Open Access

# Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications

Fixed Point Theory and Applications20092010:647085

DOI: 10.1155/2010/647085

Accepted: 7 September 2009

Published: 28 September 2009

## Abstract

The concept of weakly quasi-nonexpansive mappings with respect to a sequence is introduced. This concept generalizes the concept of quasi-nonexpansive mappings with respect to a sequence due to Ahmed and Zeyada (2002). Mainly, some convergence theorems are established and their applications to certain iterations are given.

## 1. Introduction

In 1916, Tricomi [1] introduced originally the concept of quasi-nonexpansive for real functions. Subsequently, this concept has studied for mappings in Banach and metric spaces (see, e.g., [27]). Recently, some generalized types of quasi-nonexpansive mappings in metric and Banach spaces have appeared. For example, see Ahmed and Zeyada [8], Qihou [911] and others.

Unless stated to the contrary, we assume that is a metric space. Let be any mapping and let be the set of all fixed points of . If where is the set of all real numbers and if , set . We use the symbol to denote the usual Kuratowski measure of noncompactness. For some properties of see Zeidler [12, pages 493–495]. For a given , the Picard iteration is determined by:

(I)

where is the set of all positive integers.

If is a normed space, is a convex set, and , Ishikawa [13] gave the following iteration:

(II)

for each , where and . When , it yields that . Therefore, the iteration scheme (II) becomes

(1.1)

This iteration is called Mann iteration [14].

The concepts of quasi-nonexpansive mappings, with respect to a sequence and asymptotically regular mappings at a point were given in metric spaces as follows.

Definition 1.1 (see [6]).

is said to be quasi-nonexpansive mapping if for each and for every , .

Definition 1.2 (see [8]).

The map is said to be quasi-nonexpansive with respect to if for all and for every , .

Lemma  2.1 in [8] stated that quasi-nonexpansiveness converts to quasi-nonexpansiveness with respect to (resp., , ) for each . The reverse implication is not true (see, [8, Example  2.1]). Also, the authors [8] showed that the continuity of leads to the closedness of and the converse is not true (see, [8, Example  2.2]).

Definition 1.3 (see [15]).

The mapping is called an asymptotically regular at a point if .

The following definition is given by Angrisani and Clavelli.

Definition 1.4 (see [16]).

Let be a topological space. The function is said to be a regular-global-inf (r.g.i) at if implies that there exists such that and a neighborhood of such that for each . If this condition holds for each , then is said to be an r.g.i on .

Definition 1.5 (see [17]).

Let be a convex subset of a normed space . A mapping is called directionally nonexpansive if for each and for all where denotes the segment joining and ; that is, .

Our objective in this paper is to introduce the concept of weakly quasi-nonexpansive mappings with respect to a sequence. Mainly, we establish some convergence theorems of a sequence in complete metric spaces. These theorems generalize and improve [8, Theorems  2.1 and 2.2], of [7, Theorems  1.1 and ], [5, Theorem  3.1], and [6, Proposition  1.1].

## 2. Main Result

In this section, we introduce the concept of weak quasi-nonexpansiveness of a mapping with respect to a sequence that generalizes quasi-nonexpansiveness of a mapping with respect to a sequence in [8]. We give a lemma and a counterexample to show the relation between our new concept; the previous one appeared in [8] and a monotonically decreasing sequence .

Definition 2.1.

Let be a metric space and let be a sequence in . Assume that is a mapping with satisfying . Thus, for a given there is a such that . is called weakly quasi-nonexpansive with respect to if for each there exists a such that for all with , .

We state the following lemma without proof.

Lemma 2.2.

Let be a metric space and, be a sequence in . Assume that is a mapping with satisfying . If is quasi-nonexpansive with respect to , then

(A) is weakly quasi-nonexpansive with respect to ;

(B) is a monotonically decreasing sequence in .

The following example shows that the converse of Lemma 2.2 may not be true.

Example 2.3.

