# Convergence of the Sequence of Successive Approximations to a Fixed Point

- Tomonari Suzuki
^{1}Email author

**2010**:716971

**DOI: **10.1155/2010/716971

© Tomonari Suzuki. 2010

**Received: **29 September 2009

**Accepted: **21 December 2009

**Published: **2 February 2010

## Abstract

If is a complete metric space and is a contraction on , then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations of starting from any point converges to a unique fixed point. In this paper, using the concept of -distance, we obtain simple, sufficient, and necessary conditions of the above conclusion.

## 1. Introduction

The following famous theorem is referred to as the *Banach-Caccioppoli contraction principle*. This theorem is very forceful and simple, and it became a classical tool in nonlinear analysis.

Theorem 1.1 (see Banach [1] and Caccioppoli [2]).

Let be a complete metric space and let be a self contraction on , that is, there exists such that for all . Then the following holds.

(A) has a unique fixed point , and converges to for any .

We note that the conclusion of Kannan's fixed point theorem [3] is also (A). See Kirk's survey [4]. Recently, we obtained that (A) holds if and only if is a strong Leader mapping [5, 6].

Theorem 1.2 (see [6]).

Let be a mapping on a complete metric space . Then the following are equivalent.

(i)(A) holds.

(ii) is a strong Leader mapping, that is, the following hold.

for all , where is the identity mapping on .

for all and .

The following theorem is proved in [7, 8].

Theorem 1.3 (see Rus [7] and Subrahmanyam [8]).

Let be a complete metric space and let be a continuous mapping on . Assume that there exists satisfying for all . Then the following holds.

(B) converges to a fixed point for every .

We obtained a condition equivalent to (B) in [9].

Theorem 1.4 (see [9]).

Let be a mapping on a complete metric space . Then the following are equivalent.

(i)(B) holds.

(ii)The following hold.

for all .

for all and .

We sometimes call a mapping satisfying (A) a *Picard operator* [10]. We also call a mapping satisfying (B) a *weakly Picard operator* [11–13].

We cannot tell that the conditions (ii) of Theorems 1.2 and 1.4 are simple. Motivated by this, we obtain simpler conditions which are equivalent to Conditions (A) and (B).

## 2. Preliminaries

Throughout this paper, we denote by , , and the sets of positive integers, integers and real numbers, respectively.

In 2001, Suzuki introduced the concept of -distance in order to improve results in Tataru [14], Zhong [15, 16], and others. See also [17].

Definition 2.1 (see [18]).

Let
be a metric space. Then a function
from
into
is called a
*-distance* on
if there exists a function
from
into
and the following are satisfied:

() for all ,

() and for all and , and is concave and continuous in its second variable,

() and imply for all ,

() and imply that ,

() and imply that .

The metric is a -distance on . Many useful examples and propositions are stated in [9, 18–23] and references therein. The following fixed point theorems are proved in [18].

Theorem 2.2 (see [18]).

Let be a complete metric space and let be a mapping on . Assume that there exist a -distance and such that for all . Assume the following.

(i)If , , and , then .

Then (B) holds. Moreover, if , then .

Theorem 2.3 (see[18]).

for all . Then (A) and hold.

The following lemmas are useful in our proofs.

Lemma 2.4 (see [18]).

Let be a metric space and let be a -distance on . If sequences and in satisfy and for some , then . In particular for and imply that .

Lemma 2.5 (see [18]).

Let be a metric space and let be a -distance on . If a sequence in satisfies , then is a Cauchy sequence. Moreover if a sequence in satisfies , then .

The following lemmas are easily deduced from Lemmas 2.4 and 2.5.

Lemma 2.6.

Let be a metric space and let be a -distance on . Then for every and , there exists such that and imply that .

Lemma 2.7.

Let be a metric space and let be a -distance on . Assume that a sequence in satisfies , , and . Then .

The following is proved at Page 442 of [18]. However we give a proof because we use reductio ad absurdum in [18].

