The Boyd-Wong idea extended
- Lech Pasicki^{1}Email author
https://doi.org/10.1186/s13663-016-0553-0
© Pasicki 2016
Received: 10 March 2016
Accepted: 6 May 2016
Published: 21 May 2016
Abstract
Boyd and Wong in their celebrated paper ‘On nonlinear contractions’ assumed the comparison function to be upper semicontinuous from the right. Our requirement presented in this paper is much more general and it extends also the well-known Matkowski condition.
Keywords
MSC
1 Introduction
Clearly, (3) is more general than (2). In turn, Matkowski in [3], Theorem 1.2 assumed φ to be nondecreasing and \(\lim_{n \rightarrow \infty}\varphi^{n}(\alpha) = 0\), \(\alpha> 0\). It is well known that for every function φ satisfying Matkowski’s condition we have \(\varphi (\alpha) < \alpha\), \(\alpha> 0\). Let us show that (3) extends the Matkowski condition for φ such that \(\varphi(\alpha) < \alpha\), \(\alpha> 0\). Assume φ is nondecreasing, \(\varphi(\beta) < \beta\), \(\beta > 0\), and suppose \(\varphi(\cdot) > \alpha> 0\) on an interval \((\alpha,\alpha+ \epsilon)\). Then for any \(\beta\in (\alpha,\alpha+ \epsilon)\) we have \(\alpha< \varphi(\beta) < \beta< \alpha+ \epsilon\), and consequently, \(\alpha< \varphi^{n}(\beta) < \cdots< \varphi(\beta) < \alpha+ \epsilon\), i.e. \(\lim_{n \rightarrow\infty}\varphi^{n}(\beta) \geq\alpha> 0\), a contradiction. Therefore φ must be equal to α on \((\alpha,\alpha+ \epsilon)\).
2 Definitions and auxiliary results
Let us notice that the assumption \(\varphi\in\Phi\) (or a stronger one) is present in all theorems concerning conditions (1) or (2).
Proposition 2.1
\(\Psi_{P}\subset\Phi\).
Proof
Suppose \(\alpha\leq\varphi(\alpha)\) for a \(\varphi\in\Psi_{P}\) and an \(\alpha> 0\). Then all \(a_{n}= \alpha\), \(n \in\mathbb{N}\) satisfy \(0 < a_{n+1}\leq\varphi(a_{n})\), and \(\lim_{n \rightarrow\infty}a_{n}= \alpha> 0\), a contradiction. □
Lemma 2.2
If a \(\varphi\in\Psi_{P}\), then \(\varphi\in\Phi\) and (3) is satisfied.
Proof
Suppose a \(\varphi\in\Psi_{P}\) does not satisfy (3), i.e. there exists a sequence \((x_{n})_{n \in\mathbb{N}}\) decreasing to an \(\alpha> 0\), and such that \(\varphi (x_{n}) > \alpha\), \(n \in\mathbb{N}\). Let us adopt \(a_{1} = x_{1}\). There exists an \(a_{2} \in\{x_{1},\ldots\}\) such that \(a_{2} \leq\varphi(a_{1}) < a_{1}\). If \(a_{n}\) is defined, then \(a_{n+1}\in \{x_{1},\ldots\}\) is such that \(a_{n+1}\leq\varphi(a_{n}) < a_{n}\). Our sequence \((a_{n})_{n \in \mathbb{N}}\) satisfies \(0 < \alpha< a_{n+1}\leq\varphi(a_{n})\), \(n \in\mathbb{N}\), and it does not converge to zero. Therefore, \(\varphi\notin\Psi_{P}\), a contradiction. □
Lemma 2.3
If a \(\varphi\in\Phi\) satisfies (3), then \(\varphi\in\Psi_{P}\).
Proof
Corollary 2.4
\(\Psi_{P}\) consists of all mappings \(\varphi\in\Phi\) satisfying (3).
