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Some extensions of the Meir-Keeler theorem
- Lech Pasicki^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13663-016-0593-5
© The Author(s) 2017
- Received: 9 October 2016
- Accepted: 7 December 2016
- Published: 3 January 2017
Abstract
Meir and Keeler formulated their fixed point theorem for contractive mappings with purely metric condition. This idea was extended by numerous mathematicians. In this paper, we present a simple method of proving such theorems and give new results.
Keywords
- contractive mapping
- cyclic mapping
- dislocated metric
- partial metric
- fixed point
MSC
- 47H10
- 54H25
1 Introduction
In theorems cited further, we apply notations that are better suited to the results of the next sections of our paper.
Meir and Keeler proved the following theorem.
Theorem 1.1
[1], Theorem
The result of Meir and Keeler was extended by Matkowski and Ćirić.
Theorem 1.2
Both theorems are more general than the next theorem of Boyd and Wong [4].
Theorem 1.3
[4], Theorem 1
Jachymski [6] obtained the following more general result for metric spaces.
Theorem 1.4
[6], Corollary
Let f be a selfmapping of a complete metric space \((X,\rho)\) such that \(\rho(fy,fx) < d(y,x)\) for \(x \neq y\) and \(\rho(fy,fx) \leq\varphi(\rho(y,x))\) for all \(x,y \in X\), where \(\varphi\colon[0,\infty)\to[0,\infty)\) satisfies condition (2). Then f has a unique fixed point x, and \(f^{n}x_{0}\to x\) for any \(x_{0}\in X\).
It appears that the simple reasoning presented in [5] applies to conditions of the Meir-Keeler type. Consequently, we easily obtain extensions of the well-known theorems to the case of dislocated metric spaces or partial metric spaces. In addition, new results for cyclic mappings are proved. Also, the next theorem of Proinov is strongly extended in Section 3 (see Theorems 3.6, 3.7).
Theorem 1.5
[7], Theorem 4.2
2 Lemmas
Lemma 2.1
Proof
Assume that (4) holds and suppose \(\lim_{n \rightarrow\infty }a_{n}= 0\) is false. If \(a_{n}= 0\), then (3) yields \(a_{n+k}= 0\), \(k \in\mathbb{N}\), and \(\lim_{n \rightarrow\infty} a_{n}= 0\). Therefore, \(a_{n}> 0\), \(n \in\mathbb{N}\), and \((a_{n})_{n \in\mathbb{N}}\) decreases to an \(\alpha> 0\). We have \(\alpha< a_{n+1}< a_{n} \), \(n \in\mathbb{N}\), and \(a_{n}< \alpha+ \epsilon\) for large n. Now, from (4) it follows that \(\alpha< a_{n+1} \leq\alpha\), a contradiction. In turn, assume that \(\lim_{n \rightarrow\infty}a_{n}= 0\). If \(\alpha> 0\) is such that \(a_{n}\leq \alpha\), \(n \in\mathbb{N}\), then (4) is satisfied. If there exists \(a_{n}> \alpha\), then \(a_{n+1}< a_{n}\) (see (3)), and for some n and ϵ, \(a_{n}\) is the unique element of \((a_{n})_{n \in \mathbb{N}}\) in \((\alpha,\alpha+ \epsilon)\), that is, (4) is satisfied. □
We use the term of dislocated metric (i.e., d-metric) following Hitzler and Seda [8]; d-metric differs from metric since \(p(x,y) = 0\) yields \(x=y\) (no equivalence). The topology of a d-metric space is generated by balls. If p is a d-metric, then the pair \((X,p)\) was first defined by Matthews as a metric domain (see [9], Definition, p.13).
In the present section, we put \(x_{n} = f^{n}x_{0}\), \({n \in N}\).
Lemma 2.2
Proof
We apply Lemma 2.1 to \(a_{n}= p(x_{n+1},x_{n})\), \(n \in\mathbb {N}\). □
The subsequent definition is equivalent to the classical one (see (1)) if p is a metric.
Definition 2.3
If \(f\colon X \to X\) is a contractive mapping, then (5) holds for each \(x_{0}\in X\). Now, from Lemma 2.2 we obtain the following:
Corollary 2.4
Let \((X,p)\) be a d-metric space, and let f be a contractive selfmapping on X. Then \(\lim_{n \rightarrow\infty}p(x_{n+1},x_{n}) = 0\) iff (6) holds.
Corollary 2.5
Let \((X,p)\) be a d-metric space, and let f be a selfmapping on X satisfying (8) (or (7) or (5)). Then \(\lim_{n \rightarrow\infty }p(x_{n+1},x_{n}) = 0\) iff (6) holds (\(\alpha< p(\cdots) < \alpha+ \epsilon\) can be also replaced by \(\alpha< c_{f}(\cdots) < \alpha+ \epsilon\) in (6)).
From (9b) it follows that \(p(x,y) = 0\) yields \(p(x,x) = p(y,y) = 0\), that is, \(x = y\) (see (9a)), and consequently, each partial metric is a d-metric. Therefore, all results of the present paper for d-metric spaces remain valid also for partial metric spaces, though the partial metric topology (see [10]) differs from the d-metric one.