Let be endowed with the Euclidean metric . We define the map by for each . Assume that . Then
(2.1)
Given , there exists such that for all with , there exists ,
(2.2)

Thus, is weakly quasi-nonexpansive with respect to . But, is not quasi-nonexpansive with respect to (Indeed, there exists such that for all , ). Furthermore, the sequence is monotonically decreasing in .

Before stating the main theorem, let us introduce the following lemma without proof.

Lemma 2.4.

Let be a metric space and let be a sequence in . Assume that is weakly quasi-nonexpansive with respect to with satisfying . Then, is a Cauchy sequence.

Now, we give the main theorem without proof in the following way.

Theorem 2.5.

Let be a sequence in a subset of a metric space and let be a map such that . Then

(a) if converges to a point in ;

(b) converges to a point in if , is a closed set, is weakly quasi-nonexpansive with respect to , and is complete.

As corollaries of Theorem 2.5, we have the following.

Corollary 2.6.

For each , let be a sequence in a subset of a metric space and let be a map such that . Then

(a) if converges to a point in ;

(b) converges to a point in if , is a closed set, is weakly quasi-nonexpansive with respect to and is complete.

Corollary 2.7.

For each , let be a sequence in a subset of a normed space and let be a map such that . Then

(a) if converges to a point in ;

(b) converges to a point in if , is a closed set, is weakly quasi-nonexpansive with respect to , and is a Banach space.

Corollary 2.8.

For each , let be a sequence in a subset of a normed space and let be a map such that . Then

(a) if converges to a point in ;

(b) converges to a point in if , is a closed set, is weakly quasi-nonexpansive with respect to , and is a Banach space.

Remark 2.9.
1. (I)

Theorem 2.5 generalizes and improves [8, Theorem  2.1] since is weakly quasi-nonexpansive with respect to instead of being quasi-nonexpansive with respect to .

2. (II)

Corollary 2.6 generalizes and improves [7, Theorem  1.1 page 462] for some reasons. These reasons are the following:

(1)the closedness of is superfluous;

(2) is closed instead of being continuous;

(3) is a complete metric space instead of is a Banach space;

(4) is weakly quasi-nonexpansive with respect to in lieu of being quasi-nonexpansive.

1. (III)

Corollary 2.7 (resp. Corollary 2.8) generalizes and improves [7, Theorem   page 469] (resp. of [5, Theorem  3.1 page 98]) since the reasons (1) and (2) in (II) hold and

the convexity of in Theorem  1.1 is superfluous;

is weakly quasi-nonexpansive with respect to (resp. ) instead of being quasi-nonexpansive.

1. (IV)

If we take instead of , is closed in lieu of being continuous and is weakly quasi-nonexpansive with respect to in lieu of being quasi-nonexpansive, then Corollary 2.6 generalizes and improves Kirk [6, Proposition  1.1].

In the light of Lemma 2.2 and Example 2.3, we state the following theorem.

Theorem 2.10.

Let be a sequence in a subset of a complete metric space and be a map such that is a closed set. Assume that

(i) is weakly quasi-nonexpansive with respect to ;

(ii) is a monotonically decreasing sequence in ;

(iii) ;

(iv)if the sequence satisfies , then
(2.3)

Then converges to a point in .

Proof.

From the boundedness from below by zero of the sequence and (ii), we obtain that exists. So, from (iii) and (iv), we have that or . Then (see, [18, page 37]). Therefore, by Theorem 2.5(b), the sequence converges to a point in .

Corollary 2.11.

For each , let be a sequence in a subset of a complete metric space and let be a map such that is a closed set. Assume that

(i) is weakly quasi-nonexpansive with respect to ;

(ii) is a monotonically decreasing sequence in ;

(iii) ;

(iv)if the sequence satisfies , then
(2.4)

Then converges to a point in .

Corollary 2.12.

For each let be a sequence in a subset of a Banach space and let be a map such that is a closed set. Assume that

(i) is weakly quasi-nonexpansive with respect to ;

(ii) is a monotonically decreasing sequence in ;

(iii) ;

(iv)if the sequence satisfies , then
(2.5)

Then converges to a point in .

Corollary 2.13.

