On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings
- Francisco Castro-Company^{1},
- Salvador Romaguera^{2} and
- Pedro Tirado^{2}Email author
https://doi.org/10.1186/s13663-015-0476-1
© Castro-Company et al. 2015
Received: 10 September 2015
Accepted: 29 November 2015
Published: 10 December 2015
Abstract
We present a procedure to construct a compatible metric from a given fuzzy metric space. We use this approach to obtain a characterization of a large class of complete fuzzy metric spaces by means of a fuzzy version of Caristi’s fixed point theorem, obtaining, in this way, partial solutions to a recent question posed in the literature. Some illustrative examples are also given.
Keywords
MSC
1 Introduction and preliminaries
Throughout this paper the symbols \(\mathbb{R}\), \(\mathbb{R}^{+}\), and \(\mathbb{N}\) will denote the set of all real numbers, the set of all non-negative real numbers and the set of all positive integers, respectively. Our basic reference for general topology is [1].
We start by recalling the notion of a continuous t-norm as well as some types of continuous t-norm which will be crucial throughout this paper.
According to [2], a binary operation \(*: [ 0,1 ] \times [ 0,1 ] \rightarrow [ 0,1 ] \) is a continuous t-norm if ∗ satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) \(a*1=a\) for every \(a\in[0,1]\); (iv) \(a*b\leq c*d\) whenever \(a\leq c\) and \(b\leq d\), with \(a,b,c,d\in[0,1]\).
Distinguished examples of continuous t-norm are ∧, Prod, and \(*_{L}\) (the Łukasiewicz t-norm), which are defined as follows: \(a\wedge b=\min \{a,b\}\), \(a \operatorname{Prod} b=ab\), and \(a*_{L}b=\max\{a+b-1,0\}\) for all \(a,b\in [0,1]\).
The following well-known relations hold: \(\wedge\geq \operatorname{Prod} \geq *_{L}\). In fact, ∧ ≥ ∗ for any continuous t-norm ∗.
- (A)Given \(p>0\), define for each \(a,b\in[0,1]\):Then \(*_{Y_{p}}\) is a continuous t-norm referred to in the literature as the Yager continuous t-norm (see e.g. [3]).$$a*_{Y_{p}}b=1-\min\bigl\{ 1,\bigl[(1-a)^{p}+(1-b)^{p} \bigr]^{1/p}\bigr\} . $$
- (B)Hamacher constructed in [4] a family of continuous t-norms \((*_{H_{\lambda}})_{\lambda\in\mathbb{R}^{+}}\) defined as follows. Given \(\lambda\in\mathbb{R}\), for each \(a,b\in[0,1]\), \(a*_{H_{\lambda}}b=0\) if \(a=b=\lambda=0\), andotherwise. Furthermore, we have \(a*_{H_{\lambda_{1}}}b\leq a*_{H_{\lambda_{2}}}b\) whenever \(\lambda_{1}\geq\lambda_{2}\), with \(a,b\in[0,1]\). In particular we see that \(*_{H_{1}}\) coincides with the product t-norm.$$a*_{H_{\lambda}}b=\frac{ab}{\lambda+(1-\lambda)(a+b-ab)}, $$
In this paper we shall work with fuzzy metric spaces in the sense of Kramosil and Michalek [5] (see Definition 1 below). At this point it seems suitable to remark that George and Veeramani introduced in [6] an interesting modification of Kramosil and Michalek’s notion. However, from the well-known fact that every fuzzy metric space \((X,M,*)\) in the sense of George and Veeramani can be considered as a fuzzy metric space in the sense of Kramosil and Michalek, simply putting \(M(x,y,0)=0\) for all \(x,y\in X\), we deduce that the obtained results in this paper remain valid for fuzzy metric spaces in George and Veeramani’s sense.
Definition 1
(Kramosil and Michalek [5])
- (FM1)
\(M(x,y,0)=0\);
- (FM2)
\(x=y\) if and only if \(M(x,y,t)=1\) for all \(t>0\);
- (FM3)
\(M(x,y,t)=M(y,x,t)\);
- (FM4)
\(M(x,z,t+s)\geq M(x,y,t)*M(y,z,s)\) for all \(t,s\geq0\);
- (FM5)
\(M(x,y,\_):\mathbb{R}^{+}\rightarrow[0,1]\) is left continuous.
By a fuzzy metric space we mean a triple \((X,M,*)\) such that X is a set and \((M,*)\) is a fuzzy metric on X.
It is well known that for each \(x,y\in X\), \(M(x,y,\_)\) is a non-decreasing function on \(\mathbb{R}^{+}\).
