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Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions
Fixed Point Theory and Applications volume 2008, Article number: 824607 (2008)
Abstract
This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems.
1. Introduction
Let be a real Banach space and
a nonempty closed subset of
. A mapping
is said to be pseudocontractive (see, e.g., [1]) if

holds for all .
is said to be strictly pseudocontractive if, for all
, there exists a constant
such that

Denote by the set of fixed points of
. A map
is called hemicontractive if
and for all
,
, the following inequality holds:

It is easy to see that the class of pseudocontractive mappings with fixed points is a subset of the class of hemicontractions.
There are many papers in the literature dealing with the approximation of fixed points for several classes of nonlinear mappings (see, e.g., [1–11], and the reference therein). In these works, there are two iterative methods to be used to find a point in . One is explicit and one is implicit.
The explicit one is the following well-known Mann iteration.
Let be a nonempty closed convex subset of
. For any
, the sequence
is defined by

where is a real sequence in
satisfying some assumptions.
It has been applied to many classes of nonlinear mappings to find a fixed point. However, for hemicontractive mappings and strictly pseudocontractive mappings, the iteration process of convergence is in general not strong (see a counterexample given by Chidume and Mutangadura [3]). Most recently, Marino and Xu [6] proved that the Mann iterative sequence converges weakly to a fixed point for strictly pseudocontractive mappings in a Hilbert space, while the real sequence
satisfying (i)
and (ii)
.
In order to get strong convergence for fixed points of hemicontractive mappings and strictly pseudocontractive mappings, the following Mann-type implicit iteration scheme is introduced.
Letbe a nonempty closed convex subset of
with
. For any
, the sequence
is generated by

whereis a real sequence in
satisfying suitable conditions.
Recently, in the setting of a Hilbert space, Rafiq [12] proved that the Mann-type implicit iterative sequence converges strongly to a fixed point for hemicontractive mappings, under the assumption that the domain
of
is a compact convex subset of a Hilbert space, and
for some
.
In this paper, we will study the strong convergence of the generalized Mann-type iteration scheme (see Definition 2.1) for hemicontractive and, respectively, pseudocontractive mappings. As we will see, our theorems extend the corresponding results in [12] in four aspects. (1) The space setting is a more general one: uniformly convex Banach space, which could not be a Hilbert space. (2) The requirement of the compactness on the domain of the mapping is dropped. (3) A single mapping is replaced by a family of mappings. (4) The Mann-type implicit iteration is replaced by the generalized Mann iteration. Moreover, we give answers to a question asked in [13].
2. Preliminaries and Lemmas
Definition 2.1 (generalized Mann iteration).
Let be a fixed integer,
, and
a nonempty closed convex subset of
satisfying the condition
. Let
be a family of mappings. For each
, the sequence
is defined by

where ,
,
and
are three sequences in
with
and
is bounded.
The modulus of convexity of is the function
defined by

is called uniformly convex if and only if, for all
such that
.
is called
-uniformly convex if there exists a constant
, such that
. It is well known (see [10]) that

Let be a Banach space,
and
. Then, we denote
.
Definition 2.2 (see [4]).
Let be a nondecreasing function with
and
, for all
.
-
(i)
A mapping
with
is said to satisfy condition (A) on
if there is a function
such that for all
,
.
-
(ii)
A finite family of mappings
with
are said to satisfy condition (
) if there exists a function
, such that
holds for all
.
Lemma 2.3 (see [8]).
Let be a real uniformly convex Banach space with the modulus of convexity of power type
. Then, for all
in
and
, there exists a constant
such that

where .
Remark 2.4.
If in the previous lemma, then we denote
.
Lemma 2.5.
Let be a real Banach space and
the normalized duality mapping. Then for any
in
and
, such that

Lemma 2.6 (see [7]).
Let and
be three nonnegative real sequences, satisfying

with and
. Then,
exists. In addition, if
has a subsequence converging to zero, then
.
Proposition 2.7.
If is a strict pseudocontraction, then
satisfies the Lipschitz condition

Proof.
By the definition of the strict pseudocontraction, we have

A simple computation shows the conclusion.
3. Main Results
Lemma 3.1.
Let be a uniformly convex Banach space with the convex modulus of power type
,
a nonempty closed convex subset of
satisfying
, and
hemicontractive mappings with
. Let
,
,
,
and
be the sequences in (II) and

where is the constant in Remark 2.4. Then,
(1) exists for all
,
(2) exists,
-
(3)
if
is continuous, then
, for all
.
Proof. (1) Let . By the boundedness assumption on
, there exists a constant
, for any
, such that
. From the definition of hemicontractive mappings, we have

