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Characterizations of contractive conditions by using convergent sequences
 Tomonari Suzuki^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s136630170623y
© The Author(s) 2017
 Received: 15 July 2017
 Accepted: 27 November 2017
 Published: 21 December 2017
Abstract
We give characterizations of the contractive conditions, by using convergent sequences. Since we use a unified method, we can compare the contractive conditions very easily. We also discuss the contractive conditions of integral type by a unified method.
Keywords
 contractive condition
 contraction of integral type
MSC
 54E99
 54H25
1 Introduction and preliminaries
Throughout this paper we denote by \(\mathbb {N}\) the set of all positive integers.
The fixed point theorem for contractions is referred to as the Banach contraction principle.
Theorem 1
Let \((X,d)\) be a complete metric space and let T be a contraction on X, that is, there exists \(r \in [0,1)\) such that \(d(Tx, Ty) \leq r d(x,y) \) for all \(x, y \in X\). Then T has a unique fixed point z. Moreover, \(\{ T^{n} x \}\) converges to z for any \(x \in X\).
Definition 2
 (1)
 (2)
Q is said to be MeirKeeler (MK, for short) [10] if for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that \(u < \varepsilon \) holds for any \((t,u) \in Q\) with \(t < \varepsilon + \delta \).
 (3)Q 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 \((t,u) \in Q\).
 (3i)
 (4)Q is said to be of New Type (NT, for short) [12] if there exists a function φ from \((0, \infty )\) into itself satisfying the following:
 (4i)
\(\varphi (t) < t\) for any \(t \in (0,\infty )\).
 (4ii)
For any \(\varepsilon > 0\), there exists \(\delta > 0\) such that \(\varepsilon < t < \varepsilon + \delta \) implies \(\varphi (t) \leq \varepsilon \).
 (4iii)
\(u \leq \varphi (t)\) holds for any \((t,u) \in Q\).
 (4i)
 (5)Q is said to be Matkowski (Mat, for short) [13] 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 any \(t \in (0, \infty )\).
 (5iii)
\(u \leq \varphi (t)\) holds for any \((t,u) \in Q\).
 (5i)
 (6)Q is said to be a Browder (Bro, for short) [14] if there exists a function φ from \((0, \infty )\) into itself satisfying the following:
 (6i)
φ is nondecreasing and right continuous.
 (6ii)
\(\varphi (t) < t\) holds for any \(t \in (0, \infty )\).
 (6iii)
\(u \leq \varphi (t)\) holds for any \((t,u) \in Q\).
 (6i)
There is a problem in the above list. The expressions as regards the conditions vary. So we cannot understand easily the relationship between the contractive conditions. Motivated by this, in this paper, we give characterizations (Theorems 510) of the contractive conditions, by using convergent sequences. Since we use a unified method, we can compare the contractive conditions very easily. For example, we can prove quite easily that if Q is BW and Matkowski, then Q is Browder (see Theorem 12). We also discuss the contractive conditions of integral type by a unified method.
2 New definitions
We introduce the following definitions in order to treat the contractive conditions appearing in Section 1 by a unified method.
Definition 3
 (1)
A sequence \(\{ (t_{n}, u_{n}) \}\) is said to satisfy Condition Δ if \(\{ (t_{n}, u_{n}) \}\) does not converge to \((t,t)\) for any \(t \in (0,\infty )\).
 (2)Q is said to satisfy Condition C\((0,0,0)\) if the following hold:
 (2i)
\(u < t\) holds for any \((t,u) \in Q\).
 (2ii)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing.
 (2i)
