Dislocated metric and fixed point theorems
- Lech Pasicki^{1}Email author
https://doi.org/10.1186/s13663-015-0328-z
© Pasicki 2015
Received: 21 December 2014
Accepted: 17 May 2015
Published: 10 June 2015
Abstract
The paper deals with the here defined dislocated strong quasi-metric (if \(p(y,x)= 0\), then \(x=y\); \(0 \leq p(z,x) \leq p(z,y) + p(y,x)\)) and with the well-known notion of the dislocated metric (in addition, \(p(y,x)= p(x,y)\)). In particular, the partial metric is a kind of dislocated metric. Our basic results on general contractions (also for cyclic mappings) and results of variational type can be treated as a starting point for further development.
Keywords
MSC
1 Introduction
Recent years have witnessed the appearance of many papers devoted to fixed point theorems for partial metric spaces. The aim of the present paper is to show that the dislocated (strong quasi-)metric as presented here (Definition 2.1) has a great potential. Each partial metric is a dislocated metric and examples preceding Definition 2.4 show that the dislocated metric is more general. The paper is divided into three sections.
In Section 2 the definitions of a dislocated (strong quasi-)metric and of a partial metric are presented. This section contains some examples and a comparison between the two notions. Also some additional ideas (0-completeness, Kerp) are included.
Section 3 is devoted to fixed point theorems for general contractions. The simplest requirement is condition (3.1): \(p(f(y),f(x)) \leq \varphi (p(y,x))\), for all \(x,y \in X\), where p is a dislocated metric on X, \(f : X \rightarrow X\) is a mapping, and the comparison function \(\varphi : [0,\infty) \rightarrow[0,\infty)\) belongs to a wide class of mappings defined in [1] and here. The main classical results are Theorem 3.3 (a direct extension of the celebrated theorems of Matkowski [2], Theorem 1.2, and of Boyd-Wong [3], Theorem 1), and a more general Theorem 3.5. The most sophisticated ones are the theorems for cyclic mappings (see Definition 3.6): Theorem 3.9, and a result of a new type, Theorem 3.10, which is proved with the use of cross mappings defined in [4]. Our theorems extend also some general results of Karapinar and Salimi [5], Theorems 1.8, 1.9.
Section 4 (p is a dislocated strong quasi-metric) contains theorems obtained with order reasoning for a transitive relation defined by \(y \preccurlyeq x\) iff \(\psi(y) + p(y,x) - \psi(x) \leq0\) where \(\psi: X \rightarrow\mathbb{R}\) is a mapping. The results of variational type involve the existence and properties of the smallest element in a maximal chain. Typical are Theorems 4.1 and 4.2, and an extension of the Ekeland principle (Theorem 4.5). The fixed point results are Theorem 4.3 and Theorem 4.4 (an extension of the celebrated theorems of Caristi and Takahashi).
2 Dislocated metric and dislocated strong quasi-metric
The notion of dislocated metric was introduced by Hitzler and Seda in [6].
Definition 2.1
In the previous version of the present paper (entitled ‘Near (quasi-)metric and fixed point theorems’) dislocated metric was called near metric because the author was not aware of Hitzler’s definition, and a dislocated strong quasi-metric was called a near quasi-metric.
The nonalphabetical order of \(x,y,z\) in conditions (2.1), (2.2), (2.3) is better suited to the results of Section 4 (\(y = f(x) \preccurlyeq x\) corresponds with \(\psi(y) \leq\psi(x)\)).
The topology of a d-metric (or dsq-metric) space \((X,p)\) is generated by balls \(B(x,r) = \{y \in X : p(x,y)< r\}\). Clearly, \(x \in B(x,r) \) does not necessarily hold, but the family of all balls generates the respective smallest topology for \(X = \bigcup\{ B(x,r) : x \in X, r > 0\}\) [7], Theorem 12, p.47. If \(Z = \operatorname {Ker}p\) is nonempty, then \((Z,p_{|Z \times Z})\) is a metric (or quasi-metric) subspace of \((X,p)\).
Recently, Amini-Harandi has defined metric-like mapping σ (identic with the idea of d-metric p) and metric-like space \((X,\sigma)\) [8], Definition 2.1. The topology of his space generated by σ-balls \(B_{\sigma}(x,\epsilon) = \{y \in X : |\sigma(x,y) - \sigma(x,x)| < \epsilon\}\) usually differs from the topology for a d-metric space.
