Discussion of several contractions by Jachymski’s approach
 Tomonari Suzuki^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366301605819
© Suzuki 2016
Received: 22 April 2016
Accepted: 8 September 2016
Published: 20 September 2016
Abstract
Keywords
MSC
1 Introduction
The Banach contraction principle [3, 4] is an elegant, forceful tool in nonlinear analysis and has many generalizations. See, e.g., [5–10]. For example, Boyd and Wong in [11] proved the following.
Theorem 1
(Boyd and Wong [11])
 (i)
φ is upper semicontinuous from the right.
 (ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).
 (iii)
\(d(Tx,Ty) \leq\varphi\circ d(x,y)\) for any \(x,y \in X\).
In this paper, we discuss several contractions of integral type by using Jachymski’s approach. As applications, we give alternative proofs of recent generalizations of the Banach contraction principle due to Ri [1] and Wardowski [2].
2 Preliminaries
Throughout this paper we denote by \(\mathbb {N}\) the set of all positive integers and by \(\mathbb {R}\) the set of all real numbers.
 (UR)_{ f } :

For any \(t \in Q\), there exist \(\delta> 0\) and \(\varepsilon> 0\) such that \(f(s) \leq t  \varepsilon\) holds for any \(s \in[t,t+\delta) \cap Q\).
We give some lemmas concerning (UR).
Lemma 2
 (i)
f satisfies (UR)_{ f }.
 (ii)
\(\limsup[ f(u) : u \to t, u \in Q, t \leq u ] < t \) holds for any \(t \in Q\).
 (iii)
\(\limsup[ f(u) : u \to t, u \in Q, t < u ] < t \) and \(f(t) < t \) hold for any \(t \in Q\).
Proof
Obvious. □
Lemma 3
Let f be a function from a subset Q of \(\mathbb {R}\) into \(\mathbb {R}\) such that \(f(t) < t\) for any \(t \in Q\). Assume that f is upper semicontinuous from the right. Then f satisfies (UR)_{ f }.
Proof
Obvious. □
Lemma 4
 (i)
g is upper semicontinuous from the right.
 (ii)
h and φ are right continuous.
 (iii)
\(f(t) \leq g(t) \leq h(t) < \varphi(t) < t\) holds for any \(t \in Q\).
Proof
Remark
3 Definitions
 (A1)
Let D be a subset of \((0,\infty)^{2}\).
 (A2)Let θ be a function from \((0,\infty)\) into \(\mathbb {R}\). Put \(\Theta= \theta ( (0,\infty) )\) and$$\Theta_{\leq}= \bigcup \bigl[ [t,\infty) : t \in\Theta \bigr] . $$
Jachymski in [8] discussed several contractions by using subsets of \([0,\infty)^{2}\). Since this approach seems to be very reasonable for considering future studies, we use an approach similar to Jachymski’s.
Definition 5
 (1)
D is said to be contractive (Cont for short) [3, 4] if there exists \(r \in(0,1)\) such that \(u \leq r t\) holds for any \((t,u) \in D\).
 (2)D is said to be a Browder (Bro, for short) [16] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (2i)
φ is nondecreasing and right continuous.
 (2ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).
 (2iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).
 (2i)
 (3)D is said to be BoydWong (BW for short) [11] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (3i)
φ is upper semicontinuous from the right.
 (3ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).
 (3iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).
 (3i)
 (4)
D is said to be MeirKeeler (MK for short) [17] if for any \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(u < \varepsilon\) holds for any \((t,u) \in D\) with \(t < \varepsilon+ \delta\); see also [18–20].
 (5)D is said to be Matkowski (Mat for short) [21] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (5i)
φ is nondecreasing.
 (5ii)
\(\lim_{n} \varphi^{n}(t) = 0\) for every \(t \in(0, \infty)\).
 (5iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).
 (5i)
 (6)
Remark
We give one proposition on the concept of BoydWong. Note that we can easily obtain similar results on the other concepts.
Proposition 6
Proof
The following are variants of Corollaries 9 and 14 in [14].
Proposition 7
([14])
 (i)
θ is nondecreasing and continuous.
 (ii)
There exists an upper semicontinuous function ψ from Θ into \(\mathbb {R}\) satisfying \(\psi(\tau) < \tau\) for any \(\tau\in\Theta\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
Proposition 8
([14])
 (i)
θ is nondecreasing.
 (ii)
There exists an upper semicontinuous function ψ from \(\Theta_{\leq}\) into \(\mathbb {R}\) satisfying \(\psi(\tau) < \tau\) for any \(\tau\in\Theta_{\leq}\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
Remark
 (ii)′:

There exists a function ψ from \(\Theta_{\leq}\) into \(\mathbb {R}\) such that ψ is upper semicontinuous from the right, \(\psi(\tau) < \tau\) for any \(\tau\in\Theta_{\leq}\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
4 Main results
In this section, we prove our main results. We begin with BoydWong.
Proposition 9
 (i)
θ is nondecreasing and continuous.
 (ii)
There exists a function ψ from Θ into \(\mathbb {R}\) satisfying \((\mathrm{UR})_{\psi}\) and \(\theta(u) \leq\psi\circ\theta(t) \) for any \((t,u) \in D\).
Proof
Remark
There appears \(\theta_{+}^{1}\) in Proposition 2.1 in [15].
We next discuss MeirKeeler.
Proposition 10
 (i)
θ is nondecreasing and right continuous.
 (ii)
For any \(\varepsilon\in\Theta\), there exists \(\delta> 0\) such that \(\theta(t) < \varepsilon+ \delta\) implies \(\theta(u) < \varepsilon\) for any \((t,u) \in D\).
Proof
We obtain the following, which is a generalization of Corollary 17 in [14].
Corollary 11
Assume (A1), (A2), (i) of Proposition 10, and (ii) of Proposition 9. Then D is MeirKeeler.
Let us discuss Matkowski.
Proposition 12
 (i)
θ is nondecreasing and left continuous.
 (ii)
minΘ does not exist.
 (iii)There exist a subset Q of \(\mathbb {R}\) and a nondecreasing function ψ from Q into Q satisfying \(\Theta\subset Q \subset\Theta_{\leq}\),for any \(\tau\in Q\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).$$\lim_{n \to\infty} \psi^{n} (\tau) = \inf\Theta $$
Proof
5 Counterexamples
In this section, we give counterexamples connected with the results in Section 4.
Example 13
(Example 2.3 in [15], Example 10 in [14])
Remark
Proof
Example 14
(Example 2.6 in [13], Example 11 in [14])
Remark
By Proposition 12, D is Matkowski. We define E by (2). Then \(E = \{ (2,1) \} \) holds. Hence E is contractive.
Proof
Example 15
Proof
Obvious. □
6 Applications
In this section, as applications, we give alternative proofs of some recent generalizations of the Banach contraction principle. Ri in [1] proved the following fixed point theorem.
Theorem 16
(Ri [1])
 (R1)
\(\psi(t) < t\) for any \(t \in(0,\infty)\).
 (R2)
\(\limsup_{s \to t+0} \psi(s) < t\) for any \(t \in(0,\infty)\).
 (R3)
\(d(Tx, Ty) \leq\psi ( d(x, y) )\) for any \(x, y \in X\).
We give an alternative proof of Theorem 16 by showing that a mapping T in Theorem 16 is a BoydWong contraction.
Proof of Theorem 16
By Lemma 2, the restriction ψ to \((0,\infty)\) satisfies \((\mathrm{UR})_{\psi}\). Then by Lemma 4, there exists a right continuous function φ from \((0,\infty)\) into itself satisfying \(\psi(t) < \varphi(t) < t\) for \(t \in(0,\infty)\). Thus T is a BoydWong contraction. So T has a unique fixed point. □
Wardowski in [2] proved a fixed point theorem on Fcontraction.
Theorem 17
(Wardowski [2])
 (F1)
F is strictly increasing.
 (F2)
For any sequence \(\{ \alpha_{n} \}\) of positive numbers, \(\lim_{n} \alpha_{n} = 0\) iff \(\lim_{n} F(\alpha_{n})=\infty\).
 (F3)
\(\lim_{t \to+0} t^{k} F(t) = 0\) holds.
 (F4)If \(Tx \neq Ty\), thenholds.$$F \bigl( d(Tx,Ty) \bigr) \leq F \bigl( d(x,y) \bigr)  \eta $$
Remark
 (F2)′:

\(\lim_{t \to+0} F(t) =  \infty\) holds.
We give an alternative proof of Theorem 17 by showing that mappings satisfying (F1) and (F4) are CJM contractions.
Proof of Theorem 17
Define a subset D of \((0,\infty)^{2}\) by (1). Define θ and ψ by \(\theta= F\) and \(\psi(\tau) = \tau \eta\). Then all the assumptions of Proposition 8 hold. So, by Proposition 8, D is CJM. Therefore T has a unique fixed point. □
Remark
Declarations
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
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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