Corollary 2.6
Let \((X,p)\) be a partial metric space, and let f be a selfmapping on X satisfying (10) (or (8) or (7) or (5)). Then \(\lim_{n \rightarrow\infty}p(x_{n+1} ,x_{n}) = 0\) iff (6) holds (\(\alpha< p(\cdots) < \alpha+ \epsilon\) can be also replaced by \(\alpha< m_{f}(\cdots) < \alpha+ \epsilon\) or by \(\alpha< c_{f}(\cdots) < \alpha+ \epsilon\) in (6)).
Lemma 2.7
Proof
In a similar way, we prove the following lemma suitable for cyclic mappings.
Lemma 2.8
Proof
It can be seen that Lemma 2.7 is a consequence of Lemma 2.8 for \(t=1\).
Lemma 2.9
Proof
Definition 2.10
A selfmapping f on a d-metric space \((X,p)\) is 0-continuous at x if \(\lim_{n \rightarrow\infty }p(x,x_{n}) = 0\) implies \(\lim_{n \rightarrow\infty}p(fx,fx_{n}) = 0\) for each sequence \((x_{n})_{n \in\mathbb{N}}\) in X; f is 0-continuous if it is 0-continuous at each point \(x \in X\).
Lemma 2.11
Let \((X,p)\) be a d-metric space, and let f be a selfmapping on X. If f is contractive, then f has at most one fixed point; the same holds if f satisfies (8) or (10) and p satisfies (9b) or if p is a metric and (16) holds. If f is 0-continuous at x (e.g., if f is contractive) and \(\lim_{n \rightarrow \infty}p(x,f^{n}x_{0}) = 0\), then \(x = fx\) and \(p(x,x) = 0\).
Proof
3 Theorems
Let us recall ([11], Definition 2.3) that a d-metric space \((X,p)\) is 0-complete if for each sequence \((x_{n})_{n \in\mathbb{N}}\) in X with \(\lim_{m,n \rightarrow\infty}p(x_{n},x_{m}) = 0\), there exists \(x \in X\) such that \(\lim_{n \rightarrow\infty}p(x,x_{n}) = 0\).
Now, we are ready to extend the Ćirić theorem [3] and the Matkowski Theorem 1.5.1 in [2] (here Theorem 1.2) to the case of d-metric spaces (and \(c_{f}\) in place of p).
Theorem 3.1
Proof
Our space is 0-complete, and, therefore, the sequence \((f^{n}x_{0})_{n \in\mathbb{N}}\) converges (Lemma 2.7) to a unique fixed point of f (Lemma 2.11). □
Lemma 29 from [12] and the previous theorem yield the following result.
Theorem 3.2
Let h be a selfmapping on a 0-complete d-metric space \((X,p)\) such that \(f = h^{s}\) (for some \(s \in\mathbb{N}\)) satisfies the assumptions of Theorem 3.1. Then h has a unique fixed point, say x, and \(\lim_{n \rightarrow\infty}p(x,h^{n}x_{0}) = p(x,x) = 0\), \(x_{0}\in X\).
Lemma 2.8 enables us to extend the previous theorems to the case of cyclic mappings. The idea was introduced by Kirk et al. [13], and we apply Definition 2.5 from [5]. For a fixed \(t \in\mathbb{N}\), we put \(t++ = 1\) and \(j++ = j+1\) for \(j \in\{ 1,\ldots,t-1\}\). Then \(f\colon X \to X\) is cyclic if \(X = X_{1}\cup\cdots\cup X_{t}\) and \(f(X_{j}) \subset X_{j++}\), \(j = 1,\ldots,t\).
Theorem 3.3
Proof
Our space is 0-complete, and, therefore, the sequence \((f^{n}x_{0})_{n \in\mathbb{N}}\) converges (Lemma 2.8) to a unique fixed point of f (Lemma 2.11). □
An analogue of Theorem 3.2 for cyclic mappings is the following consequence of Theorem 3.3 and of [12], Lemma 29.
Theorem 3.4
Let h be a selfmapping on a 0-complete d-metric space \((X,p)\) such that \(f=h^{s}\) (for some \(s \in\mathbb{N}\)) satisfies the assumptions of Theorem 3.3. Then h has a unique fixed point, say x, and \(\lim_{n \rightarrow\infty}p(x,h^{n}x_{0}) = p(x,x) = 0\), \(x_{0}\in X\).
Let us note that a partial metric space \((X,p)\) is 0-complete iff \((X,p)\) treated as a d-metric space is 0-complete (see [12], Corollary 4, Proposition 5).
Remark 3.5
In view of Lemma 2.7, \(c_{f}\) can be replaced by p in any of conditions of Theorems 3.1, 3.2, 3.3, and 3.4; if p is a partial metric, then in view of Lemma 2.8, \(c_{f}\) can be replaced by \(m_{f}\) in any condition of those theorems. Theorem 3.1 for \(m_{f}\) becomes an extension of a theorem of Jachymski ([6], Theorem 2) to the case of partial metric spaces.
Theorem 3.6
Theorem 3.6 with \(t=1\) yields the following one.
Theorem 3.7
We can easily present extensions of the previous theorems for \(f = h^{s}\) (see Theorems 3.2 and 3.4).
Theorem 3.7 is a further extension of Theorem 4.2 in [7] (here Theorem 1.5).
Remark 3.8
Clearly, the results of the present section stay valid if we assume that \((X,p)\) is 0-complete for orbits of f because in the proofs of our lemmas only orbits were used.
Declarations
Acknowledgements
The work has been supported by the Polish Ministry of Science and Higher Education.
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Authors’ Affiliations
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