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
the Banach contraction principle BoydWong contraction MeirKeeler contraction Matkowski contraction contraction of integral type fixed pointMSC
47H09 54H251 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
References
 Ri, SI: A new fixed point theorem in the fractal space. Indag. Math. 27, 8593 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Wardowski, D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, Article ID 94 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133181 (1922) MATHGoogle Scholar
 Caccioppoli, R: Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, 794799 (1930) MATHGoogle Scholar
 Jachymski, J: A generalization of the theorem by Rhoades and Watson for contractive type mappings. Math. Jpn. 38, 10951102 (1993) MathSciNetMATHGoogle Scholar
 Jachymski, J: Equivalent conditions and the MeirKeeler type theorems. J. Math. Anal. Appl. 194, 293303 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 23272335 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Jachymski, J: Remarks on contractive conditions of integral type. Nonlinear Anal. 71, 10731081 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Kirk, WA: Contraction mappings and extensions. In: Kirk, WA, Sims, B (eds.) Handbook of Metric Fixed Point Theory, pp. 134. Kluwer Academic, Dordrecht (2001) View ArticleGoogle Scholar
 Suzuki, T, Alamri, B: A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point II. Fixed Point Theory Appl. 2015, Article ID 59 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458464 (1969) MathSciNetView ArticleMATHGoogle Scholar
 Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29, 531536 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Suzuki, T: MeirKeeler contractions of integral type are still MeirKeeler contractions. Int. J. Math. Math. Sci. 2007, Article ID 39281 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Suzuki, T: Comments on some recent generalization of the Banach contraction principle. J. Inequal. Appl. 2016, Article ID 111 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Suzuki, T, Vetro, C: Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 6, 110120 (2010) MathSciNetView ArticleGoogle Scholar
 Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 30, 2735 (1968) MathSciNetView ArticleMATHGoogle Scholar
 Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326329 (1969) MathSciNetView ArticleMATHGoogle Scholar
 Lim, TC: On characterizations of MeirKeeler contractive maps. Nonlinear Anal. 46, 113120 (2001) MathSciNetView ArticleMATHGoogle Scholar
 Proinov, PD: Fixed point theorems in metric spaces. Nonlinear Anal. 64, 546557 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Suzuki, T: Fixed point theorem for asymptotic contractions of MeirKeeler type in complete metric spaces. Nonlinear Anal. 64, 971978 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Matkowski, J: Integrable solutions of functional equations. Diss. Math. 127, 168 (1975) MathSciNetMATHGoogle Scholar
 Ćirić, LB: A new fixedpoint theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, 2527 (1981) MathSciNetMATHGoogle Scholar
 Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol. 32. Cambridge University Press, Cambridge (1990) View ArticleMATHGoogle Scholar
 Matkowski, J: Fixed point theorems for contractive mappings in metric spaces. Čas. Pěst. Mat. 105, 341344 (1980) MathSciNetMATHGoogle Scholar