 (3)Q is said to satisfy Condition C\((0,0,1)\) if the following hold:
 (3i)
Q satisfies Condition C\((0,0,0)\).
 (3ii)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is strictly decreasing and \(\{ u_{n} \}\) is constant.
 (3i)
 (4)Q is said to satisfy Condition C\((0,0,2)\) if the following hold:
 (4i)
Q satisfies Condition C\((0,0,0)\).
 (4ii)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is strictly decreasing and \(\{ u_{n} \}\) is nondecreasing.
 (4i)
 (5)Q is said to satisfy Condition C\((0,1,0)\) if the following hold:
 (5i)
Q satisfies Condition C\((0,0,0)\).
 (5ii)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is constant and \(\{ u_{n} \}\) is strictly increasing.
 (5i)
 (6)Q is said to satisfy Condition C\((1,0,0)\) if the following hold:
 (6i)
Q satisfies Condition C\((0,0,0)\).
 (6ii)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly increasing.
 (6i)
 (7)
Let \((p,q,r) \in \{ 0,1 \}^{2} \times \{ 0,1,2 \}\). Then Q is said to satisfy Condition C\((p,q,r)\) if Q satisfies Conditions C\((p,0,0)\), C\((0,q,0)\) and C\((0,0,r)\).
Proposition 4
 (i)
Q satisfies Conditions C\((p_{1},q_{1},r_{1})\) and C\((p_{2},q_{2},r_{2})\).
 (ii)
Q satisfies Condition C\((\max \{ p_{1}, p_{2} \}, \max \{ q_{1}, q_{2} \}, \max \{ r_{1}, r_{2}\})\).
Proof
Obvious. □
3 Characterizations
In this section, we give characterizations of the contractive conditions appearing in Section 1 by a unified method.
Theorem 5
 (i)
Q is CJM.
 (ii)
Q satisfies Condition C\((0,0,0)\).
Theorem 6
 (i)
Q is MK.
 (ii)Q satisfies Condition C\((0,0,1)\), that is, the following hold:
 (a)
\(u < t\) holds for any \((t,u) \in Q\).
 (b)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is strictly decreasing and \(\{ u_{n} \}\) is nonincreasing.
 (a)
Theorem 7
 (i)
Q is BW.
 (ii)Q satisfies Condition C\((0,1,2)\), that is, the following hold:
 (a)
\(u < t\) holds for any \((t,u) \in Q\).
 (b)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is nonincreasing.
 (a)
Remark
Theorem 8
 (i)
Q is NT.
 (ii)Q satisfies Condition C\((0,1,0)\), that is, the following hold:
 (a)
\(u < t\) holds for any \((t,u) \in Q\).
 (b)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing.
 (c)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is constant and \(\{ u_{n} \}\) is strictly increasing.
 (a)
Remark
Theorem 9
 (i)
Q is Matkowski.
 (ii)Q satisfies Condition C\((1,1,0)\), that is, the following hold:
 (a)
\(u < t\) holds for any \((t,u) \in Q\).
 (b)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing.
 (c)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ provided \(\{ t_{n} \}\) is nondecreasing and \(\{ u_{n} \}\) are strictly increasing.
 (a)
Theorem 10
 (i)
Q is Browder.
 (ii)Q satisfies Condition C\((1,1,2)\), that is, the following hold:
 (a)
\(u < t\) holds for any \((t,u) \in Q\).
 (b)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Δ.
 (a)
Remark
We only give a proof of Theorem 9. The reason of this is that we can prove the other theorems easily by using the method in the proof of Theorem 9.
Lemma 11
 (i)
\(\lim_{n} \varphi ^{n}(t) = 0\) holds for any \(t \in (0, \infty )\).
 (ii)\(\varphi (t) < t\) holds for any \(t \in (0,\infty )\). For any \(\varepsilon > 0\), there exists \(\delta > 0\) such that$$ \varepsilon < t < \varepsilon + \delta \quad \textit{implies}\quad \varphi (t)\leq \varepsilon . $$
Proof
Proof of Theorem 9
 (1)
φ is nondecreasing.
 (2)
\(\lim_{n} \varphi ^{n}(t) = 0\) for any \(t \in (0, \infty )\).
 (3)
\(u \leq \varphi (t)\) holds for any \((t,u) \in Q\).

