- Research
- Open Access
- Published:

# The Boyd-Wong idea extended

*Fixed Point Theory and Applications*
**volume 2016**, Article number: 63 (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.

## Introduction

Boyd and Wong in [1] considered the condition \(\rho(f(x),f(y)) \leq\varphi(\rho(x,y))\), where \(\varphi: [0,\infty)\rightarrow[0,\infty)\) is a mapping such that \(\varphi (\alpha) < \alpha\), \(\alpha> 0\), and

holds (*i.e.*
*φ* is upper semicontinuous from the right on \((0,\infty)\)).

A part of [2], Theorem 3.3 shows that (1) can be replaced by

In the present paper we apply the following condition:

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 \((\alpha,\alpha+ \epsilon)\).

It is clear that (3) is equivalent to the following condition:

as \(\limsup_{\beta\rightarrow\alpha^{+}}\varphi(\beta) < \alpha\) yields \(\varphi(\cdot) < \alpha\) on some interval \((\alpha,\alpha+ \epsilon)\).

## Definitions and auxiliary results

It is nice if for \(f: X \rightarrow X\) the inequality

yields \(\lim_{n \rightarrow\infty}\rho(f^{n+1}(x),f^{n}(x)) = 0\). Therefore, we are interested in mappings \(\varphi: [0,\infty)\rightarrow[0,\infty)\) such that for each sequence \((a_{n})_{n \in\mathbb{N}}\) the condition \(0 < a_{n+1}\leq\varphi(a_{n})\), \(n \in\mathbb{N}\) yields \(\lim_{n \rightarrow\infty}a_{n}= 0\). The family of all such mappings was denoted in [2] by \(\boldsymbol{\Psi}_{\mathbf{P}}\), while the family of all mappings \(\varphi: [0,\infty)\rightarrow[0,\infty)\) such that \(\varphi(\alpha) < \alpha\), \(\alpha> 0\) was denoted by **Φ**.

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

Let \((a_{n})_{n \in\mathbb{N}}\) be a sequence such that \(0 < a_{n+1}\leq\varphi(a_{n})\), \(n \in\mathbb{N}\) for a \(\varphi\in\Phi\). Then we have

Therefore, \((a_{n})_{n \in\mathbb{N}}\) decreases, say to an *α*. Suppose \(\alpha> 0\). Then from (3) it follows that there exists an interval \((\alpha,\alpha+\epsilon)\) on which \(\varphi (\cdot) \leq\alpha\). For large *n* all \(a_{n}\) belong to this interval. Now, we have \(\alpha< a_{n+1}\leq \varphi (a_{n}) \leq\alpha\), a contradiction. Consequently, \(\alpha= 0\), *i.e.*
\(\varphi\in\Psi_{P}\). □

### 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.

Let *X* be a nonempty set, and \(p: X \times X \rightarrow[0,\infty)\) a mapping satisfying

Then *p* is called a *dislocated metric* (briefly a d-metric), and \((X,p)\) is called a *dislocated metric space* (briefly a d-metric space).

If \((X,p)\) is a dislocated metric space then (see [2], (2.4))

Let us recall that a d-metric space \((X,p)\) is called *0-complete* (see [2], Definition 2.3) if the following condition is satisfied:

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 \(t{+}{+} = 1\).

Our fixed point theorems concern mappings \(f: X \rightarrow X\) satisfying

or

for

where \((X,p)\) is a d-metric space.