It should be noted that also Karapinar and Salimi [5] follow the ideas of Amini-Harandi, which are better suited to partial metric spaces (see Definition 2.4).
Proposition 2.2
Let \((X,p)\) be a d-metric (or dsq-metric) space. Then \(p(\cdot,y)\) is lower semicontinuous at points of Kerp, \(y \in X\).
Proof
Let \((x_{n})_{n \in\mathbb{N}}\) be such that \(\lim_{n \rightarrow \infty}p(x,x_{n})= 0\). Then the inequality \(p(x,y) \leq \liminf_{n \rightarrow\infty}p(x_{n},y)\) is a consequence of \(p(x,y)\leq p(x,x_{n})+ p(x_{n},y)\). □
Definition 2.3
In view of (2.4) a point x as in (2.5) is unique if p is a d-metric, and then \(p(x,x)= 0\).
Another example is \(X = \{(x_{1},x_{2}) \in \mathbb {R}^{2} : x_{1}\geq-1, x_{2}\geq0 \}\) with p defined by \(p((x_{1},x_{2}), (y_{1},y_{2})) = x_{1}+ x_{2}+ y_{1} + y_{2} + 2\).
Let us recall the notions of a partial metric due to Matthews [9], and of a dualistic partial metric due to Oltra and Valero [10] and O’Neill [11].
Definition 2.4
For details concerning the topology of a (dualistic) partial metric space see, e.g. [10].
We can see that each partial metric is a d-metric ((2.6) and (2.7) for \(p \geq0\) yield (2.1)). On the other hand, \(p(x,y)= x + y + 2\) does not necessarily mean that \(p(y,y) \leq p(y,x)\), and therefore the d-metrics in our examples are not partial metrics.
A dualistic partial metric space \((X,p)\) is 0-complete (see [12], Definition 2.1, [1], Corollary 4) if for every sequence such that \(\lim_{m,n \rightarrow\infty} p(x_{n},x_{m}) = 0\) there exists an \(x \in X\) such that \(\lim_{n \rightarrow\infty}d(x,x_{n}) = 0\) and \(p(x,x)= 0\).
Now, it is clear that if a partial metric space \((X,p)\) is 0-complete, then \((X,p)\) treated as a dislocated metric space is also 0-complete.
Lemma 2.5
(cf. [13], Lemma 2.2)
Let \((X,p)\) be a d-metric space with a nonempty kernel Z. Then \((Z,p_{|Z \times Z})\) is a metric subspace of \((X,p)\); if \((X,p)\) is 0-complete, then \((Z,p_{|Z \times Z})\) is complete.
Proof
3 General contractions
It should be noted that the d-metric p defines the metric δ in the following way: \(\delta(x,x) = 0\), and \(\delta(x,y) = p(x,y)\) for \(x \neq y\). One can see that a d-metric space \((X,p)\) is 0-complete (Definition 2.3) iff the metric space \((X,\delta )\) is complete.
Our theorems in the present section are new also if p is a metric. The special features of the mapping φ will enable one to give proofs of these theorems using metric δ but such full proofs would be unnecessarily complex (see also [14]).
Let us consider \(\boldsymbol{\Psi}_{\mathbf{P}}\) (\(\Phi_{P}\subset\Psi_{P} \subset \Phi\)) consisting of mappings φ for which every sequence \((a_{n})_{n \in\mathbb{N}}\) such that \(0 < a_{n+1}\leq \varphi (a_{n} )\), \(n \in\mathbb{N}\) converges to zero. If \(\varphi \in\Phi\) satisfies (3.4), then \(\varphi \in\Psi_{P}\) (see the proof of [1], Proposition 16). It was noted in [1] that Theorems 28 and 31 of that paper are valid also for \(\Phi_{P}\) replaced by \(\Psi_{P}\), as \(\varphi (0)\) is meaningless.
Example
- (i)
\(\varphi (x) < 1/2\), \(x \in \mathbb {R}\),
- (ii)
\(\varphi (x) < 1/(n+1)\), \(x \leq1/n\),
- (iii)
\(\limsup_{\beta\rightarrow(1/n)^{+}} \varphi (\beta) = 1/n\), \(1 < n \in \mathbb {N}\).