Each fuzzy metric \((M,*)\) on a set X induces a topology \(\tau_{M}\) on X which has as a base the family of open sets \(\{B_{M}(x,\varepsilon ,t):\varepsilon\in(0,1),t>0\}\), where \(B_{M}(x,\varepsilon,t)=\{y\in X:M(x,y,t)>1-\varepsilon\}\) for all \(\varepsilon\in(0,1)\), \(t>0\).
A Cauchy sequence in a fuzzy metric space \((X,M,*)\) is a sequence \((x_{n})_{n\in\mathbb{N}}\) in X such that for each \(\varepsilon\in(0,1)\) and \(t>0\) there exists an \(n_{0}\in\mathbb{N}\) satisfying \(M(x_{n},x_{m},t)>1-\varepsilon\) whenever \(n,m\geq n_{0}\).
A fuzzy metric space \((X,M,*)\) is said to be complete if every Cauchy sequence \((x_{n})_{n\in\mathbb{N}}\) converges with respect to the topology \(\tau_{M}\), i.e., if there exists \(y\in X\) such that for each \(t>0\), \(\lim_{n}M(y,x_{n},t)=1\) (see e.g. [6]).
It is well known (see e.g. [7]) that every fuzzy metric space is metrizable, i.e., given a fuzzy metric space \((X,M,*)\) there exists a metric on X whose induced topology coincides with the topology \(\tau_{M}\). A short and easy proof of this result consists in showing that the countable family \(\{U_{n}:n\in\mathbb{N}\}\), where \(U_{n}=\{(x,y)\in X\times X:M(x,y,1/n)>1-1/n\}\) for all \(n\in\mathbb{N}\), is a base for a uniformity on X whose induced topology coincides with the topology \(\tau_{M}\), and then to apply the famous Kelley metrization lemma [1], p.185.
However, the important problem of obtaining a general procedure to construct a visual and manageable compatible metric for any fuzzy metric space, in such a way that the fixed point theory for fuzzy metric spaces could be deduced from the classical fixed point theory for metric spaces, remains unsolved.
Radu obtained some partial but interesting solutions to this problem. Thus, he proved in [8] the following theorem.
Theorem 1
[8]
Theorem 1 was successfully applied (see e.g. [9–15]) to deduce several fixed point theorems for complete fuzzy metric spaces from the corresponding results for complete metric spaces. See also [16–18] and the references therein, for some recent contributions to the fixed point theory in fuzzy metric spaces and related structures.
Later on, Radu [20], Theorem 2.1.7, obtained a substantial improvement of Hicks’ result, which is established below in a slightly different form.
Theorem 2
[20]
- (R1)
μ is continuous on \(\mathbb{R}^{+}\);
- (R2)
\(\mu(t)=0\Longleftrightarrow t=0\);
- (R3)
\(\mu(t+s)\geq\mu(t)+\mu(s)\) for all \(t,s\geq0\);
- (R4)
\(M(x,y,t)>1-\mu(t)\), \(M(y,z,s)>1-\mu(s)\Longrightarrow M(x,z,t+s)>1-\mu(t+s)\).
Here, we shall present a modification of Theorem 2 which can be applied to certain cases and instances where Radu’s theorem does not work. From our approach we deduce a fixed point theorem of Caristi type for multivalued mappings which is valid for any complete fuzzy metric space \((X,M,*)\), as well as a characterization of those complete fuzzy metric spaces \((X,M,*)\) with \(* \geq *_{Y_{p}}\), \(p>0\), in terms of Caristi’s fixed point theorem. These results provide partial solutions to a question posed in [9]. Several illustrative examples are also given.
2 Constructing metrics from fuzzy metrics
We start this section by establishing our promised modification of Theorem 2 above (a background in this direction may be found in [21], Lemma 1.7).
Theorem 3
- (c1)
α is strictly increasing on \([0,1]\);
- (c2)
\(0<\alpha(t)\leq t\) for all \(t\in(0,1)\) and \(\alpha (t)>1\) for all \(t>1\);
- (c3)
\((1-\alpha(t))*(1-\alpha(s))\geq1-\alpha(t+s)\) for all \(t,s\in[0,1]\).
Proof
We first note that \(\alpha(0)=0\) because, by (c1) and (c2), \(\alpha(0)<\alpha(t)\leq t\) for all \(t\in(0,1)\).
Now we prove that \(d_{\alpha}\) is a metric on X such that \(d_{\alpha }(x,y)\leq1\) for all \(x,y\in X\).
