- Research Article
- Open access
- Published:
Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense
Fixed Point Theory and Applications volume 2010, Article number: 186874 (2010)
Abstract
A new nonlinear mapping is introduced. The convergence of Ishikawa iterative processes for the class of asymptotically pseudocontractive mappings in the intermediate sense is studied. Weak convergence theorems are established. A strong convergence theorem is also established without any compact assumption by considering the so-called hybrid projection methods.
1. Introduction and Preliminaries
Throughout this paper, we always assume that is a real Hilbert space, whose inner product and norm are denoted by
and
. The symbols
and
are denoted by strong convergence and weak convergence, respectively.
denotes the weak
-limit set of
. Let
be a nonempty closed and convex subset of
and
a mapping. In this paper, we denote the fixed point set of
by
.
Recall that is said to benonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ1_HTML.gif)
is said to beasymptotically nonexpansive if there exists a sequence
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ2_HTML.gif)
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex and bounded subset of a real uniformly convex Banach space and
is an asymptotically nonexpansive mapping on
, then
has a fixed point.
is said to beasymptotically 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%2F186874/MediaObjects/13663_2009_Article_1220_Equ3_HTML.gif)
Observe that if we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ4_HTML.gif)
then as
. It follows that (1.3) is reduced to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ5_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 close convex 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 bestrictly pseudocontractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ6_HTML.gif)
The class of strict pseudocontractions was introduced by Browder and Petryshyn [4] in a real Hilbert space. Marino and Xu [5] proved that the fixed point set of strict pseudocontractions is closed convex, and they also obtained a weak convergence theorem for strictly pseudocontractive mappings by Mann iterative process; see [5] for more details.
Recall that is said to be aasymptotically strict pseudocontraction if there exist a constant
and a sequence
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ7_HTML.gif)
The class of asymptotically strict pseudocontractions was introduced by Qihou [6] in 1996 (see also [7]). Kim and Xu [8] proved that the fixed point set of asymptotically strict pseudocontractions is closed convex. They also obtained that the class of asymptotically strict pseudocontractions is demiclosed at the origin; see [8, 9] for more details.
Recently, Sahu et al. [10] 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 if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ8_HTML.gif)
where and
such that
as
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ9_HTML.gif)
It follows that as
Then, (1.8) is reduced to the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ10_HTML.gif)
They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by considering the so-called hybrid projection methods; see [10] for more details.
Recall that is said to beasymptotically pseudocontractive if there exists a sequence
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ11_HTML.gif)
The class of asymptotically pseudocontractive mapping was introduced by Schu [11] (see also [12]). In [13], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [13] for more details. In 1991, Schu [11] established the following classical results.
Theorem JS.
Let be a Hilbert space:
closed bounded and covnex;
;
completely continuous, uniformly
-Lipschitzian and asymptotically pseudocontractive with sequence
;
for all
;
;
,
are sequences in
;
for all
, some
and some
;
; for all
, define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ12_HTML.gif)
then converges strongly to some fixed point of
.
Recently, Zhou [14] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point. Moreover, the fixed point set is closed and convex.
In this paper, we introduce and consider the following mapping.
Definition 1.1.
A mapping is said to be aasymptotically pseudocontractive mapping in the intermediate sense if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ13_HTML.gif)
where is a sequence in
such that
as
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ14_HTML.gif)
It follows that as
Then, (1.13) is reduced to the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ15_HTML.gif)
In real Hilbert spaces, we see that (1.15) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ16_HTML.gif)
We remark that if for each
, then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.
In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of mappings which are asymptotically pseudocontractive in the intermediate sense.
In order to prove our main results, we also need the following lemmas.
Lemma 1.2 (see [15]).
Let ,
and
be three nonnegative sequences satisfying the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ17_HTML.gif)
where is some nonnegative integer. If
and
, then
exists.
Lemma 1.3.
