 Research
 Open Access
Comments on some recent generalization of the Banach contraction principle
 Tomonari Suzuki^{1}Email author
https://doi.org/10.1186/s1366001610575
© Suzuki 2016
 Received: 2 December 2015
 Accepted: 30 March 2016
 Published: 6 April 2016
Abstract
We study Browder and CJM contractions of integral type. As a result, we give an alternative proof of some recent generalization of the Banach contraction principle by Jleli and Samet.
Keywords
 the Banach contraction principle
 Browder contraction
 CJM contraction
 contraction of integral type
 fixed point
MSC
 47H09
 54H25
1 Introduction
We have introduced many types of contractions. We give the definitions of three of them. Define Φ as follows: \(\varphi\in\varPhi\) iff φ is a nondecreasing, right continuous function from \([0, \infty)\) into itself satisfying \(\varphi(t) < t\) for any \(t > 0\). It is obvious that \(\varphi(0) = 0\) holds.
Definition 1

T is said to be a (usual) contraction (C, for short) [1, 2] if there exists \(r \in[0,1)\) such that \(d(Tx,Ty) \leq r d(x,y) \) for any \(x,y \in X\).

T is said to be a Browder contraction (BroC, for short) [3] if there exists \(\varphi\in\varPhi\) such that \(d(Tx,Ty) \leq\varphi\circ d(x,y)\) for any \(x,y \in X\).
We have studied contractions of integral type in [11, 12]. Very recently, Jleli and Samet [13] introduced the following new type of contractions.
Theorem 2
(Jleli and Samet [13])
 (θ1):

θ is nondecreasing.
 (θ2):

For any sequence \(\{ t_{n} \}\) in \((0,\infty)\), \(\lim_{n} \theta(t_{n}) = 1\) iff \(\lim_{n} t_{n} = 0\).
 (θ3):

There exist \(r \in(0,1)\) and \(\ell\in(0,\infty]\) such that \(\lim_{t \to+0} (\theta(t)  1 )/t^{r} = \ell\).
Remark

Considering the domain and range of θ and (θ1), it is obvious that (θ2) is equivalent to \(\inf \{ \theta(t) : t \in(0,\infty) \} = 1\).

The underlying space of Theorem 2 is a generalized metric space. This interesting concept was introduced by Branciari [14]. See also [15–18] and others. However, we omit the statement of the definition of a generalized metric space because it is not essential in this paper.
In this paper, motivated by Theorem 2, we deepen the study of contractions of integral type. We also give an alternative proof of Theorem 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. For a function f, we denote by \(\operatorname {Dom}(f)\) the domain of f.
 (U)_{ f } :

For any \(t \in \operatorname {Dom}(f)\), there exist \(\delta> 0\) and \(\varepsilon> 0\) such that \(f(s) \leq t  \varepsilon\) holds for any \(s \in(t\delta,t+\delta) \cap \operatorname {Dom}(f)\).
 (A1)
Let Y be an arbitrary set and let h be a function from Y into \([0,\infty)\). Let S be a mapping on Y satisfying that \(h(x) = 0\) implies \(h(Sx) = 0\) for any \(x \in Y\).
 (A2)Let θ be a function from \((0,\infty)\) into \(\mathbb {R}\). Put \(\Theta= \theta ( (0,\infty) )\) and$$\Theta_{\leq}= \bigcup \bigl\{ [t,\infty) : t \in\Theta \bigr\} . $$
The purpose of this study is to obtain the mathematical structure of contractions mentioned in Section 1. So, we simplify our setting as follows:
Definition 3

S is said to be a Browder contraction if there exists \(\varphi\in\varPhi\) such thatfor any \(x \in Y\).$$ h(Sx) \leq\varphi\circ h(x) $$

S is said to be a CJM contraction if the following hold:
 (j)
For any \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(h(x) < \varepsilon+ \delta\) implies \(h(Sx) \leq\varepsilon\).
 (jj)
\(h(x) > 0 \) implies \(h(Sx) < h(x) \).
 (j)
Remark
Let \((X,d)\) and T be as in Definition 1. Put \(Y = X \times X\) and define h and S by \(h ( (x, y) ) = d(x, y)\) and \(S(x, y) = (Tx, Ty)\). Then the concept of Browder contraction in Definition 3 becomes that in Definition 1 and the concept of CJM contraction in Definition 3 becomes that in Definition 1.
We give some lemmas concerning (U)_{ f }.
Lemma 4
Remark

There exists a sequence \(\{ u_{n} \}\) in \(\operatorname {Dom}(f)\) such that \(\{ u_{n} \}\) converges to t and \(\{ f(u_{n}) \}\) converges to γ.

\(\limsup_{n} f(u_{n}) \leq\gamma\) holds for any sequence \(\{ u_{n} \}\) in \(\operatorname {Dom}(f)\) converging to t.
Proof of Lemma 4
Obvious. □
Lemma 5
Let f be an upper semicontinuous function from a subset of \(\mathbb {R}\) into \(\mathbb {R}\) such that \(f(t) < t\) for any \(t \in \operatorname {Dom}(f)\). Then f satisfies (U)_{ f }.
Proof
Obvious. □
Lemma 6
 (i)
φ satisfies (U)_{ φ }.
 (ii)
φ is strictly increasing and Lipschitz continuous.
 (iii)
\(\psi(t) < \varphi(t)\) holds for any \(t \in D\).
Proof
3 Browder contraction
In this section, we discuss Browder contractions.
Lemma 7
Proof