The subsequent two lemmas for (3.1), (3.2) are partial extensions of [1], Lemmas 25, 26, proved for partial metric spaces (see also [15], Lemmas 1, 2).
Lemma 3.1
Proof
Lemma 3.2
Let \((X,p)\) be a 0-complete d-metric space, and let \(f : X \rightarrow X\) be a mapping satisfying condition (3.1) or (3.2), for all \(x,y \in X\) and a \(\varphi \in \Phi\). If for \(x_{n}= f^{n}(x_{0})\), \(\lim_{m,n \rightarrow\infty}p(x_{n},x_{m}) = 0\) holds, then there exists a unique fixed point x of f, \(\lim_{n \rightarrow\infty}p(x,x_{n})= 0\) (i.e. \(x = \lim_{n \rightarrow\infty}x_{n}\) in \((X,p)\)), and \(p(x,x)= 0\).
Proof
Now, we are ready to prove the following extension of [1], Theorem 27 (the proof is almost the same as in [1]).
Theorem 3.3
Proof
Let us recall the following.
Lemma 3.4
[1], Lemma 29
Let \(f : X \rightarrow X\) be a mapping such that \(f^{t}\) for a \(t \in \mathbb {N}\) has a unique fixed point, say x. Then x is the unique fixed point of f. If, in addition, \(x \in\lim_{n \rightarrow\infty} (f^{t})^{n}(x_{0})\), \(x_{0}\in X\), then \(x \in\lim_{n \rightarrow\infty}f^{n}(x_{0})\), \(x_{0}\in X\) holds.
Now, Theorem 3.3 and Lemma 3.4 yield the following.
Theorem 3.5
Let \((X,p)\) be a 0-complete d-metric space, and let \(f : X \rightarrow X\) be a mapping satisfying condition (3.1) or (3.2), for all \(x,y \in X\) with f replaced by \(f^{t}\) for a \(t \in \mathbb {N}\), and a \(\varphi \in\Phi\) having property (3.4) or a \(\varphi \in\Psi _{P}\) such that (3.6) holds. Then f has a unique fixed point; if \(x = f(x)\), then x satisfies \(p(x,x)= 0\) and \(\lim_{n \rightarrow\infty}p(x,f^{n}(x_{0})) = 0\), \(x_{0}\in X\).
Kirk et al. [16] suggested the idea of cyclic mappings which was later formalized by Rus in [17] as the cyclic representation of \(X = X_{1}\cup \cdots\cup X_{t}\) with respect to f. The next definition means the same, but it is more compact.
Definition 3.6
A mapping \(f : X \rightarrow X\) is called cyclic on \(X_{1},\ldots,X_{t}\) (for a \(t > 1\)) if \(\emptyset\neq X = X_{1}\cup\cdots\cup X_{t}\), and \(f(X_{j}) \subset X_{j++} \), \(j = 1,\ldots,t\), where \(j++ = j+1\) for \(j = 1,\ldots,t-1\), and \(t++ = 1\).
Clearly, \(X_{j}\neq\emptyset\) for a j in Definition 3.6, and hence \(X_{j}\neq\emptyset\), \(j = 1,\ldots,t\).
The proof of Lemma 3.1 works also for the following.
Lemma 3.7
Let \(p : X \times X \rightarrow[0,\infty)\) be a mapping, and let \(f : X \rightarrow X\) be cyclic on \(X_{1},\ldots,X_{t}\). Assume that (3.1) or (3.2) is satisfied for all \(x \in X_{j}\), \(y \in X_{j++}\), \(j=1,\ldots,t\), and a \(\varphi \in\Phi\). Then condition (3.5) holds, and if \(\varphi \in\Psi_{P}\), then \(\lim_{n \rightarrow\infty }p(f^{n+1}(x),f^{n}(x)) = 0\), \(x \in X\).
If we consider n such that \(x \in X_{j}\) and \(x_{n}\in X_{j++}\) for a \(j \in\{1,\ldots,t\}\), then the proof of Lemma 3.2 yields the following.