For \(\varepsilon \in (0,\infty )\), there exists \(\delta > 0\) such that$$ \varepsilon < t < \varepsilon + \delta \quad \mbox{implies}\quad \varphi (t) \leq \varepsilon . $$
By Proposition 4, we can prove the following.
Theorem 12
 (i)
Q is Browder.
 (ii)
Q is BW and Matkowski.
4 Integral type
There is another merit in our approach.
In [5, 15, 18, 19], we also studied contractions of integral type for several contractive conditions stated in Section 1. We note that we have used various methods to prove theorems there. Motivated by this fact, in this paper, we study contractions of integral type by a unified method.
Lemma 13
If R satisfies Condition C\((0,0,0)\), then Q also satisfies Condition C\((0,0,0)\).
Proof

There exists a sequence \(\{ (t_{n}, u_{n}) \}\) in Q such that \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing and \(\lim_{n} t_{n} = \lim_{n} u_{n} = \tau \) holds for some \(\tau \in (0,\infty )\).

\(\lim [\theta (t) : t \to \tau +0 ] = \theta (s)\) holds for some \(s \in (\tau ,\infty )\).

\(\lim [\theta (t) : t \to \tau +0 ] < \theta (s)\) holds for any \(s \in (\tau ,\infty )\).
Remark
Compare this proof with the proof of Theorem 2.7 in [18]. In our new proof, the reason why we do not need any continuity of θ is quite clear.
Lemma 14
If R satisfies Condition C\((0,0,1)\) and θ is right continuous, then Q also satisfies Condition C\((0,0,1)\).
Proof
Remark
Compare this proof with the proof of Theorem 2.1 in [18]. In our new proof, the reason why we need the right continuity of θ is quite clear.
Lemma 15
If R satisfies Condition C\((0,1,0)\) and θ is left continuous, then Q also satisfies Condition C\((0,1,0)\).
Proof
Remark
Compare this proof with the proof of Proposition 2.1 in [19]. In our new proof, the reason why we need the left continuity of θ is quite clear.
Lemma 16
If R satisfies Condition C\((0,1,2)\) and θ is continuous, then Q also satisfies Condition C\((0,1,2)\).
Proof
Remark
Compare this proof with the proofs of Proposition 8 in [15] and Proposition 9 in [5]. In our new proof, the reason why we need the continuity of θ is quite clear.
Lemma 17
If R satisfies Condition C\((1,0,0)\), then Q also satisfies Condition C\((1,0,0)\).
Proof

\(\theta (s) = \lim [\theta (t) : t \to \tau 0 ]\) holds for some \(s \in (0,\tau )\).

\(\theta (s) < \lim [\theta (t) : t \to \tau 0 ]\) holds for any \(s \in (0,\tau )\).
Theorem 18
If R is CJM, then Q is also CJM.
Proof
By Theorem 5, R satisfies Condition C\((0,0,0)\). So by Lemma 13, Q satisfies Condition C\((0,0,0)\). By Theorem 5 again, Q is CJM. □
Theorem 19
If R is MK and θ is right continuous, then Q is also MK.
Proof
By Theorem 6, R satisfies Condition C\((0,0,1)\). So by Lemma 14, Q satisfies Condition C\((0,0,1)\). By Theorem 6 again, Q is MK. □
Theorem 20
If R is BW and θ is continuous, then Q is also BW.
Proof
By Theorem 7, R satisfies Condition C\((0,1,2)\). So by Lemma 16, Q satisfies Condition C\((0,1,2)\). By Theorem 7 again, Q is BW. □
Theorem 21
If R is NT and θ is left continuous, then Q is also NT.
Proof
By Theorem 8, R satisfies Condition C\((0,1,0)\). So by Lemma 15, Q satisfies Condition C\((0,1,0)\). By Theorem 8 again, Q is NT. □
Theorem 22
If R is Matkowski and θ is left continuous, then Q is also Matkowski.
Proof
By Theorem 9, R satisfies Condition C\((1,1,0)\). So by Lemmas 15 and 17, Q satisfies Condition C\((1,1,0)\). By Theorem 9 again, Q is Matkowski. □
Theorem 23
If R is Browder and θ is continuous, then Q is also Browder.
5 Conclusions
In this paper, we give characterizations of the contractive conditions, by using convergent sequences (see Theorems 510). Since we use a unified method, we can compare the contractive conditions very easily (see Theorem 12). We also discuss the contractive conditions of integral type by a unified method (see Theorems 1823).
Declarations
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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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