A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point II
 Tomonari Suzuki^{1, 2}Email author and
 Badriah Alamri^{2}
https://doi.org/10.1186/s1366301503029
© Suzuki and Alamri; licensee Springer. 2015
Received: 20 January 2015
Accepted: 24 March 2015
Published: 24 April 2015
Abstract
Using the concept of a BoydWong contraction, we obtain a simple, sufficient, and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point.
Keywords
MSC
1 Introduction
The Banach contraction principle is a very forceful tool in nonlinear analysis.
Theorem 1
(Banach [1] and Caccioppoli [2])
 (A)
T has a unique fixed point z and \(\{ T^{n} x \}\) converges to z for any \(x \in X\).
In [3, 4] we studied (A) and obtained the following. See also [5, 6].
Theorem 2
 (B)T is a strong Leader mapping, that is, the following hold:

For \(x, y \in X\) and \(\varepsilon> 0\), there exist \(\delta> 0\) and \(\nu\in\mathbb{N}\) such thatfor all \(i, j \in\mathbb{N}\cup\{ 0 \}\), where \(T^{0}\) is the identity mapping on X.$$d\bigl(T^{i} x, T^{j} y\bigr) < \varepsilon+ \delta\quad \Longrightarrow\quad d\bigl(T^{i+\nu}x, T^{j+\nu}y\bigr) < \varepsilon $$

For \(x, y \in X\), there exist \(\nu\in\mathbb{N}\) and a sequence \(\{ \alpha_{n} \}\) in \((0, \infty)\) such thatfor all \(i, j \in\mathbb{N}\cup\{ 0 \}\) and \(n \in\mathbb{N}\).$$d\bigl(T^{i} x, T^{j} y\bigr) < \alpha_{n}\quad \Longrightarrow\quad d\bigl(T^{i+\nu}x, T^{j+\nu}y\bigr) < 1/n $$

 (C)
There exist a τdistance p and \(r \in(0,1)\) such that \(p(Tx, T^{2} x) \leq r p(x, Tx) \) and \(p(Tx, Ty) < p(x, y) \) for all \(x, y \in X \) with \(x \neq y\).
We cannot tell that Theorem 2 is simple. Motivated by this fact, in this paper, we obtain a simpler condition equivalent to (A).
In 1969, Boyd and Wong proved a very interesting fixed point theorem. See [7]. The concept of a BoydWong contraction plays an important role in this paper. Indeed, using this concept, we give a condition equivalent to (A); see Theorem 9 below. We will find that Theorem 3 is an essential generalization of Theorem 1 in some sense; see Theorem 11 and Example 12 below.
Theorem 3
(Boyd and Wong [8])
 (i)
φ is upper semicontinuous.
 (ii)
\(\varphi(t) < t\) for every \(t \in(0, \infty)\).
 (iii)
\(d(Tx, Ty) \leq\psi ( d(x,y) ) \) for all \(x, y \in X\).
Later, in 1975, Matkowski proved the following generalization of Theorem 1. Interestingly, while Theorem 3 and Theorem 4 look similar, we will find that Theorem 4 is similar to Theorem 1, not Theorem 3, in some sense; see Theorem 13 below.
Theorem 4
(Matkowski [9])
 (i)
ψ is nondecreasing.
 (ii)
\(\lim_{n} \psi^{n}(t) = 0\) for every \(t \in(0, \infty)\).
 (iii)
\(d(Tx, Ty) \leq\psi ( d(x,y) ) \) for all \(x, y \in X\).
We introduce two more interesting theorems. Theorem 5 is a generalization of Theorem 3 and Theorem 6 is a generalization of Theorems 4 and 5.
Theorem 5
(Meir and Keeler [10])
Theorem 6
(Ćirić [11], Jachymski [12] and Matkowski [13, 14])
 (i)
For every \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(d(x, y) < \varepsilon+ \delta\) implies \(d(Tx, Ty) \leq\varepsilon\).
 (ii)
\(x \neq y \) implies \(d(Tx, Ty) < d(x,y) \).
2 Preliminaries
Throughout this paper, we denote by \(\mathbb{N}\), \(\mathbb{Z}\), and \(\mathbb{R}\) the sets of positive integers, integers, and real numbers, respectively. For \(t \in\mathbb{R}\), we denote by \([t]\) the maximum integer not exceeding t. For an arbitrary set A, we denote by ♯A the cardinal number of A.
In the proof of our main result, we use the following.
Lemma 7
Proof
We denote by Ψ the set of all functions ψ satisfying (i) and (ii) of Theorem 4.
Lemma 8
Let φ and ψ belong to Ψ. Then a function η from \([0, \infty)\) defined by \(\eta(t) = \max\{ \varphi(t), \psi(t) \}\) also belongs to Ψ.
Proof
3 Main results
We prove our main results.
Theorem 9
 (D)
There exists a complete metric ρ on X such that \(\rho\geq d\) and \(T \in\operatorname{BWC}(X,\rho)\).
Proof
(D) ⇒ (A): We assume (D). Then Theorem 3 shows that there exists a unique fixed point z of T and that for every \(x \in X\), \(\{ T^{n} x \}\) converges to z in \((X,\rho)\). Since the topology of \((X, \rho)\) is stronger than that of \((X, d)\), \(\{ T^{n} x \}\) converges to z in \((X,d)\). Hence (A) holds.

