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Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 385986 (2010)
Abstract
We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.
1. Introduction and Preliminaries
Let be a nonempty subset of a metric space
and let
be a map. Denote the set of fixed points of
by
The map
is said to be (i) quasi-nonexpansive if
and
for all
and
, (ii)
-Lipschitz if for some
we have
for all
for
it becomes nonexpansive, and (iii) generalized nonexpansive (cf. [1] and the references therein) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ1_HTML.gif)
for all where
with
The concept of quasi-nonexpansiveness is more general than that of nonexpansiveness. A nonexpansive map with at least one fixed point is quasi-nonexpansive but there are quasi-nonexpansive maps which are not nonexpansive [2].
Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive maps have been extensively studied in uniformly convex Banach spaces [1, 3–6]. Senter and Dotson [7] established convergence of Mann type iterates of quais-nonexpansive maps under a condition in uniformly convex Banach spaces. In 1973, Goebel et al. [8] proved that generalized nonexpansive self maps have fixed points in uniformly convex Banach spaces. Based on their work, Bose and Mukerjee [1] proved theorems for the convergence of Mann type iterates of generalized nonexpansive maps and obtained a result of Kannan [9] under relaxed conditions. Maiti and Ghosh [6] generalized the results of Bose and Mukerjee [1] for Ishikawa iterates by using modified conditions of Senter and Dotson [7] (see, also [10]). For the sake of completeness, we state the result of Kannan [9] and its generalization by Bose and Mukerjee [1].
Theorem 1.1 (see [9]).
Let be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space. Let
be a map of
into itself such that
(i) for all
,
(ii) where
is any nonempty convex subset of
which is mapped into itself by
and
is the diameter of
Then the sequence defined by
converges to the fixed point of
where
is any arbitrary point of
Theorem 1.2 (see [1]).
Let be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space. Let
be a map of
into itself such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ2_HTML.gif)
for all where
and
Define a sequence
in
for
, for all
, where
Then
converges to a fixed point of
.
In Theorem 1.2, taking , and
for all
it becomes Theorem 1.1 without requiring condition (ii).
In 1970, Takahashi [11] introduced a notion of convexity in a metric space as follows: a map
is a convex structure in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ3_HTML.gif)
for all and
A metric space together with a convex structure is said to be convex metric space. A nonempty subset
of a convex metric space is convex if
for all
and
In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see [11]). Later on, Shimizu and Takahashi [12] obtained some fixed point theorems for nonexpansive maps in convex metric spaces. This notion of convexity has been used in [13–15] to study Mann and Ishikawa iterations in convex metric spaces. For other fixed point results in the closely related classes of spaces, namely, hyperbolic and hyperconvex metric spaces, we refer to [16–19].
In the sequel, we assume that is a nonempty convex subset of a convex metric space
and
is a selfmap on
. For an initial value
we define the Ishikawa iteration scheme in
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ4_HTML.gif)
where and
are control sequences in
If we choose then (1.3) reduces to the following Mann iteration scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ5_HTML.gif)
where is a control sequence in
If is a normed space with
as its convex subset, then
is a convex structure in
consequently (1.3) and (1.4), respectively, become
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ6_HTML.gif)
where and
are control sequences in
A convex metric space is said to be uniformly convex [11] if for arbitrary positive numbers
and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ7_HTML.gif)
whenever and
In 1989, Maiti and Ghosh [6] generalized the two conditions due to Senter and Dotson [7]. We state all these conditions in convex metric spaces:
Letbe a map with nonempty fixed point set
and
. Then
is said to satisfy the following Condotions.
Condition 1.
If there is a nondecreasing function with
and
for all
such that
for
.
Condition 2.
If there exists a real number such that
for
.
Condition 3.
If there is a nondecreasing function with
and
for all
such that
for
and all corresponding
where
.
Condition 4.
If there exists a real number such that
for
and all corresponding
where
Note that if satisfies Condition 1 (resp., 3), then it satisfies Condition 2 (resp., 4). We also note that Conditions 1 and 2 become Conditions A and B, respectively, of Senter and Dotson [7] while Conditions 3 and 4 become Conditions I and II, respectively, of Maiti and Ghosh [6] in a normed space. Further, Conditions 3 and 4 reduce to Conditions 1 and 2, respectively, when
In this note, we present results under relaxed control conditions which generalize the corresponding results of Kannan [9], Bose and Mukerjee [1], and Maiti and Ghosh [6] from uniformly convex Banach spaces to uniformly convex metric spaces. We present sufficient conditions for the convergence of Ishikawa iterates of Lipschitz maps to their fixed points in convex metric spaces and improve [3, Lemma 2]. A necessary and sufficient condition is obtained for the convergence of a sequence to fixed point of a generalized nonexpansive map in metric spaces.
We need the following fundamental result for the developmant of our results.
Theorem 1.3 (see [20]).
Let be a uniformly convex metric space with a continuous convex structure
Then for arbitrary positive numbers
and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ8_HTML.gif)
for all and
2. Convergence Analysis
We prove a lemma which plays key role to establish strong convergence of the iterative schemes (1.3) and (1.4).
Lemma 2.1.
Let be a uniformly convex metric space. Let
be a nonempty closed convex subset of
a quasi-nonexpansive map and
as in (1.3). If
and
then
Proof.
For , we consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ9_HTML.gif)
This implies that the sequence is nonincreasing and bounded below. Thus
exists. We may assume that
For any , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ10_HTML.gif)
Since exists, so
is bounded and hence
exists. We show that
. Assume that
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ11_HTML.gif)
Hence by Theorem 1.3, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ12_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ13_HTML.gif)
Taking and summing up the
terms on the both sides in the above inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ14_HTML.gif)
Let . Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ15_HTML.gif)
This is contradiction and hence
In the light of above result, we can construct subsequences and
of
and
, respectively, such that
and hence
Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.
