorem 2.8
- (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.
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
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