For each , let be a sequence in a subset of a Banach space and let be a map such that is a closed set. Assume that

(i) is weakly quasi-nonexpansive with respect to ;

(ii) is a monotonically decreasing sequence in ;

(iii) ;

(iv)if the sequence satisfies , then
(2.6)

Then converges to a point in .

Remark 2.14.

From Lemma 2.2, we find that [8, Theorem  2.2] is a special case of Theorem 2.10. Also, Corollary 2.11 generalizes and improves [7, Theorem  1.2 page 464] for the same reasons in Remark 2.9(II).

We establish another consequence of Theorem 2.5 as follows.

Theorem 2.15.

Let be a sequence in a subset of a complete metric space . Furthermore, let be a mapping such that is a closed set. Assume that the conditions (i) and (ii) in Theorem 2.10 hold and

(iii)the sequence contains a convergent subsequence converging to such that there exists a continuous mapping satisfying and for some .

Then and .

Proof.

From (ii), one can deduce that exists, say equal . Suppose that does not belong to . So, we have from (iii) that for some ,
(2.7)
This contradiction implies that . Then,
(2.8)

From Theorem 2.5(b), we obtain that .

Corollary 2.16.

For each , let be a sequence in a subset of a complete metric space . Furthermore, let be a mapping such that is a closed set. Assume that the conditions (i) and (ii) in Corollary 2.11 hold and

(iii)the sequence contains a convergent subsequence converging to such that there exists a continuous mapping satisfying and for some .

Then and .

Corollary 2.17.

For each , let be a sequence in a subset of a complete metric space . Furthermore, let be a mapping such that is a closed set. Assume that the conditions (i) and (ii) in Corollary 2.12 hold and

(iii)the sequence contains a convergent subsequence converging to such that there exists a continuous mapping satisfying and for some .

Then and .

Corollary 2.18.

For each , let be a sequence in a subset of a complete metric space . Furthermore, let be a mapping such that is a closed set. Assume that the conditions (i) and (ii) in Corollary 2.13 hold and

(iii)the sequence contains a convergent subsequence converging to such that there exists a continuous mapping satisfying and for some .

Then and .

Remark 2.19.

Theorem  1.3 in [7] is a special case of Corollary 2.16 for the same reasons in Remark 2.9(II) and for the generalization of the conditions (1.6) and (1.7) in [7, Theorem  1.3] to the condition (iii) in Corollary 2.16.

From [17, Corollary  2.4] and Theorem 2.5(b), one can prove the following theorem.

Theorem 2.20.

Let be a mapping of a complete metric space satisfying

(i) for some and for all ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) is a sequence in such that and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Corollary 2.21.

Let be a mapping of a complete metric space satisfying

(i) for some and for all ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) is a sequence satisfying for each and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Corollary 2.22.

Let be a mapping of a Banach space satisfying

(i) for some and for all ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) is a sequence in such that for each and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Corollary 2.23.

Let be a mapping of a Banach space satisfying

(i) for some and for all ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) is a sequence in such that for each and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Theorem 2.24.

Let be a bounded closed convex subset of a Banach space Suppose that satisfies

(i) is directionally nonexpansive on

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) satisfies and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Proof.

The conclusion is obtained by combining [17, Theorem  3.3] and Theorem 2.5(b).

Corollary 2.25.

Let be a bounded closed convex subset of a Banach space Suppose that satisfies

(i) is directionally nonexpansive on ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) for each satisfies and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Corollary 2.26.

Let be a bounded closed convex subset of a Banach space . Suppose that satisfies

(i) is directionally nonexpansive on ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) for each satisfies and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Corollary 2.27.

Let be a bounded closed convex subset of a Banach space . Suppose that satisfies

(i) is directionally nonexpansive on ;

(ii) for some and for all ;

(iii) is an r.g.i. on ;

(iv) for each satisfies and is weakly quasi-nonexpansive with respect to .

Then converges to a point in .

Remark 2.28.

It is worth to mention that Corollaries 2.12, 2.13, 2.17, 2.18, 2.21–2.23, 2.25–2.27 are new results.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Assiut University

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