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.
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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
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
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
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:
where 
and 
are control sequences in 
If we choose 
then (1.3) reduces to the following Mann iteration scheme:
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
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
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:
Let
be 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
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
This implies that the sequence 
is nonincreasing and bounded below. Thus 
exists. We may assume that
For any 
, we have that
Since
exists, so 
is bounded and hence 
exists. We show that 
. Assume that 
Then
Hence by Theorem 1.3, there exists 
such that
That is,
Taking 
and summing up the 
terms on the both sides in the above inequality, we have
Let 
. Then, we have
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
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
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
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
which implies
Thus 
is quasi-nonexpansive.
For any 
we also observe that
If 
where 
then
Using (2.14) in (2.13), we have
Also it is obvious that
Combining (2.16) and (2.17), we get that
Now inserting (2.15) in (2.18), we derive
That is,
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
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
That is,
Since 
therefore there exists 
such that 
for all 
This implies that
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:
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:
That is,
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