Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications
© M. A. Ahmed and F. M. Zeyada. 2010
Received: 20 June 2009
Accepted: 7 September 2009
Published: 28 September 2009
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.
In 1916, Tricomi  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 , 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:
If is a normed space, is a convex set, and , Ishikawa  gave the following iteration:
This iteration is called Mann iteration .
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 ).
Definition 1.2 (see ).
Lemma 2.1 in  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  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 ).
The following definition is given by Angrisani and Clavelli.
Definition 1.4 (see ).
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 ).
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 . We give a lemma and a counterexample to show the relation between our new concept; the previous one appeared in  and a monotonically decreasing sequence .
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.
The following example shows that the converse of Lemma 2.2 may not be true.
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.
Now, we give the main theorem without proof in the following way.
As corollaries of Theorem 2.5, we have the following.
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.
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 .
We establish another consequence of Theorem 2.5 as follows.
Theorem 1.3 in  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.
The conclusion is obtained by combining [17, Theorem 3.3] and Theorem 2.5(b).
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.
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