Let \(x,y\in X\). Since \(M(x,y,0)=0<1-\alpha(0)\) we deduce that \(d_{\alpha}(x,y)\geq0\). Moreover, \(d_{\alpha}(x,y)\leq1\) because, by (c2), \(\alpha (t)>1\) for all \(t>1\).
We also have \(d_{\alpha}(x,x)=0\) because for each \(t>0\), \(M(x,x,t)=1>1-\alpha(t)\).
Furthermore, we have \(d_{\alpha}(x,y)=d_{\alpha}(y,x)\) because \(M(x,y,t)=M(y,x,t)\) for all \(t>0\).
It remains to prove that \(d_{\alpha}\) satisfies the triangle inequality. To this end, let \(x,y,z\in X\). If \(d_{\alpha}(x,z)+d_{\alpha}(z,y)\geq 1\) we immediately obtain \(d_{\alpha}(x,y)\leq d_{\alpha}(x,z)+d_{\alpha }(z,y)\), because \(d_{\alpha}(x,y)\leq1\).
It follows from (2) that \(d_{\alpha}(x,y)\leq d_{\alpha }(x,z)+d_{\alpha}(z,y)+2\varepsilon\). Since ε is arbitrary, we conclude that \(d_{\alpha}(x,y)\leq d_{\alpha}(x,z)+d_{\alpha}(z,y)\).
We have proved that \(d_{\alpha}\) is a metric on X.
Now suppose that α is left continuous on \((0,1]\). If \(d_{\alpha }(x,y)<\varepsilon\), with \(\varepsilon\in(0,1)\), then \(M(x,y,t)>1-\alpha (\varepsilon)\) by the definition of \(d_{\alpha}\). Conversely, if \(M(x,y,\varepsilon)>1-\alpha(\varepsilon)\), then \(d_{\alpha}(x,y)\leq \varepsilon\) by the definition of \(d_{\alpha}\). In that case, if \(d_{\alpha}(x,y)=\varepsilon\), left continuity of \(M(x,y,\_)\) and of α at ε, provide a contradiction. So \(d_{\alpha }(x,y)<\varepsilon\), and thus we have shown the equivalence (1).
From this equivalence it immediately follows that the uniformities, and hence, the topologies induced by \((M,*)\) and \(d_{\alpha}\) coincide. In particular, a sequence in X is a Cauchy sequence in \((X,M,*)\) if and only if it is a Cauchy sequence in \((X,d_{\alpha})\). Consequently \((X,M,*)\) is complete if and only if \((X,d_{\alpha})\) is complete. This concludes the proof. □
Example 1
Example 2
Although Theorem 2 can we also applied to Examples 1 and 2 above, the following provides an instance where Theorem 3 works but not Theorem 2.
Example 3
Let \((X,M,*)\) be a fuzzy metric space such that \(*\geq *_{H_{\lambda}}\) for some \(\lambda\in(0,2]\). Let \(\alpha:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) defined as \(\alpha(t)=\lambda t/2\) if \(t\in [0,1]\), and \(\alpha(t)=(2t+1)/3\) if \(t>1\). It is clear that α is left continuous and strictly increasing on \(\mathbb{R}^{+}\), and it satisfies condition (c2) of Theorem 3.
3 Application to the fixed point theory of multivalued mappings on fuzzy metric spaces
Given a non-empty set X we shall denote by \(\mathcal{P}_{0}(X)\) the collection of all non-empty subsets of X.
Let \((X,d)\) be a metric space. A multivalued mapping \(T:X\rightarrow \mathcal{P}_{0}(X)\) is said to be a Caristi multivalued mapping (on \((X,d)\)) if there is a lower semicontinuous function \(\varphi:X\rightarrow \mathbb{R}^{+}\) such that for each \(x\in X\) there is \(y_{x}\in Tx\) satisfying \(d(x,y_{x})\leq\varphi(x)-\varphi(y_{x})\).
In particular, a self-mapping T of a metric space \((X,d)\) is said to be a Caristi mapping if there is a lower semicontinuous function \(\varphi :X\rightarrow\mathbb{R}^{+}\) such that \(d(x,Tx)\leq\varphi(x)-\varphi(Tx)\), for all \(x\in X\).
Caristi proved in [22] his celebrated theorem that every Caristi mapping on a complete metric space has a fixed point.
Kirk proved in [23] that Caristi’s fixed point theorem actually characterizes the metric completeness.
On the other hand, it is well known that Caristi’s fixed point theorem admits an easy a natural multivalued generalization (see e.g. [24]).
These results are usually combined as follows.
Theorem 4
- (1)
\((X,d)\) is complete.