Theorem 2.2.
Let be a uniformly convex complete metric space with continuous convex structure and let
be its nonempty closed convex subset. Let
be a continuous quasi-nonexpansive map of
into itself satisfying Condition 3. If
is as in (1.3), where
and
, then
converges to a fixed point of
.
Proof.
In Lemma 2.1, we have shown that Therefore
. This implies that the sequence
is nonincreasing and bounded below. Thus
exists. Now by Condition 3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ16_HTML.gif)
Using the properties of we have
. As
exists, therefore
Now, we show that is a Cauchy sequence. For
there exists a constant
such that for all
we have
In particular,
That is,
There must exist
such that
Now, for
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ17_HTML.gif)
This proves that is a Cauchy sequence in
. Since
is a closed subset of a complete metric space
therefore it must converge to a point
in
.
Finally, we prove that is a fixed point of
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ18_HTML.gif)
therefore As
is closed, so
Choose for all
in the above theorem; it reduces to the following Mann type convergence result.
Theorem 2.3.
Let be a uniformly convex complete metric space with continuous convex structure and let
be its nonempty closed convex subset. Let
be a continuous quasi-nonexpansive map of
into itself satisfying Condition 1. If
is as in (1.4), where
, then
converges to a fixed point of
.
Next we establish strong convergence of Ishikawa iterates of a generalized nonexpansive map.
Theorem 2.4.
Let and
be as in Theorem 2.3. Let
be a continuous generalied nonexpansive map of
into itself with at least one fixed point. If
is as in (1.3), where
and
then
converges to a fixed point of
.
Proof.
Let be any fixed point of
Then setting
in (*), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ19_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ20_HTML.gif)
Thus is quasi-nonexpansive.
For any we also observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ21_HTML.gif)
If where
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ23_HTML.gif)
Using (2.14) in (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ24_HTML.gif)
Also it is obvious that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ25_HTML.gif)
Combining (2.16) and (2.17), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ26_HTML.gif)
Now inserting (2.15) in (2.18), we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ27_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ28_HTML.gif)
where Thus
satisfies Condition 4 (and hence Condition 3). The result now follows from Theorem 2.2.
Remark 2.5.
In the above theorem, we have assumed that the generalied nonexpansive map has a fixed point. It remains an open questions: what conditions on
, and
in (*) are sufficient to guarantee the existence of a fixed point of
even in the setting of a metric space.
Choose for all
in Theorem 2.4 to get the following Mann type convergence result.
Theorem 2.6.
Let and
be as in Theorem 2.4. If
is as in (1.4), where
then
converges to a fixed point of
.
Proof.
For for all
,
the inequality (2.20) in the proof of Theorem 2.4 becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ29_HTML.gif)
Thus satisfies Condition 2 (and hence Condition 1) and so the result follows from Theorem 2.3.
The analogue of Kannan result in uniformly convex metric space can be deduced from Theorem 2.6 (by taking , and
for all
) as follows.
Theorem 2.7.
Let be a uniformly convex complete metric space with continuous convex structure and let
be its nonempty closed convex subset. Let
be a continuous map of
into itself with at least one fixed point such that
for all
. Then the sequence
where
and
converges to a fixed point of
Next we give sufficient conditions for the existence of fixed point of a -Lipschitz map in terms of the Ishikawa iterates.
Theorem 2.8.
Let be a convex metric space and let
be its nonempty convex subset. Let
be a
-Lipschitz selfmap of
Let
be the sequence as in (1.3), where
and
satisfy (i)
for all
(ii)
and (iii)
If
and
then
is a fixed point of
Proof.
Let Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ30_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ31_HTML.gif)
Since therefore there exists
such that
for all
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ32_HTML.gif)
Taking on both the sides in the above inequality and using the condition
, we have
Finally, using a generalized nonexpansive map on a metric space
, we provide a necessary and sufficient condition for the convergence of an arbitrary sequence
in
to a fixed point of
in terms of the approximating sequence
Theorem 2.9.
Suppose that is a closed subset of a complete metric space
and
is a continuous map such that for some
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ33_HTML.gif)
for all Then a sequence
in
converges to a fixed point of
if and only if
Proof.
Suppose that First we show that
is a Cauchy sequence in
To acheive this goal, consider:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ34_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F385986/MediaObjects/13663_2009_Article_1270_Equ35_HTML.gif)
Since and
therefore from the above inequality, it follows that
is a Cauchy sequence in
In view of closedness of
this sequence converges to an element
of
Also
gives that
Now using the continuity of
we have
Hence
is a fixed point of
Conversely, suppose that converges to a fixed point
of
Using the continuity of
we have that
Thus
Remark 2.10.
Theorem 2.8 improves Lemma 2 in [3] from real line to convex metric space setting. Theorem 2.9 is an extension of Theorem 4 in [21] to metric spaces. If we choose in Theorem 2.9, it is still an improvement of [21, Theorem 4].
Remark 2.11.
We have proved our results (2.1)–(2.8) in convex metric space setting. All these results, in particular, hold in Banach spaces if we set
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Acknowledgment
The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals for support during this research.
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Khan, A., Fukhar-ud-din, H. & Domlo, A. Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces. Fixed Point Theory Appl 2010, 385986 (2010). https://doi.org/10.1155/2010/385986
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DOI: https://doi.org/10.1155/2010/385986