Hitzler and Seda in [4] introduced the following notion of dislocated metric space.
The first idea of cyclic mappings is due to Kirk, Srinivasan and Veeramani [5]. The subsequent definition refines [2], Definition 3.6 in such a way that the case of \(X = X_{1}\) is included.
Definition 2.5
A mapping \(f: X \rightarrow X\) is called cyclic on \(X_{1},\ldots,X_{t}\) (for a \(t \geq1\)) 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 \(j < t\), and \(t{+}{+} = 1\).
3 Theorems
The theorems of the present section look like some theorems from [2], but condition (3) matters a lot. Our first theorem extends [2], Theorem 3.3.
Theorem 3.1
Let \((X,p)\) be a 0-complete d-metric space, and let \(f: X \rightarrow X\) be a mapping satisfying condition (6) or (7), for all \(x,y \in X\) and a \(\varphi\in \Phi\) such that (3) holds. Then f has a unique fixed point; if \(x = f(x)\), then \(\lim_{n \rightarrow\infty }p(x,f^{n}(x_{0})) = p(x,x)= 0\) (i.e. \(x \in\operatorname{Ker}p\)), \(x_{0}\in X\).
Proof
Now, Theorem 3.1, and [6], Lemma 29 yield the following extension of [2], Theorem 3.5.
Theorem 3.2
Let \((X,p)\) be a 0-complete d-metric space, and let \(f: X \rightarrow X\) be a mapping satisfying condition (6) or (7), for all \(x,y \in X\) with f replaced by \(f^{s} \) for an \(s \in\mathbb{N}\), and a \(\varphi\in\Phi\) having property (3). Then f has a unique fixed point; if \(x = f(x)\), then \(\lim_{n \rightarrow\infty}p(x,f^{n}(x_{0})) = p(x,x)= 0\), \(x_{0}\in X\).
A refinement of the proof of Theorem 3.1, yields the following extension of [2], Theorem 3.9.
Theorem 3.3
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 (6) or (7) is satisfied for all \(x \in X_{j}\), \(y \in X_{j++}\), \(j=1,\ldots,t\) and a \(\varphi\in\Phi\) having property (3). Then f has a unique fixed point; if \(x = f(x)\), then \(\lim_{n \rightarrow\infty}p(x,f^{n}(x_{0})) = p(x,x)= 0\), \(x_{0}\in X\).
Proof
Clearly Theorem 3.3 is more general than Theorem 3.1. The proof of Theorem 3.1 is easier, it helps to understand the idea of the proof of Theorem 3.3, and therefore, it is also presented.
Now, Theorem 3.3, and [6], Lemma 29 yield the following.
Theorem 3.4
Let \((X,p)\) be a 0-complete d-metric space, and let \(f: X \rightarrow X\) be a mapping such that \(f^{s}\) is cyclic on \(X_{1},\ldots,X_{t}\) for an \(s \in\mathbb{N}\). Assume that (6) or (7) is satisfied for all \(x \in X_{j}\), \(y \in X_{j++}\), \(j=1,\ldots,t\) with f replaced by \(f^{s}\), and a \(\varphi\in\Phi\) having property (3). Then f has a unique fixed point; if \(x = f(x)\), then \(\lim_{n \rightarrow\infty}p(x,f^{n}(x_{0})) = p(x,x)= 0\), \(x_{0}\in X\).
Remark 3.5
Let us note that [2], Lemmas 3.2, 3.8 stay valid if we assume that \((X,p)\) is 0-complete for orbits of f, i.e. (5) holds for \(x_{n}= f^{n}(x_{0} )\), \(x_{m} = f^{m}(x_{0})\), \(m,n \in\mathbb{N}\), \(x_{0}\in X\). Consequently, theorems of Section 3 stay valid if the assumption that \((X,p)\) is 0-complete is replaced by the requirement that \((X,p)\) is 0-complete for orbits of f.
Declarations
Acknowledgements
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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