- (2)
Every Caristi multivalued mapping \(T:X\rightarrow\mathcal {P}_{0}(X)\) has a fixed point, i.e., there is \(z\in X\) such that \(z\in Tz\).
- (3)
Every Caristi mapping \(T:X\rightarrow X\) has a fixed point.
In a recent paper [9] it was obtained a fuzzy version of Theorem 4. To this end, the authors of [9] introduced the following notion.
The notion of a fuzzy Caristi mapping for a self-mapping \(T:X\rightarrow X\) is defined in the obvious manner (see [9], Definition 2).
Then in Theorem 3 of [9] the following was proved.
Theorem 5
[9]
- (1)
\((X,M,*)\) is complete.
- (2)
Every fuzzy Caristi multivalued mapping \(T:X\rightarrow \mathcal{P}_{0}(X)\) has a fixed point.
- (3)
Every fuzzy Caristi’s mapping \(T:X\rightarrow X\) has a fixed point.
Remark 1
Actually Theorem 5 was proved in [9] for multivalued mappings from X to the set \(\mathcal{C}_{0}(X)\) of all non-empty closed subsets of \((X,M,*)\). However, the proof remains valid, without changes, for the case that T take values in \(\mathcal{P}_{0}(X)\).
We are going to improve Theorem 5 in two directions, which will provide partial solutions to a question posed in [9], p.1220.
Definition 2
The notion of an α-fuzzy Caristi mapping for a self-map \(T:X\rightarrow X\) is defined in the obvious manner.
Then we obtain the following.
Theorem 6
Let \((X,M,*)\) be a complete fuzzy metric space. Then every α-fuzzy Caristi multivalued mapping has a fixed point.
Proof
We now show that T is a Caristi multivalued mapping on \((X,d_{\alpha})\). Indeed, by Definition 2, for each \(x\in X\) there exists \(y_{x}\in Tx\) for which condition (3) is satisfied. Hence, for each \(t>0\) such that \(M(x,y,t)\leq1-\alpha(t)\), we deduce that \(\varphi(x)-\varphi(y_{x})\geq t\). Consequently, \(d_{\alpha}(x,y)\leq \varphi(x)-\varphi(y_{x})\).
Finally, T has a fixed point by Theorem 4, (1) ⟹ (2). □
Theorem 7
- (1)
\((X,M,*)\) is complete.
- (2)
Every α-fuzzy Caristi multivalued mapping has a fixed point.
- (3)
Every α-fuzzy Caristi mapping has a fixed point.
Proof
(1) ⟹ (2) follows from Theorem 6, and (2) ⟹ (3) is obvious.
We conclude the paper with an example that illustrates our results in this section.
Example 4
Let \(X=[0,1]\) and let ∗ be a continuous t-norm such that \(* \geq *_{Y_{p}}\), \(p\in(0,1]\). Define \(M:X\times X\times\mathbb{R}^{+}\rightarrow[0,1]\) as \(M(x,y,0)=0\), \(M(x,y,t)=x*y\) if \(x\neq y\) and \(t>0\), and \(M(x,x,t)=1\) for all \(t>0\). It is well known, and easy to check, that \((X,M,*)\) is a complete fuzzy metric space. For each \(x\in X\backslash \{1\}\), fix a subset \(A_{x}\) of X such that \(x\notin A_{x}\) and \(1\in A_{x} \). Define \(T:X\rightarrow\mathcal{P}_{0}(X)\) as \(T1=\{1\}\) and \(Tx=A_{x}\) for all \(x\in X\backslash\{1\}\). We show that T is an α-fuzzy Caristi multivalued mapping on X for \(\varphi:X\rightarrow\mathbb{R}^{+}\) given by \(\varphi(1)=0\) and \(\varphi(x)=1\) for all \(x\in X\backslash \{1\}\) and \(\alpha:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) given by \(\alpha(t)=t^{1/p} \) for all \(t\geq0\). Indeed, let \(x\in X\backslash\{1\}\). Take \(y_{x}=1\in T_{x}\). If \(\varphi(x)-\varphi(y_{x})< t\), then \(1< t\), and thus, \(1< t^{1/p}\). Hence \(M(x,y_{x},t)\geq0>1-t^{1/p}=1-\alpha(t)\). Therefore T is an α-fuzzy Caristi multivalued mapping, and all conditions of Theorem 6 are satisfied (note that we can also apply Theorem 7, (1) ⟹ (2)).
Declarations
Acknowledgements
The authors thank the referees for several useful suggestions. Salvador Romaguera and Pedro Tirado acknowledge the support of the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.
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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