 Research
 Open Access
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
 Banach contraction principle
 BoydWong contraction
 contraction
 fixed point
 successive approximation
MSC
 54H25
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
References
 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
 Suzuki, T: A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point. Proc. Am. Math. Soc. 136, 40894093 (2008). MR2425751 View ArticleMATHGoogle Scholar
 Suzuki, T: Convergence of the sequence of successive approximations to a fixed point. Fixed Point Theory Appl. 2010, Article ID 716971 (2010). MR2595834 View ArticleGoogle Scholar
 Leader, S: Equivalent Cauchy sequences and contractive fixed points in metric spaces. Stud. Math. 76, 6367 (1983). MR0728197 MATHMathSciNetGoogle Scholar
 Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253, 440458 (2001). MR1808147 View ArticleMATHMathSciNetGoogle Scholar
 Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 23272335 (1997). MR1389524 View ArticleMATHMathSciNetGoogle Scholar
 Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458464 (1969). MR0239559 View ArticleMATHMathSciNetGoogle Scholar
 Matkowski, J: Integrable solutions of functional equations. Diss. Math. 127, 168 (1975). MR0412650 MathSciNetGoogle Scholar
 Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326329 (1969). MR0250291 View ArticleMATHMathSciNetGoogle Scholar
 Ćirić, LB: A new fixedpoint theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, 2527 (1981). MR0672538 Google Scholar
 Jachymski, J: Equivalent conditions and the MeirKeeler type theorems. J. Math. Anal. Appl. 194, 293303 (1995). MR1353081 View ArticleMATHMathSciNetGoogle Scholar
 Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol. 32. Cambridge University Press, Cambridge (1990). MR1067720 View ArticleMATHGoogle Scholar
 Matkowski, J: Fixed point theorems for contractive mappings in metric spaces. Čas. Pěst. Mat. 105, 341344 (1980). MR0597909 MATHMathSciNetGoogle Scholar
 Bessaga, C: On the converse of the Banach ‘fixedpoint principle’. Colloq. Math. 7, 4143 (1959). MR0111015 MATHMathSciNetGoogle Scholar
 Jachymski, J: A short proof of the converse to the contraction principle and some related results. Topol. Methods Nonlinear Anal. 15, 179186 (2000). MR1786260 MATHMathSciNetGoogle Scholar