## 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

It is sufficient to prove that \(\lim_{m,n \rightarrow\infty }p(x_{n},x_{m})= 0\) holds for \(x_{n}= f^{n} (x_{0})\), \(n \in\mathbb{N}\) (see [2], Lemma 3.2). From the fact that \(\varphi\in\Psi_{P}\) (Lemma 2.3) it follows that \(\lim_{n \rightarrow\infty}p(x_{n+1},x_{n}) = 0\) (see [2], Lemma 3.1). Suppose that there exists an infinite set \(K \subset\mathbb{N}\) such that for each \(k \in K\) there exists an \(n \in\mathbb{N}\) for which \(p(x_{n+1+k},x_{k}) > \alpha> 0\) holds. Let \(n = n(k) > 0\) be the smallest numbers satisfying this inequality for \(k \in K\). For simplicity let us adopt \(x = f^{k}(x_{0})\) (\(x_{-1}= f^{k-1}(x_{0})\)), and \(x_{m} = f^{m}(x)\), \(m \in\mathbb{N}\). From

(see (7)) we get \(p(x_{n+1},x) \leq\varphi (p(x_{n},x_{-1}))\), for large *k* (or from (6) directly), as

The inequality

yields \(p(x_{n},x_{-1}) < \alpha+\epsilon\), for large *k*. Consequently, from (3) and \(\varphi(\beta) < \alpha\), \(\beta\leq\alpha\), we obtain

for large *k*, a contradiction, *i.e.*
\(\lim_{m,n \rightarrow \infty}p(x_{n},x_{m})= 0\). □

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

It is sufficient to prove that \(\lim_{m,n \rightarrow\infty }p(x_{n},x_{m})= 0\) holds for \(x_{n}= f^{n} (x_{0})\), \(n \in\mathbb{N}\) (see [2], Lemma 3.8). From the fact that \(\varphi\in\Psi_{P}\) (Lemma 2.3) it follows that \(\lim_{n \rightarrow\infty}p(x_{n+1},x_{n}) = 0\) (see [2], Lemma 3.7). Suppose that there exists an infinite set \(K \subset \mathbb{N}\) such that for each \(k \in K\) there exists an \(n \in\mathbb{N}\) for which \(p(x_{(n+1)t+k+1},x_{k}) > \alpha> 0\) holds. Let \(n = n(k) > 0\) be the smallest numbers satisfying this inequality for \(k \in K\). For simplicity let us adopt \(x = f^{k}(x_{0})\) (\(x_{-1}= f^{k-1}(x_{0})\)), and \(x_{m} = f^{m}(x)\), \(m \in\mathbb{N}\). Clearly, \(x \in X_{j}\) yields \(x_{nt+1},x_{(n+1)t+1}\in X_{j++}\). In view of (7) we have

which, for large *k* (or from (6) directly) gives

as

Now,

yields \(p(x_{(n+1)t},x_{-1}) < \alpha+ \epsilon\), for large *k*. Consequently, from (3) and \(\varphi(\beta) < \alpha\), \(\beta\leq\alpha\), we obtain

for large *k*, a contradiction. Now, it is clear that \(\lim_{m,n \rightarrow\infty} p(x_{m+nt+1},x_{m}) = 0\). One step more is necessary for \(t > 1\). We have

for any \(s \in\{2,\ldots,t\}\), *i.e.*
\(\lim_{m,n \rightarrow \infty}p(x_{n},x_{m}) = 0\). □

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*.

## References

- 1.
Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc.

**20**, 458-464 (1969) - 2.
Pasicki, L: Dislocated metric and fixed point theorems. Fixed Point Theory Appl.

**2015**, 82 (2015) - 3.
Matkowski, J: Integrable solutions of functional equations. Diss. Math.

**127**, 1-68 (1975) - 4.
Hitzler, P, Seda, AK: Dislocated topologies. J. Electr. Eng.

**51**(12), 3-7 (2000) - 5.
Kirk, WA, Srinivasan, PS, Veeramani, P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory

**4**, 79-89 (2003) - 6.
Pasicki, L: Fixed point theorems for contracting mappings in partial metric spaces. Fixed Point Theory Appl.

**2014**, 185 (2014)

## Acknowledgements

The work has been supported by the Polish Ministry of Science and Higher Education.

## Author information

## Additional information

### Competing interests

The author declares that he has no competing interests.

## Rights and permissions

**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.

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### MSC

- 47H10
- 54H25

### Keywords

- generalized contraction
- cyclic mapping
- dislocated metric
- fixed point