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Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense
Fixed Point Theory and Applications volume 2010, Article number: 281070 (2010)
Abstract
We study the convergence of Ishikawa iteration process for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.
1. Introduction and Preliminaries
Throughout this paper, we always assume that is a real Hilbert space with inner product
and norm
and
denote weak and strong convergence, respectively.
denotes the weak
-limit set of
, that is,
. Let
be a nonempty closed convex subset of
. It is well known that for every point
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ1_HTML.gif)
for all .
is called the metric projection of
onto
.
is a nonexpansive mapping of
onto
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ2_HTML.gif)
Let be a mapping. In this paper, we denote the fixed point set of
by
. Recall that
is said to be uniformly
-Lipschitzian if there exists a constant
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ3_HTML.gif)
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ4_HTML.gif)
is said to be asymptotically nonexpansive if there exists a sequence
in
with
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ5_HTML.gif)
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ6_HTML.gif)
Observe that if we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ7_HTML.gif)
then as
. It follows that (1.6) is reduced to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ8_HTML.gif)
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if is a nonempty closed convex bounded subset of a uniformly convex Banach space
and
is asymptotically nonexpansive in the intermediate sense, then
has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.
Recall that is said to be a
-strict pseudocontraction if there exists a constant
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ9_HTML.gif)
is said to be an asymptotically
-strict pseudocontraction with sequence
if there exist a constant
and a sequence
with
as
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ10_HTML.gif)
The class of asymptotically -strict pseudocontractions was introduced by Qihou [4] in 1996 (see also [5]). Kim and Xu [6] studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically
-strict pseudocontractive mapping with sequence
is a uniformly
-Lipschitzian mapping with
.
Recently, Sahu et al. [7] introduced a class of new mappings: asymptotically -strict pseudocontractive mappings in the intermediate sense. Recall that
is said to be an asymptotically
-strict pseudocontraction in the intermediate sense with sequence
if there exist a constant
and a sequence
with
as
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ11_HTML.gif)
Throughout this paper, we assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ12_HTML.gif)
It follows that as
and (1.11) is reduced to the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ13_HTML.gif)
They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see [7] for more details.
In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense.
In order to prove our main results, we also need the following lemmas.
Let ,
, and
be three sequences of nonnegative numbers satisfying the recursive inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ14_HTML.gif)
If ,
and
, then
exists.
Lemma 1.2 (see [10]).
Let be a bounded sequence in a reflexive Banach space
. If
, then
.
Lemma 1.3 (see [11]).
Let be a nonempty closed convex subset of a real Hilbert space
. Given
and
, then
if and only if
, for all
Lemma 1.4 (see [11]).
For a real Hilbert space , the following identities hold:
(i),  for all
,
(ii) for all
, for all
;
(iii)(Opial condition) If is a sequence in
weakly convergent to
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ15_HTML.gif)
Lemma 1.5 (see [7]).
Let be a nonempty subset of a Hilbert space
and
an asymptotically
-strict pseudocontractive mapping in the intermediate sense with sequence
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ16_HTML.gif)
Lemma 1.6.
Let be a nonempty subset of a Hilbert space
and
an asymptotically
-strict pseudocontractive mapping in the intermediate sense with sequence
. Let
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ17_HTML.gif)
Proof.
If , for
, we obtain from Lemma 1.5 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ18_HTML.gif)
Lemma 1.7 (see [7]).
Let be a nonempty subset of a Hilbert space
and
a uniformly continuous asymptotically
-strict pseudocontractive mapping in the intermediate sense with sequence
. Let
be a sequence in
such that
and
as
, then
as
.
Lemma 1.8 (see [7, Proposition ]).
Let be a nonempty closed convex subset of a Hilbert space
and
a continuous asymptotically
-strict pseudocontractive mapping in the intermediate sense. Then
is demiclosed at zero in the sense that if
is a sequence in
such that
and
, then
.
Lemma 1.9 (see [7]).
Let be a nonempty closed convex subset of a Hilbert space
and
a continuous asymptotically
-strict pseudocontractive mapping in the intermediate sense. Then
is closed and convex.
2. Main Results
Theorem 2.1.
Let be a nonempty closed convex subset of a Hilbert space
and
a uniformly continuous asymptotically
-strict pseudocontractive mapping in the intermediate sense with sequence
such that
. Let
be a sequence in
generated by the following Ishikawa iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ19_HTML.gif)
where and
are sequences in
. Assume that the following restrictions are satisfied:
(i) and
,
(ii) for some
and
.
Then the sequence given by (2.1) converges weakly to an element of
.
Proof.