Lemma 2.8 (see [18]).

Then , for all ; and is concave and continuous.

Proof.

Since and are arbitrary, we obtain . Thus, .

The following is obvious.

Lemma 2.9.

Then the following hold.

(i)For every and , if and only if for and .

(ii) for with .

(iii) for every .

## 3. Condition (B)

In this section, we discuss Condition (B).

Theorem 3.1.

for all . Then (B) holds. Moreover, if , then .

Proof.

and hence, . By Lemma 2.7, we obtain . By Theorem 2.2, we obtain the desired result.

As a direct consequence of Theorem 3.1, we obtain the following.

Corollary 3.2.

for all . Then (B) holds.

Corollary 3.2 characterizes Condition (B).

Theorem 3.3.

Let be a mapping on a metric space such that (B) holds. Then there exist a -distance and satisfying (3.3).

Proof.

We put for with and . We have defined . We note that implies that .

where . We note that implies either of the following.

(i) .

(ii)There exist , , and such that , , , and . (In this case, , , , and hold.)

Then satisfies ( 2) and by Lemma 2.8. In order to show ( 3), we assume that and . Then without loss of generality, we may assume that . Thus for . It is obvious that for with . We consider the following two cases.

(i)There exists such that for .

(ii)There exists a subsequence of such that .

Therefore we have shown ( 3). Let us prove ( 4). We assume that and . Without loss of generality, we may assume that . We consider the following two cases.

(i)There exists such that for .

(ii)There exists a subsequence of such that .

which imply ( 5). Therefore we have shown that is a -distance on .

Therefore (3.3) holds.

Remark 3.4.

We have proved that, for every , there exists a -distance satisfying (3.3).

Combining Theorem 6 in [9], we obtain the following.

Corollary 3.5.

Let be a mapping on a complete metric space . Then the following are equivalent.

(i)(B) holds.

(ii)There exists a -distance on satisfying the following.

for all with .

for all and with .

(iii)There exist a -distance and such that and for all .

## 4. Condition (A)

In this section, we discuss Condition (A).

Define a relation on as follows: if and only if either or holds.

Theorem 4.1.

for all . Then (A) holds.

Proof.

which implies that . Since , we obtain by Lemma 2.4. Thus the fixed point is unique.

Theorem 4.2.

for all with . Then (A) holds.

Proof.

In the case where consists of one element, the conclusion obviously holds. So we consider the other case. Assume that , , and . We consider the following two cases:

(i) for sufficiently large ,

(ii)there exists a sequence of such that .

which implies a contradiction. Thus the fixed point is unique.

We shall show that Theorems 4.1 and 4.2 characterize Condition (A).

Theorem 4.3.

Let be a mapping on a metric space such that (A) holds. Then there exist a -distance and satisfying (4.1).

Proof.

Let , , , and be as in the proof of Theorem 3.3. Then holds. Fix . We consider the following two cases:

(i) and

(ii)either or

Therefore (4.1) holds.

Theorem 4.4.

Let be a mapping on a metric space such that (A) holds. Then there exist a -distance and satisfying (4.3) for all with .

Proof.

The proof of Theorem 4.3 works.

Combining Theorem 7 in [9], we obtain the following.

Corollary 4.5.

Let be a mapping on a complete metric space . Then the following are equivalent.

(i)(A) holds.

(ii)There exists a -distance on satisfying the following.

for all with .

for all and with .

(iii)There exist a -distance and such that and for all .

(iv)There exist a -distance and such that and for all with .

## 5. Additional Result

Since Theorem 2.2 deduces Corollary 3.2, we can tell that Theorem 2.2 characterizes Condition (B). However, the following example tells that Theorem 2.3 does not characterize Condition (A).

Example 5.1.

Then (A) holds. However, is not a contraction with respect to any -distance .

Proof.

This is a contradiction.

## Declarations

### Acknowledgments

The author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science. The author wishes to express his gratitude to the referees for careful reading and giving a historical comment.

## Authors’ Affiliations

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