Lemma 3.8
Let \((X,p)\) be a 0-complete d-metric space, and let \(f : X \rightarrow X\) be cyclic on \(X_{1},\ldots,X_{t}\). Assume that (3.1) or (3.2) is satisfied for all \(x \in X_{j} \), \(y \in X_{j++}\), \(j=1,\ldots,t\), and a \(\varphi \in\Phi\). If for \(x_{n}= f^{n}(x_{0})\), \(\lim_{m,n \rightarrow\infty}p(x_{n},x_{m}) = 0\) holds, then there exists a unique fixed point x of f, \(\lim_{n \rightarrow\infty}p(x,x_{n})= 0\) (i.e. \(x = \lim_{n \rightarrow\infty}x_{n}\) in \((X,p)\)), and \(p(x,x)= 0\).
Lemmas 3.7 and 3.8 yield the following extension of Theorem 3.3.
Theorem 3.9
Let \((X,p)\) be a 0-complete d-metric space, and let \(f : X \rightarrow X\) be cyclic on \(X_{1},\ldots,X_{t}\). Assume that (3.1) or (3.2) is satisfied for all \(x \in X_{j} \), \(y \in X_{j++}\), \(j=1,\ldots,t\), and a \(\varphi \in\Phi\) having property (3.4) or a \(\varphi \in\Psi _{P}\) such that (3.6) holds. Then f has a unique fixed point; if \(x = f(x)\), then \(p(x,x)= 0\) and \(\lim_{n \rightarrow\infty}p(x,f^{n} (x_{0} )) = 0\), \(x_{0}\in X\).
Proof
Karapinar and Salimi [5], Definition 1.7, have defined the notion of a generalized ϕ-ψ-contractive mapping. It appears that \(\varphi = (id + \psi )^{-1}\circ(id + \psi- \phi) \in\Phi_{P}\) because condition (3.4) is satisfied. Therefore, [5], Theorem 1.9, is a particular case of Theorem 3.3, and [5], Theorem 1.8, is a consequence of Theorem 3.9 (see also the example preceding Lemma 3.1).
Let us present cyclic mappings of the second type, i.e. those for (3.1) or (3.2) with \(x,y \in X_{j}\), \(j = 1,\ldots,t\). It is convenient to apply the idea of cross mappings introduced in [4].
We can see that for \(E_{j} \subset X_{j}\), \(j = 1,\ldots,t\) the composition \(F_{t} \circ F_{t-1} \circ\cdots\circ F_{1}\) has a fixed point in \(X_{1} \) iff F has a fixed point. This concept is very efficient for multivalued mappings (see [4], Section 3). Let us apply cross mappings to prove the following.
Theorem 3.10
Let \((X,p)\) be a 0-complete d-metric space, and let \(f : X \rightarrow X\) be cyclic on 0-complete sets \(X_{1},\ldots,X_{t}\). Assume that (3.1) or (3.2) is satisfied for all \(x,y \in X_{j}\), \(j=1,\ldots,t\), and a nondecreasing \(\varphi \in\Psi_{P}\). Then f has a unique fixed point; if \(x = f(x)\), then \(p(x,x)= 0\) and \(\lim_{n \rightarrow\infty}p(x,f^{n}(x_{0})) = 0\), \(x_{0}\in X\).
Proof
There exist many papers concerning cyclic mappings (see, e.g., the references of [18]), and it is very likely that Theorems 3.9 and 3.10 are just a starting point for further development.
4 Variational results
In this section p is a dislocated strong quasi-metric (i.e. a dsq-metric).
Let us prove the following analog of [19], Theorem 21.
Theorem 4.1
- (i)
\(\psi(x) = \inf\{\psi(z) : z \in A\}\) and \(x \in B\),
- (ii)
\(\psi(x) + p(x,x_{0}) - \psi(x_{0}) = \inf\{\psi(z) + p(z,x_{0}) - \psi(x_{0}) : z \in A \} \leq0\),
- (iii)
\(0 < \psi(y) + p(y,x) - \psi(x)\), for each \(y \in X \setminus\{x\}\),
- (iv)
\(x \in \operatorname {Ker}p\) (i.e. \(p(x,x)= 0\) if p is a d-metric).
Proof
The ‘order’ reasoning fails for a quasi-metric (\(x = y\) iff \(q(x,y) = q(y,x) = 0\), \(q(z,x) \leq q(z,y) + q(y,x)\)), as \(q(y,x) = 0\) does not necessarily yield \(q(x,y) = 0\).
A reasoning similar to the one presented in the above proof yields the following analog of [19], Theorem 22.
