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Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications
Fixed Point Theory and Applications volume 2010, Article number: 647085 (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., [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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ2_HTML.gif)
Given , there exists
such that for all
with
, there exists
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ3_HTML.gif)
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.
-
(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
.
-
(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.
-
(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.
-
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ7_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ8_HTML.gif)
This contradiction implies that . Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647085/MediaObjects/13663_2009_Article_1321_Equ9_HTML.gif)
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.
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Ahmed, M., Zeyada, F. Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications. Fixed Point Theory Appl 2010, 647085 (2009). https://doi.org/10.1155/2010/647085
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DOI: https://doi.org/10.1155/2010/647085