In a real Hilbert space, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ18_HTML.gif)
From now on, we always use to denotes
.
Lemma 1.4.
Let be a nonempty close convex subset of a real Hilbert space
and
a uniformly
-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences
and
as defined in (1.15). Then
is a closed convex subset of
.
Proof.
To show that is convex, let
and
. Put
, where
. Next, we show that
Choose
and define
for each
. From the assumption that
is uniformly
-Lipschitz, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ19_HTML.gif)
For any , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ20_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ21_HTML.gif)
Letting and
in (1.21), respectively, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ22_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ23_HTML.gif)
Letting in (1.23), we obtain that
. Since
is uniformly
-Lipschitz, we see that
This completes the proof of the convexity of
. From the continuity of
, we can also obtain the closedness of
. The proof is completed.
Lemma 1.5.
Let be a nonempty close convex subset of a real Hilbert space
and
a uniformly
-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that
is nonempty. Then
is demiclosed at zero.
Proof.
Let be a sequence in
such that
and
as
Next, we show that
and
. Since
is closed and convex, we see that
It is sufficient to show that
Choose
and define
for arbitrary but fixed
From the assumption that
is uniformly
-Lipschitz, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ24_HTML.gif)
It follows from the assumption that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ25_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ26_HTML.gif)
Since and (1.25), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ27_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ28_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ29_HTML.gif)
Substituting (1.27) and (1.28) into (1.29), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ30_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ31_HTML.gif)
Letting in (1.31), we see that
. Since
is uniformly
-Lipschitz, we can obtain that
This completes the proof.
2. Main Results
Theorem 2.1.
Let be a nonempty closed convex bounded subset of a real Hilbert space
and
a uniformly
-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences
and
defined as in (1.15). Assume that
is nonempty. Let
be a sequence generated in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ32_HTML.gif)
where and
are sequences in
. Assume that the following restrictions are satisfied:
(a),
, where
for each
(b) for some
and some
then the sequence generated by (*) converges weakly to fixed point of
.
Proof.
Fix . From (1.16) and Lemma 1.3, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ34_HTML.gif)
From (2.1) and (2.2), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ35_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ36_HTML.gif)
From condition (b), we see that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ37_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ38_HTML.gif)
In view of Lemma 1.2, we see that exists. For any
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ39_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ40_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ41_HTML.gif)
Thanks to (2.8), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ42_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ43_HTML.gif)
From (2.8) and (2.10), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ44_HTML.gif)
Since is bounded, we see that there exists a subsequence
such that
. From Lemma 1.5, we see that
.
Next we prove that converges weakly to
. Suppose the contrary. Then we see that there exists some subsequence
such that
converges weakly to
and
. From Lemma 1.5, we can also prove that
. Put
Since
satisfies Opial property, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ45_HTML.gif)
This derives a contradiction. It follows that . This completes the proof.
Next, we modify Ishikawa iterative processes to obtain a strong convergence theorem without any compact assumption.
Theorem 2.2.
Let be a nonempty closed convex bounded subset of a real Hilbert space
,
the metric projection from
onto
and
a uniformly
-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences
and
as defined in (1.15). Let
for each
Assume that
is nonempty. Let
and
be sequences in
. Let
be a sequence generated in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ46_HTML.gif)
where for each
. Assume that the control sequences
and
are chosen such that
for some
and some
Then the sequence
generated in (**) converges strongly to a fixed point of
.
Proof.
The proof is divided into seven steps.
Step 1.
Show that is closed and convex for each
It is obvious that is closed and convex and
is closed for each
. We, therefore, only need to prove that
is convex for each
. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ47_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ48_HTML.gif)
It is easy to see that is convex for each
. Hence, we obtain that
is closed and convex for each
This completes Step 1.
Step 2.
Show that for each
.