Let . From (1.13) and Lemma 1.4, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ20_HTML.gif)
Without loss of generality, we may assume that for all
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ21_HTML.gif)
it follows from Lemma 1.6 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ22_HTML.gif)
By (2.2) and (2.4), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ23_HTML.gif)
where . It follows from (2.5) and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ24_HTML.gif)
From the condition (ii) and , we see that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ25_HTML.gif)
By (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ26_HTML.gif)
In view of Lemma 1.1 and the condition (i), we obtain that exists. For any
, it is easy to see from (2.6) and (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ27_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ28_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ29_HTML.gif)
From (2.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ30_HTML.gif)
Since is uniformly continuous, we obtain from (2.10), (2.12) and Lemma 1.7 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ31_HTML.gif)
By the boundedness of , there exist a subsequence
of
such that
. Observe that
is uniformly continuous and
as
, for any
we have
as
. From Lemma 1.8, we see that
.
To complete the proof, it suffices to show that consists of exactly one point, namely,
. Suppose there exists another subsequence
of
such that
converges weakly to some
and
. As in the case of
, we can also see that
. It follows that
and
exist. Since
satisfies the Opial condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ32_HTML.gif)
which is a contradiction. We see and hence
is a singleton. Thus,
converges weakly to
by Lemma 1.2.
Corollary 2.2.
Let be a nonempty closed convex subset of a Hilbert space
and
a uniformly continuous asymptotically
-strict pseudocontractive mapping with sequence
such that
. Let
be a sequence in
generated by the following Ishikawa iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ33_HTML.gif)
where and
are sequences in
. Assume that the following restrictions are satisfied:
(i),
(ii) for some
and
.
Then the sequence given by (2.15) converges weakly to an element of
.
Next, we modify Ishikawa iterative process to get a strong convergence theorem.
Theorem 2.3.
Let be a nonempty closed convex subset of a Hilbert space
and
a uniformly continuous asymptotically
-strict pseudocontractive mapping in the intermediate sense with sequence
such that
and bounded. Let
and
are sequences in
. Let
be a sequence in
generated by the modified Ishikawa iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ34_HTML.gif)
where ,
,
and
for each
. Assume that the control sequences
and
are chosen such that
for some
and
. Then the sequence
given by (2.16) converges strongly to an element of
.
Proof.
We break the proof into six steps.
Step 1 ( is closed and convex for each
).
It is obvious that is closed and convex and
is closed for each
. Note that the defining inequality in
is equivalent to the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ35_HTML.gif)
it is easy to see that is convex for each
. Hence,
is closed and convex for each
.
Step 2 ( for each
).
Let . Following (2.6), (2.7) and the algorithm (2.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ36_HTML.gif)
where ,
,
and
for each
. Hence
for each
.
Next, we show that for each
. We prove this by induction. For
, we have
. Assume that
for some
. Since
is the projection of
onto
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ37_HTML.gif)
By the induction consumption, we know that . In particular, for any
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ38_HTML.gif)
This implies that . That is,
. By the principle of mathematical induction, we get
and hence
for all
. This means that the iteration algorithm (2.16) is well defined.
Step 3 ( exists and
is bounded).
In view of (2.16), we see that and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ39_HTML.gif)
for each . We, therefore, obtain that the sequence
is nondecreasing. Noticing that
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ40_HTML.gif)
This shows that the sequence is bounded. Therefore, the limit of
exists and
is bounded.
Step 4 ().
Observe that and
which imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ41_HTML.gif)
Using Lemma 1.4, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ42_HTML.gif)
Hence, we obtain that as
.
Step 5 ( as
).
In view of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ43_HTML.gif)
On the other hand, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ44_HTML.gif)
Combing (2.25) and (2.26) and noting , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ45_HTML.gif)
From the assumption and (2.7), we see that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ46_HTML.gif)
For any , it follows from the definition of
and (2.27) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ47_HTML.gif)
Noting that as
and Step 4, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ48_HTML.gif)
It follows from Step 4, (2.30) and Lemma 1.7 that as
.
Step 6 ( as
, where
).
Since is reflexive and
is bounded, we get that
is nonempty. First, we show that
is a singleton. Assume that
is subsequence of
such that
. Observe that
is uniformly continuous and
as
, for any
we have
as
. From Lemma 1.8, we see that
.
Since , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ49_HTML.gif)
for each . Observe that
as
. By the weak lower semicontinuity of norm, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ50_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F281070/MediaObjects/13663_2010_Article_1246_Equ52_HTML.gif)
Hence by the uniqueness of the nearest point projection of
onto
. Since
is an arbitrary weakly convergent subsequence, it follows that
and hence
. It is easy to see as (2.34) that
. Since
has the Kadec-Klee property, we obtain that
, that is,
as
. This completes the proof.
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Acknowledgments
This research is supported by Fundamental Research Funds for the Central Universities (ZXH2009D021) and supported by the Science Research Foundation Program in Civil Aviation University of China (no. 09CAUC-S05) as well.
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Zhao, J., He, S. Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense. Fixed Point Theory Appl 2010, 281070 (2010). https://doi.org/10.1155/2010/281070
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DOI: https://doi.org/10.1155/2010/281070