Theorem 4.2
- (i)
\(\psi(x) = \inf\{\psi(z) : z \in A\}\),
- (ii)
\(\psi(x) + p(x,x_{0}) - \psi(x_{0}) = \inf\{\psi(z) + p(z,x_{0}) - \psi(x_{0}) : z \in A \} \leq0\),
- (iii)
\(0 < \psi(y) + p(y,x) - \psi(x)\), for each \(y \in X \setminus\{x\}\),
- (iv)
\(x \in \operatorname {Ker}p\) (i.e. \(p(x,x)= 0\) if p is a d-metric).
Proof
In view of Kuratowski’s lemma [7], p.33, there exists a maximal chain A containing \(x_{0}\). Now, for A we follow the proof of Theorem 4.1, omitting the last sentence. □
We also have the following analog of [19], Theorem 23.
Theorem 4.3
Proof
If \(x_{0}\notin F(x_{0})\), then Theorem 4.2 applies and (iii) contradicts (4.2) for \(x \notin F(x)\). □
The subsequent theorem extends the theorems of Caristi [20], Theorem (2.1)′, and Takahashi [21], Theorem 5.
Theorem 4.4
The subsequent theorem extends Ekeland’s variational principle [22], Theorem 1, (cf. [19], Theorem 25, and [23], Theorem 3).
Theorem 4.5
- (a)
there exists an \(x \in X\) such that \(\psi(x) \leq\psi (x_{0})\) and \(\psi(x) - p(y,x) < \psi(y)\), for each \(y \in X \setminus\{x\}\),
- (b)
for any \(\epsilon> 0\) and each \(x_{0}\in X\) with \(p(x_{0} ,x_{0}) = 0\) there exists an \(x \in X\) such that \(\psi(x) \leq\psi(x_{0})\), and \(\psi(x) - \epsilon p(y,x) < \psi(y)\), for each \(y \in X \setminus\{x\}\); if, in addition, \(\psi(x_{0}) \leq\epsilon+ \inf\{\psi(z) : z \in X \}\) holds, then \(p(x,x_{0}) \leq1\).
Proof
Kirk and Saliga [24], p.2769, say that for a Hausdorff space \((X,\tau)\) a mapping \(\psi: X \rightarrow \mathbb {R}\) is τ-lower semicontinuous from above if given any sequence \((x_{n})_{n \in\mathbb{N}}\) in X, the conditions: \((\psi(x_{n}))_{n \in\mathbb{N}}\) decreases to α and \(\lim_{n \rightarrow\infty}x_{n}= x\), yield \(\psi(x) \leq\alpha\). The proof of Theorem 4.1 works without any change if we relax in such a way the assumption of lower semicontinuity of ψ at points of Kerp (we may then say that ψ is lower semicontinuous from above at the points of Kerp). Consequently, all results of the present section stay valid with this weaker assumption.
It should be noted that the dislocated (strong quasi-)metric p defines the (quasi-)metric δ in the following way: \(\delta(x,x) = 0\), and \(\delta(x,y) = p(x,y)\) for \(x \neq y\). One can see that a dislocated (strong quasi-)metric space \((X,p)\) is 0-complete (Definition 2.3) iff the (quasi-)metric space \((X,\delta)\) is complete (for the quasi-metric, see [19], Definitions 14, 7).
Consequently, if a proof of a fixed point theorem for complete metric spaces is based on a sequence \((x_{n})_{n \in\mathbb{N}}\) that converges to a fixed point, \(\delta (x_{n},x_{n+1}) \neq0\), \(n \in\mathbb{N} \) (and \(\delta(x_{n},x_{n}) = 0\) can be disregarded), then the same proof works for 0-complete d-metric spaces, and further, for 0-complete partial metric spaces.
Another method is to prove that \(Z = \operatorname {Ker}p\) is nonempty, \(f_{|Z} : Z \rightarrow Z\), and to apply Lemma 2.5 (see comments on Theorems 2.3 and 2.4 [13]).
Declarations
Acknowledgements
The author is obliged to Pascal Hitzler and Anthony Karel Seda for their old idea of dislocated metric (thankfully known to the referee), which is identical to the ‘new’ concept of near metric. That is why the former version of this manuscript entitled ‘Near (quasi-)metric and fixed point theorems’ (sent to FPTAppl in December 2014) had to be rearranged to the present form). The work has been supported by the Polish Ministry of Science and Higher Education.
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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