\(Tx = z\);

\(Tx \neq z\) and \(k(x) = 0\);

\(Tx \neq z\) and \(k(x) \in\mathbb{N}\).
From Theorem 9, we obtain the following.
Corollary 10
 (E)
There exists a complete metric ρ on X such that the topology of \((X,\rho)\) is stronger than that of \((X,d)\) and \(T \in\operatorname{BWC}(X,\rho)\).
 (F)
There exists a complete metric ρ on X such that the topology of \((X,\rho)\) is stronger than that of \((X,d)\) and \(T \in\operatorname{MKC}(X,\rho)\).
 (G)
There exists a complete metric ρ on X such that the topology of \((X,\rho)\) is stronger than that of \((X,d)\) and \(T \in\operatorname{CJMC}(X,\rho)\).
Proof
It is obvious that (E) ⇒ (F) ⇒ (G). The proof of (G) ⇒ (A) is almost the same as that of (D) ⇒ (A). Since (E) is weaker than (D), we have (A) ⇒ (E) by Theorem 9. □
We note that (E) is much simpler than (B) and (C).
4 Additional results
In the previous section, we have showed that (D) is equivalent to (A). In this section, we will show that the condition (H) on contractions is not equivalent to (A).
Theorem 11
 (H)
There exists a complete metric ρ on X such that the topology of \((X,\rho)\) is stronger than that of \((X,d)\) and \(T \in\operatorname{Cont}(X,\rho)\).
Proof
The proof of (D) ⇒ (A) works. □
The following example tells that (A) ⇒ (H) does not hold.
Example 12
([4])
Proof
The Matkowski contraction version of (H) is equivalent to (H) itself.
Theorem 13
 (I)
There exists a complete metric ρ on X such that the topology of \((X,\rho)\) is stronger than that of \((X,d)\) and \(T \in\operatorname{MC}(X,\rho)\).
Proof
(H) ⇒ (I): Obvious.

\(t_{n+1} = \varphi(t_{n})\) for every \(n \in\mathbb{Z}\),

\(\{ t_{n} \}_{n \in\mathbb{Z}}\) is a strictly decreasing sequence,

\(\{ t_{n} \}_{n \in\mathbb{Z}}\) converges to 0 as n tends to ∞,

\(\{ t_{n} \}_{n \in\mathbb{Z}}\) converges to ∞ as n tends to −∞.
The following result due to Bessaga [15] (see also [16]) shows that the topological condition appearing in condition (H) cannot be removed, because otherwise the convergence of iterates in the metric space \((X, d)\) cannot be ensured.
Theorem 14
(Bessaga [15])
 (J)
There exists a complete metric ρ on X such that \(T \in\operatorname{Cont}(X,\rho)\).
 (K)
There exists a unique fixed point z of T and the set of periodic points of T is \(\{ z \}\).
If X is a metric space, then (J) is strictly weaker than (A) because (K) is strictly weaker than (A).
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (3513035HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The first author is supported in part by GrantinAid for Scientific Research 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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