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A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2011, Article number: 859795 (2011)
Abstract
The modified Ishikawa iterative process is investigated for the class of total asymptotically pseudocontractive mappings. A weak convergence theorem of fixed points is established in the framework of Hilbert spaces.
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
.
and
are denoted by strong convergence and weak convergence, respectively. Let
be a nonempty closed convex subset of
and
a mapping. In this paper, we denote the fixed point set of
by
.
is said to be a contraction if there exists a constant
such that

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.
is said to be a weak contraction if

where is a continuous and nondecreasing function such that
is positive on
,
, and
. We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.
is said to be nonexpansive if

is said to be asymptotically nonexpansive if there exists a sequence
with
as
such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex bounded subset of a real uniformly convex Banach space and
is an asymptotically nonexpansive mapping on
, then
has a fixed point.
is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

Observe that if we define

then as
. It follows that (1.5) is reduced to

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [4] (see also [5]). It is known [6] 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 may not be Lipschitz continuous; see [5, 7].
is said to be total asymptotically nonexpansive if

where is a continuous and strictly increasing function with
and
and
are nonnegative real sequences such that
and
as
. The class of mapping was introduced by Alber et al. [8]. From the definition, we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings and the class of asymptotically nonexpansive mappings in the intermediate sense as special cases; see [9, 10] for more details.
is said to be strictly pseudocontractive if there exists a constant
such that

The class of strict pseudocontractions was introduced by Browder and Petryshyn [11] in a real Hilbert space. In 2007, Marino and Xu [12] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings; see [12] for more details.
is said to be an asymptotically strict pseudocontraction if there exist a constant
and a sequence
with
as
such that

The class of asymptotically strict pseudocontractions was introduced by Qihou [13] in 1996. Kim and Xu [14] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [14] for more details.
is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a constant
and a sequence
with
as
such that

Put

It follows that as
. Then, (1.11) is reduced to the following:

The class of mappings was introduced by Sahu et al. [15]. They proved that the class of asymptotically strict pseudocontractions in the intermediate sense is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [15] for more details.
is said to be asymptotically pseudocontractive if there exists a sequence
with
as
such that

It is not hard to see that (1.14) is equivalent to

The class of asymptotically pseudocontractive mapping was introduced by Schu [16] (see also [17]). In [18], Rhoades gave an example to showed that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [18] for more details. Zhou [19] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point.
is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence
with
as
such

Put

It follows that as
. Then, (1.16) is reduced to the following:

It is easy to see that (1.18) is equivalent to

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [20]. Weak convergence theorems of fixed points were established based on iterative methods; see [20] for more details.
In this paper, we introduce the following mapping.
Definition 1.1.
Recall that is said to be total asymptotically pseudocontractive if there exist sequences
and
with
and
as
such that

where is a continuous and strictly increasing function with
.
It is easy to see that (1.20) is equivalent to the following:

Remark 1.2.
If , then (1.20) is reduced to

Remark 1.3.
Put

If , then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.
Recall that the modified Ishikawa iterative process which was introduced by Schu [16] generates a sequence in the following manner:

where is a mapping,
is an initial value, and
and
are real sequences in
.
If for each
, then the modified Ishikawa iterative process (1.24) is reduced to the following modified Mann iterative process:

The purpose of this paper is to consider total asymptotically pseudocontractive mappings based on the modified Ishikawa iterative process. Weak convergence theorems are established in real Hilbert spaces.
In order to prove our main results, we also need the following lemmas.
Lemma 1.4.
In a real Hilbert space, the following inequality holds:

Lemma 1.5 (see [21]).
Let ,
, and
be three nonnegative sequences satisfying the following condition:

where is some nonnegative integer. If
and
, then
exists.
2. Main Results
Now, we are ready to give our main results.
Theorem 2.1.
Let be a nonempty closed convex subset of a real Hilbert space
and
a uniformly
-Lipschitz and total asymptotically pseudocontractive mapping as defined in (1.20). Assume that
is nonempty and there exist positive constants
and
such that
for all
. Let
be a sequence generated in the following manner:

where and
are sequences in
. Assume that the following restrictions are satisfied:
(a) and
,
(b) for some
and some
.
Then, the sequence generated in (2.1) converges weakly to fixed point of
.
Proof.
Fix . Since
is an increasing function, it results that
if
and
if
. In either case, we can obtain that

In view of Lemma 1.4, we see from (2.2) that

where for each
. Notice from Lemma 1.4 that

Since is an increasing function, it results that
if
and
if
. In either case, we can obtain that

This implies from (2.3) and (2.4) that

where for each
. It follows that

From the restriction (b), we see that there exists such that

It follows from (2.7) that

Notice that and
. In view of Lemma 1.5, we see that
exists. For any
, we see that

from which it follows that

Note that

In view of (2.11), we obtain that

Note that

Combining (2.11) and (2.13) yields that

Since is bounded, we see that there exists a subsequence
such that
. Next, we claim that
. Choose
and define
for arbitrary but fixed
. From the assumption that
is uniformly
-Lipschitz, we see that

It follows from (2.15) that

Since is an increasing function, it results that
if
and
if
. In either case, we can obtain that

This in turn implies that

Since , we see from (2.17) that

On the other hand, we have

Note that

Substituting (2.20) and (2.21) into (2.22), we arrive at

This implies that

Letting in (2.24), we see that
. Since
is uniformly
-Lipschitz, we can obtain that
.
Next, we prove that converges weakly to
. Suppose the contrary. Then, we see that there exists some subsequence
such that
converges weakly to
, where
. It is not hard to see that that
. Put
. Since
enjoys Opial property, we see that

This derives a contradiction. It follows that . This completes the proof.
Remark 2.2.
Demiclosedness principle of the class of total asymptotically pseudocontractive mappings can be deduced from Theorem 2.1.
Remark 2.3.
Since the class of total asymptotically pseudocontractive mappings includes the class of strict pseudocontractions, the class of asymptotically strict pseudocontractions, the class of pseudocontractive mappings, the class of asymptotically pseudocontractive mappings and the class of asymptotically pseudocontractive mappings in the intermediate sense as special cases, Theorem 2.1 improves the corresponding results in Marino and Xu [12], Kim and Xu [14], Sahu et al. [15], Schu [16], Zhou [19], and Qin et al. [20].
Remark 2.4.
It is of interest to improve the main results of this paper to a Banach space.
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The authors thank the referees for useful comments and suggestions.
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Qin, X., Cho, S. & Kang, S. A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2011, 859795 (2011). https://doi.org/10.1155/2011/859795
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DOI: https://doi.org/10.1155/2011/859795
Keywords
- Hilbert Space
- Banach Space
- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space