Using Lemmas 2.3, 2.5, and (3.2), we obtain

Hence,

It follows from (II) and Lemma 2.5 that

By the condition , we may assume that

Therefore,

Substituting (3.7) into (3.4), we get

Assumptions (i) and (ii) imply that there exists a positive integer such that for every
,

Hence, for all ,

where

From (3.9) and conditions (i) and (ii), it follows that

By Lemma 2.6, we see that exists and the sequence
is bounded.
(2) It is easy to verify that exists.
(3) By the boundedness of , there exists a constant
such that
, for all
. From (3.10), we get, for
,

which implies

Thus,

It implies that

Therefore, by (3.7), we have

Using (II), we obtain

By a combination with the continuity of (
, we get

It is clear that for each , there exists
such that
. Consequently,

This completes the proof.
Theorem 3.2.
Let the assumptions of Lemma 3.1 hold, and let be continuous. Then,
converges strongly to a common fixed point of
if and only if
.
Proof.
The necessity is obvious.
Now, we prove the sufficiency. Since , it follows from Lemma 3.1 that
.
For any , we have

Hence, we get

So, is a Cauchy sequence in
. By the closedness of
, we get that the sequence
converges strongly to
. Let a sequence
, for some
, be such that
converges strongly to
. By the continuity of
, we obtain

Therefore, . This implies that
is closed. Therefore,
is closed. By
, we get
. This completes the proof.
Theorem 3.3.
Let the assumptions of Lemma 3.1 hold. Let be continuous and
satisfy condition
. Then,
converges strongly to a common fixed point of
.
Proof.
Since satisfies condition
, and
for each
, it follows from the existence of
that
. Applying the similar arguments as in the proof of Theorem 3.2, we conclude that
converges strongly to a common fixed point of
. This completes the proof.
As a direct consequence of Theorem 3.3, we get the following result.
Corollary 3.4 (see [12, Theorem 3]).
Let H be a real Hilbert space, a nonempty closed convex subset of
satisfying
, and
continuous hemicontractive mapping which satisfies condition (A). Let
be a real sequence in
with
. For any
, the sequence
is defined by

Then, converges strongly to a fixed point of
.
Proof.
Employing the similar proof method of Lemma 3.1, we obtain by (3.10)

This implies

By , we have
. Equation (3.7) implies that
. Since
satisfies condition (A) and the limit
exists, we get
. The rest of the proof follows now directly from Theorem 3.2. This completes the proof.
Remark 3.5.
Theorems 3.2 and 3.3 extend [12, Theorem 3] essentially since the following hold.
-
(i)
Hilbert spaces are extended to uniformly convex Banach spaces.
-
(ii)
The requirement of compactness on domain
on [12, Theorem 3] is dropped.
-
(iii)
A single mapping is replaced by a family of mappings.
-
(iv)
The Mann-type implicit iteration is replaced by the generalized Mann iteration. So the restrictions of
with
for some
are relaxed to
. The error term is also considered in the iteration (II).
Moreover, if , then
is well defined by (II). Hence, Theorems 3.2 and 3.3 are also answers to the question proposed by Qing [13].
Theorem 3.6.
Let and
be as the assumptions of Lemma 3.1. Let
be strictly pseudocontractive mappings with
being nonempty. Let
,
,
,
, and
be the sequences in (II) and

where is the constant in Remark 2.4. Then,
(1) converges strongly to a common fixed point of
if and only if
.
-
(2)
If
satisfies condition (
) , then
converges strongly to a common fixed point of
.
Remark 3.7.
Theorem 3.6 extends the corresponding result [6, Theorem 3.1].
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Acknowledgments
The authors would like to thank the referees very much for helpful comments and suggestions. The work was supported partly by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.
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Hu, LG., Xiao, TJ. & Liang, J. Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions. Fixed Point Theory Appl 2008, 824607 (2008). https://doi.org/10.1155/2008/824607
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DOI: https://doi.org/10.1155/2008/824607
Keywords
- Strong Convergence
- Real Banach Space
- Common Fixed Point
- Nonempty Closed Convex Subset
- Real Sequence