\(h(Sx) = 0\),

\(h(Sx) > 0\).
Proposition 8
 (i)
θ is nondecreasing and continuous.
 (ii)There exists a function ψ from Θ into \(\mathbb {R}\) satisfying (U)_{ ψ } andfor any \(x \in Y\) with \(h(x) > 0 \) and \(h(Sx) > 0\).$$ \theta\circ h(Sx) \leq\psi\circ\theta\circ h(x) $$(1)
Proof
By Lemma 5, we obtain the following.
Corollary 9
The following examples tell that the continuity of θ in Proposition 8 is needed. In Example 10, θ is right continuous, however, it is not left continuous. On the other hand, in Example 11, θ is left continuous, however, it is not right continuous.
Example 10
(Example 2.3 in [12])
Example 11
(Example 2.6 in [11])
4 CJM contraction
In this section, we discuss CJM contractions.
The following is a modification of Proposition 2.6 in [12].
Proposition 12
 (i)
θ is nondecreasing.
 (ii)
For any \(\varepsilon\in\Theta_{\leq}\), there exists \(\delta> 0\) such that \(h(x) > 0 \), \(h(Sx) > 0\), and \(\theta\circ h(x) < \varepsilon+ \delta\) imply \(\theta\circ h(Sx) \leq\varepsilon\) for all \(x \in Y\).
 (iii)
\(h(x) > 0 \) and \(h(Sx) > 0 \) imply \(\theta\circ h(Sx) < \theta\circ h(x) \).
Proof

\(\beta< \theta(\varepsilon+ \gamma) \) holds for any \(\gamma> 0\).

There exists \(\delta_{2} > 0\) such that \(\beta= \theta(\varepsilon+ \delta_{2}) \).
The proof of the following is obvious, however, we give a proof in order to show how differently Θ and \(\Theta_{\leq}\) work.
Corollary 13
Proof
By Lemma 5, we obtain the following.
Corollary 14
 (ii)
There exists an upper semicontinuous function ψ from \(\Theta_{\leq}\) into \(\mathbb {R}\) satisfying \(\psi(t) < t\) for any \(t \in\Theta_{\leq}\) and (1) for any \(x \in Y\) with \(h(x) > 0 \) and \(h(Sx) > 0\).
There is a counterexample if we replace \(\Theta_{\leq}\) with Θ in Corollary 14.
Example 15
Remark
We note that θ is left continuous, however, θ is not right continuous.
Proof
We assume additionally that θ is right continuous. Then we can prove the following.
Proposition 16
 (i)
θ is nondecreasing and right continuous.
 (ii)
For any \(\varepsilon\in\Theta\), there exists \(\delta> 0\) such that \(h(x) > 0 \), \(h(Sx) > 0\), and \(\theta\circ h(x) < \varepsilon+ \delta\) imply \(\theta\circ h(Sx) \leq\varepsilon\) for all \(x \in Y\).
Proof
Corollary 17
Assume (A1), (A2), (i) in Proposition 16 and (ii) in Proposition 8. Then S is a CJM contraction.
Proof
We can prove (iii) in Proposition 16 (in Proposition 12) as in the proof of Corollary 13. Also, we can prove (ii) in Proposition 16 as in the proof of Corollary 13. Therefore by Proposition 16, S is a CJM contraction. □
By Lemma 5, we also obtain the following.
Corollary 18
Assume (A1), (A2), (i) in Proposition 16 and (ii) in Corollary 9. Then S is a CJM contraction.
Finally, using Corollary 14, we give an alternative proof of Theorem 2.
Proof of Theorem 2
Let Y, h, and S be as in Example 10. Define a function ψ from \((1,\infty)\) into \((1,\infty)\) by \(\psi(t) = t^{k}\). Since ψ is continuous and \(\psi(t) < t\) for any \(t \in(1,\infty)\), all the assumption of Corollary 14 are satisfied. So by Corollary 14, S is a CJM contraction. By Theorem 13 in [16], T has a unique fixed point z. Moreover, \(\lim_{n} d(T^{n} x, z) = 0\) for any \(x \in X\). □
Remark
 (i)
We do not need (θ2) and (θ3) in Theorem 2.
 (ii)
If we assume additionally that θ is continuous, then T is a Browder contraction.
5 Some comments
As stated in Section 1, Lemma 5 in Jachymski [8] gives seven equivalent conditions connected with BroC. Finally, by Proposition 8 and Corollary 9 we can obtain the following, which is similar to Lemma 5 in [8].
Lemma 19
 (i)
There exists \(\varphi\in\varPhi\) such that for any \((t,u) \in D\), \(u \leq\varphi(t)\).
 (ii)
There exist a nondecreasing, continuous function θ from \((0,\infty)\) into \(\mathbb {R}\) and a function ψ from \(\theta ( (0,\infty) )\) into \(\mathbb {R}\) satisfying (U)_{ ψ } such that for any \((t,u) \in D\), \(\theta(u) \leq\psi\circ\theta(t)\).
 (iii)
There exist a nondecreasing, continuous function θ from \((0,\infty)\) into \(\mathbb {R}\) and an upper semicontinuous function ψ from \(\theta ( (0,\infty) )\) into \(\mathbb {R}\) satisfying \(\psi(t) < t\) such that for any \((t,u) \in D\), \(\theta(u) \leq\psi\circ\theta(t)\).
Declarations
Acknowledgements
The author is grateful to the referees for many suggestions to improve the exposition of the paper. The author is supported in part by JSPS KAKENHI Grant Number 25400141 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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