- Research Article
- Open Access

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

- MA Ahmed
^{1}Email author and - FM Zeyada
^{1}

**2010**:647085

https://doi.org/10.1155/2010/647085

© M. A. Ahmed and F. M. Zeyada. 2010

**Received:**20 June 2009**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.

## Keywords

- Banach Space
- Differential Geometry
- Normed Space
- Generalize Type
- Convergence Theorem

## 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., [2–7]). Recently, some generalized types of quasi-nonexpansive mappings in metric and Banach spaces have appeared. For example, see Ahmed and Zeyada [8], Qihou [9–11] 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

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.

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.

(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.

the convexity of in Theorem 1.1 is superfluous;

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

- (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) ;

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) ;

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) ;

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) ;

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 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

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

## References

- Tricomi F:
**Un teorema sulla convergenza delle successioni formate della successive iterate di una funzione di una variable reale.***Giornale di Matematiche di Battaglini*1916,**54:**1–9.MATHGoogle Scholar - Das KM, Singh SP, Watson B:
**A note on Mann iteration for quasinonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*1981,**5**(6):675–676. 10.1016/0362-546X(81)90083-3MathSciNetView ArticleMATHGoogle Scholar - Dotson WG Jr.:
**On the Mann iterative process.***Transactions of the American Mathematical Society*1970,**149:**65–73. 10.1090/S0002-9947-1970-0257828-6MathSciNetView ArticleMATHGoogle Scholar - Ghosh MK, Debnath L:
**Approximation of the fixed points of quasi-nonexpansive mappings in a uniformly convex Banach space.***Applied Mathematics Letters*1992,**5**(3):47–50. 10.1016/0893-9659(92)90037-AMathSciNetView ArticleMATHGoogle Scholar - Ghosh MK, Debnath L:
**Convergence of Ishikawa iterates of quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1997,**207**(1):96–103. 10.1006/jmaa.1997.5268MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**Remarks on approximation and approximate fixed points in metric fixed point theory.***Annales Universitatis Mariae Curie-Skłodowska. Sectio A*1997,**51**(2):167–178.MathSciNetMATHGoogle Scholar - Petryshyn WV, Williamson TE Jr.:
**Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1973,**43:**459–497. 10.1016/0022-247X(73)90087-5MathSciNetView ArticleMATHGoogle Scholar - Ahmed MA, Zeyada FM:
**On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2002,**274**(1):458–465. 10.1016/S0022-247X(02)00242-1MathSciNetView ArticleMATHGoogle Scholar - Qihou L:
**Iterative sequences for asymptotically quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2001,**259**(1):1–7. 10.1006/jmaa.2000.6980MathSciNetView ArticleMATHGoogle Scholar - Qihou L:
**Iterative sequences for asymptotically quasi-nonexpansive mappings with error member.***Journal of Mathematical Analysis and Applications*2001,**259**(1):18–24. 10.1006/jmaa.2000.7353MathSciNetView ArticleMATHGoogle Scholar - Qihou L:
**Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space.***Journal of Mathematical Analysis and Applications*2002,**266**(2):468–471. 10.1006/jmaa.2001.7629MathSciNetView ArticleMATHGoogle Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems*. Springer, New York, NY, USA; 1986:xxi+897.View ArticleGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**Nonexpansive mappings and asymptotic regularity.***Nonlinear Analysis: Theory, Methods & Applications*2000,**40**(1–8):323–332.MathSciNetView ArticleMATHGoogle Scholar - Angrisani M, Clavelli M:
**Synthetic approaches to problems of fixed points in metric space.***Annali di Matematica Pura ed Applicata. Serie Quarta*1996,**170:**1–12. 10.1007/BF01758980MathSciNetView ArticleMATHGoogle Scholar - Kirk WA, Saliga LM:
**Some results on existence and approximation in metric fixed point theory.***Journal of Computational and Applied Mathematics*2000,**113**(1–2):141–152. 10.1016/S0377-0427(99)00249-6MathSciNetView ArticleMATHGoogle Scholar - Royden HL:
*Real Analysis*. The Macmillan, New York, NY, USA; 1963:xvi+284.MATHGoogle Scholar

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