 Research
 Open access
 Published:
Characterizations of contractive conditions by using convergent sequences
Fixed Point Theory and Applications volumeÂ 2017, ArticleÂ number:Â 30 (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.
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 \(x \in X\).
Theorem 1 has many generalizations. In other words, we have studied many contractive conditions. Using the subsets Q of \([0,\infty )^{2}\) defined by
HegedÃ¼s and SzilÃ¡gyi in [3] studied some contractive conditions. See Jachymski [4] and the references therein. Using the subsets Q of \((0,\infty )^{2}\) defined by
we studied contractive conditions in [5]. Using subsets of \((0,\infty )^{2}\), we list some contractive conditions, which are weaker than or equivalent to the Browder contractive condition.
Definition 2
Let Q be a subset of \((0,\infty )^{2}\).

(1)
Q is said to be CJM [6â€“9] if the following hold:

(1i)
For any \(\varepsilon > 0\), there exists \(\delta > 0\) such that \(u \leq \varepsilon \) holds for any \((t,u) \in Q\) with \(t < \varepsilon + \delta \).

(1ii)
\(u < t\) holds for any \((t,u) \in Q\).

(1i)

(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 Ï† from \((0, \infty )\) into itself satisfying the following:

(3i)
Ï† 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 Ï† 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 Ï† from \((0, \infty )\) into itself satisfying the following:

(5i)
Ï† 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 Ï† from \((0, \infty )\) into itself satisfying the following:

(6i)
Ï† 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)
We know the following implications:
We note that for each implication, there exists a counterexample for its converse implication.
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
Let Q be a subset of \((0,\infty )^{2}\).

(1)
A sequence \(\{ (t_{n}, u_{n}) \}\) is said to satisfy Condition Î” 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 Î” 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 Î” 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 Î” 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 Î” 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 Î” 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
Let \(p_{1},p_{2},q_{1},q_{2} \in \{ 0, 1 \}\) and let \(r_{1},r_{2} \in \{ 0, 1, 2 \}\). Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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.â€ƒâ–¡
3 Characterizations
In this section, we give characterizations of the contractive conditions appearing in Section 1 by a unified method.
Theorem 5
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(i)
Q is CJM.

(ii)
Q satisfies Condition C\((0,0,0)\).
Theorem 6
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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 Î” provided \(\{ t_{n} \}\) is strictly decreasing and \(\{ u_{n} \}\) is nonincreasing.

(a)
Theorem 7
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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 Î” provided \(\{ t_{n} \}\) is nonincreasing.

(a)
Remark
In order to prove (ii) â‡’ (i), we define a function Ï† from \((0,\infty )\) into itself by
where Ïˆ is a function from \((0,\infty )\) into itself defined by
See Lemma 4 in [5].
Theorem 8
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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 Î” provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing.

(c)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Î” provided \(\{ t_{n} \}\) is constant and \(\{ u_{n} \}\) is strictly increasing.

(a)
Remark
In order to prove (ii) â‡’ (i), we define a function Ï† from \((0,\infty )\) into itself by
Theorem 9
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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 Î” provided \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing.

(c)
Every sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfies Condition Î” provided \(\{ t_{n} \}\) is nondecreasing and \(\{ u_{n} \}\) are strictly increasing.

(a)
Theorem 10
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(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 Î”.

(a)
Remark
In order to prove (ii) â‡’ (i), we define a function Ï† from \((0,\infty )\) into itself by
where Ïˆ is a function from \((0,\infty )\) into itself defined by (1). See Lemma 6 in [15]. See also the proof of Proposition 1 in [16].
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
Let Ï† be a nondecreasing function from \((0,\infty )\) into itself. Then the following are equivalent:

(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
We first show (i) â‡’ (ii). We assume (i). Arguing by contradiction, there exists \(\tau \in (0,\infty )\) satisfying \(\varphi (\tau ) \geq \tau \). Since Ï† is nondecreasing, we have \(\varphi ^{2}(\tau ) \geq \varphi (\tau ) \). Continuing this argument, we can show that \(\{ \varphi ^{n}(\tau ) \}\) is nondecreasing. Since \(\varphi (\tau ) \geq \tau > 0\) holds, \(\{ \varphi ^{n}(\tau ) \}\) cannot converge to 0. This is a contradiction. So we have shown \(\varphi (t) < t\) for any \(t \in (0,\infty )\). Also, arguing by contradiction, we assume that there exists \(\varepsilon > 0\) such that for any \(\delta > 0\), there exists t satisfying
Since Ï† is nondecreasing, we have
Fix \(\tau > \varepsilon \). Then we have \(\varphi (\tau ) > \varepsilon \). Hence \(\varphi ^{2}(\tau ) > \varepsilon \) holds. Continuing this argument, we have \(\varphi ^{n}(\tau ) > \varepsilon \) for any \(n \in \mathbb {N}\). Therefore \(\{ \varphi ^{n}(\tau ) \}\) cannot converge to 0, which implies a contradiction. Therefore we have shown (ii).
Let us prove (ii) â‡’ (i): We assume (ii). Arguing by contradiction, we assume that there exists \(t \in (0,\infty )\) such that \(\{ \varphi ^{n}(t) \}\) does not converge to 0. Since \(\{ \varphi ^{n}(t) \}\) is strictly decreasing, \(\{ \varphi ^{n}(t) \}\) converges to some \(\varepsilon \in (0,\infty )\). We can choose \(\delta > 0\) satisfying
Choose \(\nu \in \mathbb {N}\) satisfying \(\varphi ^{\nu }(t) < \varepsilon + \delta \). Then we have \(\varphi ^{\nu +1}(t) \leq \varepsilon \), which implies a contradiction. Therefore we have shown (i).â€ƒâ–¡
Proof of Theorem 9
We first prove (i) â‡’ (ii). We assume (i). Then there exists a function Ï† from \((0, \infty )\) into itself satisfying the following:

(1)
Ï† 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 any \((t,u) \in Q\), we have by Lemma 11
Thus (a) of Theorem 9 holds. Also, arguing by contradiction, we assume that (b) of Theorem 9 does not hold. Then there exists a sequence \(\{ (t_{n}, u_{n}) \}\) in Q such that \(\{ (t_{n}, u_{n}) \}\) converges to \((\tau ,\tau )\) for some \(\tau \in (0,\infty )\) and \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly decreasing. By Lemma 11 again, there exists \(\delta > 0\) such that
For sufficiently large \(n \in \mathbb {N}\), since \(\tau < t_{n} < \tau + \delta \) holds, we have
Since \(\{ u_{n} \}\) is strictly decreasing, \(\{ u_{n} \}\) does not converge to Ï„, which implies a contradiction. We have shown (b) of Theorem 9. Also, arguing by contradiction, we assume that (c) of Theorem 9 does not hold. Then there exists a sequence \(\{ (t_{n}, u_{n}) \}\) in Q such that \(\{ (t_{n}, u_{n}) \}\) converges to \((\tau ,\tau )\) for some \(\tau \in (0,\infty )\), \(\{ t_{n} \}\) is nondecreasing and \(\{ u_{n} \}\) is strictly increasing. By (1) and (3), we have
As n tends to âˆž, we obtain \(\tau \leq \varphi (\tau ) \). By Lemma 11, this is a contradiction. We have shown (a)(c) of Theorem 9.
Let us prove (ii) â‡’ (i). We assume (a)(c) of Theorem 9. Define a function Ï† from \((0,\infty )\) into itself by
where \(\sup \varnothing =  \infty \). It is obvious that Ï† is nondecreasing. It is also obvious that \(u \leq \varphi (t)\) holds for any \((t,u) \in Q\). Since Q satisfies Condition C\((0,0,0)\), we have
Arguing by contradiction, we assume \(\varphi (\tau ) = \tau \) for some \(\tau \in (0,\infty )\). Then there exists a sequence \(\{ (t_{n}, u_{n}) \}\) in Q satisfying \(t_{n} \leq \tau \) for \(n \in \mathbb {N}\) and \(\lim_{n} u_{n} = \tau \). Since \(u_{n} < t_{n} \leq \tau \) holds for \(n \in \mathbb {N}\), we can choose a subsequence \(\{ f(n) \}\) of the sequence \(\{ n \}\) in \(\mathbb {N}\) such that \(\{ t_{f(n)} \}\) is nondecreasing and \(\{ u_{f(n)} \}\) is strictly increasing. Since \(\{ t_{f(n)} \}\) converges to Ï„, we obtain a contradiction. Therefore we have shown
for any \(s \in (0,\infty )\). We will show the following:

For \(\varepsilon \in (0,\infty )\), there exists \(\delta > 0\) such that
$$ \varepsilon < t < \varepsilon + \delta \quad \mbox{implies}\quad \varphi (t) \leq \varepsilon . $$
Arguing by contradiction, we assume that there exists \(\varepsilon > 0\) such that for any \(\delta \in (0,\varepsilon )\), there exists t satisfying
So we can choose a sequence \(\{ t_{n} \}\) in \((\varepsilon ,2 \varepsilon )\) such that \(\{ t_{n} \}\) is strictly decreasing, \(\{ t_{n} \}\) converges to Îµ and \(\varepsilon < \varphi (t_{n})\) for \(n \in \mathbb {N}\). Noting \(t_{n}/2 < \varepsilon \), we have
So there exists \((t'_{n}, u'_{n}) \in Q\) satisfying \(u'_{n} > \varepsilon \) and \(t'_{n} \leq t_{n}\). Since \((t'_{n}, u'_{n}) \in Q\) holds, we have
Hence \(\{ t'_{n} \}\) and \(\{ u'_{n} \}\) converge to Îµ. So we can choose a subsequence \(\{ f(n) \}\) of \(\{ n \}\) such that \(\{ t'_{f(n)} \}\) and \(\{ u'_{f(n)} \}\) are strictly decreasing. This contradicts that Q satisfies Condition C\((0,0,0)\). By Lemma 11, we obtain (2).â€ƒâ–¡
By Proposition 4, we can prove the following.
Theorem 12
Let Q be a subset of \((0,\infty )^{2}\). Then the following are equivalent:

(i)
Q is Browder.

(ii)
Q is BW and Matkowski.
Proof
By Proposition 4 and Theorems 7, 9 and 10, both are equivalent to Condition C\((1,1,2)\).â€ƒâ–¡
4 Integral type
There is another merit in our approach.
Branciari in [17] introduced contractions of integral type as follows: A mapping T on a metric space \((X,d)\) is a Branciari contraction if there exist \(r \in [0, 1)\) and a locally integrable function f from \([0, \infty )\) into itself such that
holds for all \(s >0\) and \(x, y \in X\). Jachymski in [4] proved that the concepts of the Branciari contraction and the Browder contraction are equivalent.
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.
Throughout this section, we let Q be a subset of \((0,\infty )^{2}\). Let Î¸ be a nondecreasing function from \((0,\infty )\) into itself and define a subset R of \((0,\infty )^{2}\) by
Lemma 13
If R satisfies Condition C\((0,0,0)\), then Q also satisfies Condition C\((0,0,0)\).
Proof
We assume that R satisfies Condition C\((0,0,0)\). Fix \((t,u) \in Q\). Arguing by contradiction, we assume \(u \geq t\). Then since Î¸ is nondecreasing, we have \(\theta (u) \geq \theta (t)\), which implies a contradiction. Therefore we have shown
Also, arguing by contradiction, we assume the following:

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 \(\tau \in (0,\infty )\).
We consider the following two cases:

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

\(\lim [\theta (t) : t \to \tau +0 ] < \theta (s)\) holds for any \(s \in (\tau ,\infty )\).
In the first case, we have \(\theta (t_{n}) = \theta (u_{n}) \) for sufficiently large \(n \in \mathbb {N}\), which implies a contradiction. In the second case, taking subsequences, without loss of generality, we may assume that \(\{ \theta (t_{n}) \}\) and \(\{ \theta (u_{n}) \}\) are strictly decreasing. We have
which also implies a contradiction. Therefore Q satisfies Condition C\((0,0,0)\).â€ƒâ–¡
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 Î¸ is quite clear.
Lemma 14
If R satisfies Condition C\((0,0,1)\) and Î¸ is right continuous, then Q also satisfies Condition C\((0,0,1)\).
Proof
We assume that R satisfies Condition C\((0,0,1)\). Then by Lemma 13, Q satisfies Condition C\((0,0,0)\). Arguing by contradiction, we assume that there exists a sequence \(\{ (t_{n},\tau ) \}\) in Q such that \(\{ t_{n} \}\) is strictly decreasing and \(\lim_{n} t_{n} = \tau \) holds. Since Î¸ is right continuous, we have
Since \(\theta (\tau ) < \theta (t_{n})\) holds for \(n \in \mathbb {N}\), taking a subsequence, without loss of generality, we may assume that \(\{ \theta (t_{n}) \}\) is strictly decreasing. Hence R does not satisfy Condition C\((0,0,1)\), which implies a contradiction.â€ƒâ–¡
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 Î¸ is quite clear.
Lemma 15
If R satisfies Condition C\((0,1,0)\) and Î¸ is left continuous, then Q also satisfies Condition C\((0,1,0)\).
Proof
We assume that R satisfies Condition C\((0,1,0)\). Then by Lemma 13, Q satisfies Condition C\((0,0,0)\). Arguing by contradiction, we assume that there exists a sequence \(\{ (\tau , u_{n}) \}\) in Q such that \(\{ u_{n} \}\) is strictly increasing and \(\lim_{n} u_{n} = \tau \) holds. Since Î¸ is left continuous, we have
Since \(\theta (u_{n}) < \theta (\tau )\) holds for \(n \in \mathbb {N}\), taking a subsequence, without loss of generality, we may assume that \(\{ \theta (u_{n}) \}\) is strictly increasing. Hence R does not satisfy Condition C\((0,1,0)\), which implies a contradiction.â€ƒâ–¡
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 Î¸ is quite clear.
Lemma 16
If R satisfies Condition C\((0,1,2)\) and Î¸ is continuous, then Q also satisfies Condition C\((0,1,2)\).
Proof
We assume that R satisfies Condition C\((0,1,2)\). Then by Lemma 13, Q satisfies Condition C\((0,0,0)\). Arguing by contradiction, we assume that there exists a sequence \(\{ (t_{n},u_{n}) \}\) in Q such that \(\{ t_{n} \}\) is nonincreasing, \(\{ u_{n} \}\) is nondecreasing and \(\lim_{n} t_{n} = \lim_{n} u_{n} = \tau \) holds. Since Î¸ is nondecreasing, we note that \(\{ \theta (t_{n}) \}\) is nonincreasing and \(\{ \theta (u_{n}) \}\) is nondecreasing. Since Î¸ is continuous, we have
Hence R does not satisfy Condition C\((0,1,2)\), which implies a contradiction.â€ƒâ–¡
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 Î¸ is quite clear.
Lemma 17
If R satisfies Condition C\((1,0,0)\), then Q also satisfies Condition C\((1,0,0)\).
Proof
We assume that R satisfies Condition C\((1,0,0)\). Then by Lemma 13, Q satisfies Condition C\((0,0,0)\). Arguing by contradiction, we assume that there exists a sequence \(\{ (t_{n}, u_{n}) \}\) in Q such that \(\{ t_{n} \}\) and \(\{ u_{n} \}\) are strictly increasing and \(\lim_{n} t_{n} = \lim_{n} u_{n} = \tau \) holds for some \(\tau \in (0,\infty )\). We consider the following two cases:

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

\(\theta (s) < \lim [\theta (t) : t \to \tau 0 ]\) holds for any \(s \in (0,\tau )\).
In the first case, we have \(\theta (t_{n}) = \theta (u_{n}) \) for sufficiently large \(n \in \mathbb {N}\), which implies a contradiction. In the second case, taking subsequences, without loss of generality, we may assume that \(\{ \theta (t_{n}) \}\) and \(\{ \theta (u_{n}) \}\) are strictly increasing. Since Î¸ is nondecreasing, we have
Hence R does not satisfy Condition C\((1,0,0)\), which implies a contradiction.â€ƒâ–¡
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.â€ƒâ–¡
Theorem 19
If R is MK and Î¸ 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.â€ƒâ–¡
Theorem 20
If R is BW and Î¸ 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.â€ƒâ–¡
Theorem 21
If R is NT and Î¸ 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.â€ƒâ–¡
Theorem 22
If R is Matkowski and Î¸ 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.â€ƒâ–¡
Theorem 23
If R is Browder and Î¸ is continuous, then Q is also Browder.
Proof
By Theorem 10, R satisfies Condition C\((1,1,2)\). So by Lemmas 16 and 17, Q satisfies Condition C\((1,1,2)\). By Theorem 10 again, Q is Browder.â€ƒâ–¡
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).
References
Banach, S: Sur les opÃ©rations dans les ensembles abstraits et leur application aux Ã©quations intÃ©grales. Fundam. Math. 3, 133181 (1922)
Caccioppoli, R: Un teorema generale sullâ€™esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, 794799 (1930)
HegedÃ¼s, M, SzilÃ¡gyi, T: Equivalent conditions and a new fixed point theorem in the theory of contractive type mappings. Math. Jpn. 25, 147157 (1980). MR0571276
Jachymski, J: Remarks on contractive conditions of integral type. Nonlinear Anal. 71, 10731081 (2009). MR2527526
Suzuki, T: Discussion of several contractions by Jachymskiâ€™s approach. Fixed Point Theory Appl. 2016, 91 (2016). MR3548683
Ä†iriÄ‡, LjB: A new fixedpoint theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, 2527 (1981). MR0672538
Jachymski, J: Equivalent conditions and the MeirKeeler type theorems. J.Â Math. Anal. Appl. 194, 293303 (1995). MR1353081
Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol.Â 32, Cambridge University Press, Cambridge (1990). MR1067720
Matkowski, J: Fixed point theorems for contractive mappings in metric spaces. ÄŒas. PÄ›st. Mat. 105, 341344 (1980). MR0597909
Meir, A, Keeler, E: A theorem on contraction mappings. J.Â Math. Anal. Appl. 28, 326329 (1969). MR0250291
Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458464 (1969). MR239559
Suzuki, T: A generalization of HegedÃ¼sSzilÃ¡gyiâ€™s fixed point theorem in complete metric spaces. Fixed Point Theory Appl. (in press)
Matkowski, J: Integrable Solutions of Functional Equations. Dissertationes Mathematicae, vol.Â 127. Instytut Matematyczny Polskiej Akademii Nauk, Warsaw (1975). MR0412650
Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30, 2735 (1968). MR0230180
Suzuki, T: Comments on some recent generalization of the Banach contraction principle. J.Â Inequal. Appl. 2016, Article ID 111 (2016). MR3483303
Suzuki, T: Some notes on MeirKeeler contractions and Lfunctions. Bull. Kyushu Inst. Technol., Pure Appl. Math. 53, 113 (2006). MR2237618
Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J.Â Math. Math. Sci. 29, 531536 (2002). MR1900344
Suzuki, T: MeirKeeler contractions of integral type are still MeirKeeler contractions. Int. J.Â Math. Math. Sci. 2007, Article ID 39281 (2007). MR2285999
Suzuki, T, Vetro, C: Three existence theorems for weak contractions of Matkowski type. Int. J.Â Math. Stat. 6, 110120 (2010). MR2520397
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that he has no competing interests.
Additional information
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Cite this article
Suzuki, T. Characterizations of contractive conditions by using convergent sequences. Fixed Point Theory Appl 2017, 30 (2017). https://doi.org/10.1186/s136630170623y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136630170623y