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

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.

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

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. Zeidler E: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer, New York, NY, USA; 1986:xxi+897.View ArticleGoogle Scholar
  13. 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
  14. 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
  15. Kirk WA: Nonexpansive mappings and asymptotic regularity. Nonlinear Analysis: Theory, Methods & Applications 2000,40(1–8):323–332.MathSciNetView ArticleMATHGoogle Scholar
  16. 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
  17. 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
  18. Royden HL: Real Analysis. The Macmillan, New York, NY, USA; 1963:xvi+284.MATHGoogle Scholar

Copyright

© M. A. Ahmed and F. M. Zeyada. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.