Let . From Lemma 1.3 and the algorithm (**), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ51_HTML.gif)
Substituting (2.17) and (2.18) into (2.16), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ52_HTML.gif)
where for each
. This implies that
for each
. That is,
for each
Next, we show that for each
We prove this by inductions. It is obvious that
. Suppose that
for some
. Since
is the projection of
onto
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ53_HTML.gif)
By the induction assumption, we know that . In particular, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ54_HTML.gif)
which implies that . That is,
. This proves that
for each
. Hence,
for each
. This completes Step 2.
Step 3.
Show that exists.
In view of the algorithm (**), we see that and
which give that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ55_HTML.gif)
This shows that the sequence is nondecreasing. Note that
is bounded. It follows that
exists. This completes Step 3.
Step 4.
Show that as
Note that and
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ56_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ57_HTML.gif)
Hence, we have as
This completes Step 4.
Step 5.
Show that as
In view of , we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ58_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ59_HTML.gif)
Combining (2.25) and (2.26) and noting , we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ60_HTML.gif)
From the assumption, we see that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ61_HTML.gif)
For any , it follows from (2.27) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ62_HTML.gif)
Note that as
Thanks to Step 4, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ63_HTML.gif)
This completes Step 5.
Step 6.
Show that as
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ64_HTML.gif)
From Step 5, we can conclude the desired conclusion. This completes Step 6.
Step 7.
Show that , where
as
Note that Lemma 1.5 ensures that . From
and
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F186874/MediaObjects/13663_2009_Article_1220_Equ65_HTML.gif)
From Lemma of Yanes and Xu [16], we can obtain Step 7. This completes the proof.
Remark 2.3.
The results of Theorem 2.2 are more general which includes the corresponding results of Kim and Xu [17], Marino and Xu [5], Qin et al. [18], Sahu et al. [10], Zhou [14, 19] as special cases.
References
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloquium Mathematicum 1993,65(2):169–179.
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel Journal of Mathematics 1974, 17: 339–346. 10.1007/BF02757136
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055
Liu QH: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Analysis: Theory, Methods & Applications 1996,26(11):1835–1842. 10.1016/0362-546X(94)00351-H
Chang S-S, Huang J, Wang X, Kim JK: Implicit iteration process for common fixed points of strictly asymptotically pseudocontractive mappings in Banach spaces. Fixed Point Theory and Applications 2008, 2008:-12.
Kim T-H, Xu H-K: Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions. Nonlinear Analysis: Theory, Methods & Applications 2008,68(9):2828–2836. 10.1016/j.na.2007.02.029
Qin X, Cho YJ, Kang SM, Shang M: A hybrid iterative scheme for asymptotically -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1902–1911. 10.1016/j.na.2008.02.090
Sahu DR, Xu H-K, Yao J-C: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Analysis: Theory, Methods & Applications 2009,70(10):3502–3511. 10.1016/j.na.2008.07.007
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 1991,158(2):407–413. 10.1016/0022-247X(91)90245-U
Kim JK, Nam YM: Modified Ishikawa iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in Banach spaces. Bulletin of the Korean Mathematical Society 2006,43(4):847–860.
Rhoades BE: Comments on two fixed point iteration methods. Journal of Mathematical Analysis and Applications 1976,56(3):741–750. 10.1016/0022-247X(76)90038-X
Zhou H: Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3140–3145. 10.1016/j.na.2008.04.017
Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309
Yanes CM, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018
Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1140–1152. 10.1016/j.na.2005.05.059
Qin X, Su Y, Shang M: Strong convergence theorems for asymptotically nonexpansive mappings by hybrid methods. Kyungpook Mathematical Journal 2008,48(1):133–142.
Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,343(1):546–556. 10.1016/j.jmaa.2008.01.045
Acknowledgment
This work was supported by the Kyungnam University Research Fund 2009.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Qin, X., Cho, S. & Kim, J. Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense. Fixed Point Theory Appl 2010, 186874 (2010). https://doi.org/10.1155/2010/186874
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/186874