We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.
1. Introduction and Preliminaries
Throughout this paper, we assume that 
is a real Banach space with the normalized duality mapping 
from 
into 
give by
(11)where 
denotes the dual space of 
and 
denotes the generalized duality pairing. We assume that 
is a nonempty closed convex subset of 
and 
a mapping. A point 
is a fixed point of 
provided 
. Denote by 
the set of fixed points of 
, that is, 
. Recall that 
is nonexpansive if 
for all 
One classical way to study nonexpansive mappings is to use contractions to approximate
a nonexpansive mapping (see [1, 2]). More precisely, take 
and define a contraction 
by
(12)where 
is a fixed point. Banach's contraction mapping principle guarantees that 
has a unique fixed point 
in 
. It is unclear, in general, what is the behavior of 
as 
even if 
has a fixed point. However, in the case of 
having a fixed point, Browder [1] proved that if 
is a Hilbert space, then 
converges strongly to a fixed point of 
that is nearest to 
. Reich [2] extended Broweder's result to the setting of Banach spaces and proved that if 
is a uniformly smooth Banach space, then 
converges strongly to a fixed point of 
and the limit defines the (unique) sunny nonexpansive retraction from 
onto 
.
Recall that the normal Mann iterative process was introduced by Mann [3] in 1953. The normal Mann iterative process generates a sequence 
in the following manner:
(13)where the sequence 
is in the interval (0,1). If 
is a nonexpansive mapping with a fixed point and the control sequence 
is chosen so that 
then the sequence 
generated by normal Mann's iterative process (1.3) converges weakly to a fixed point
of 
(this is also valid in a uniformly convex Banach space with the Fréchet differentiable
norm [4]). In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm
has only weak convergence, in general, even for nonexpansive mappings. Therefore,
many authors try to modify normal Mann's iteration process to have strong convergence
for nonexpansive mappings (see, e.g., [5–8] and the references therein).
Recently, Kim and Xu [5] introduced the following iteration process:
(14)where 
is a nonexpansive mapping of 
into itself and 
is a given point. They proved that the sequence 
defined by (1.4) converges strongly to a fixed point of 
provided the control sequences 
and 
satisfy appropriate conditions.
Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, [9]. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see, e.g., [10]).
In this paper, we consider the mapping 
defined by
(15)where 
are sequences in 
. Such a mapping 
is called the 
-mapping generated by 
and 
. Nonexpansivity of each 
ensures the nonexpansivity of 
. Moreover, in [11], it is shown that 
Motivated by Atsushiba and Takahashi [11], Kim and Xu [5], and Shang et al. [7], we study the following iterative algorithm:
(16)where 
is defined by (1.5) and 
is given point. We prove, under certain appropriate assumptions on the sequences
and 
, that 
defined by (1.6) converges to a common fixed point of the finite family nonexpansive
mappings without any commutative assumptions.
In order to prove our main results, we need the following definitions and lemmas.
Recall that if 
and 
are nonempty subsets of a Banach space 
such that 
is nonempty closed convex and 
, then a map 
is sunny (see [12, 13]) provided 
for all 
and 
whenever 
A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive.
Sunny nonexpansive retractions play an important role in our argument. They are characterized
as follows [12, 13]: if 
is a smooth Banach space, then 
is a sunny nonexpansive retraction if and only if there holds the inequality 
for all 
and 
Reich [2] showed that if 
is uniformly smooth and 
is the fixed point set of a nonexpansive mapping from 
into itself, then there is a sunny nonexpansive retraction from 
onto 
and it can be constructed as follows.
Lemma 1.1.
Let 
be a uniformly smooth Banach space and let 
be a nonexpansive mapping with a fixed point. For each fixed 
and 
, the unique fixed point 
of the contraction 
converges strongly as 
to a fixed point of 
. Define 
by 
. Then 
is the unique sunny nonexpansive retract from 
onto 
, that is, 
satisfies the property 
for all
and 
Lemma 1.2 (See [14]).
Let 
and 
be bounded sequences in a Banach space 
and let 
be a sequence in [0,1] with 
. Suppose 
for all integers 
and 
Then 
Lemma 1.3.
In a Banach space 
, there holds the inequality 
for all 
where 
.
Lemma 1.4 (See [15]).
Assume that 
is a sequence of nonnegative real numbers such that 
where 
is a sequence in (0,1) and 
is a sequence such that 
and 
or 
Then 
2. Main Results
Theorem 2.1.
Let 
be a closed convex subset of a uniformly smooth and strictly convex Banach space
. Let 
be a nonexpansive mapping from 
into itself for 
. Assume that 
. Given a point 
and given sequences 
and 
in (0,1), the following conditions are satisfied:
(i)
(ii)
(iii)
Let 
be the composite process defined by (1.6). Then 
converges strongly to 
, where 
and 
is the unique sunny nonexpansive retraction from 
onto 
.
Proof.
We divide the proof into four parts.
Step 1.
First we observe that sequences 
and 
are bounded.
Indeed, take a point 
and notice that
(21)It follows that
(22)By simple inductions, we have 
which gives that the sequence 
is bounded, so is 
.
Step 2.
In this part, we will claim that 
as 
Put 
. Now, we compute 
that is,
(23)Observing that
(24)we have
(25)From the proof of Yao [8], we have
(26)where 
is an appropriate constant. Substituting (2.6) into (2.5), we have
(27)Observing the conditions (i)–(iii), we get 
We can obtain 
easily by Lemma 1.2. Observe that (2.3) yields 
Therefore, we have
(28)Step 3.
We will prove 
.
Observing that 
and the condition (i), we can easily get
(29)On the other hand, we have 
Combining (2.8) with (2.9), we have
(210)Notice that 
This implies 
From the condition (iii) and (2.10), we obtain
(211)Step 4.
Finally, we will show 
as 
.
First, we claim that
(212)where 
with 
being the fixed point of the contraction 
Then 
solves the fixed point equation 
Thus we have
(213)It follows from Lemma 1.3 that
(214)where
(215)It follows from (2.14) that
(216)Letting 
in (2.16) and noting (2.15) yield
(217)where 
is an appropriate constant. Taking 
in (2.17), we have
(218)On the other hand, we have
(219)It follows that
(220)Noticing that 
is norm-to-norm uniformly continuous on bounded subsets of 
and from (2.18), we have 
It follows that
(221)Hence, (2.12) holds. Now, from Lemma 1.3, we have
(222)Applying Lemma 1.4 to (2.22) we have 
as 
Remark 2.2.
Theorem 2.1 improves the results of Kim and Xu [5] from a single nonexpansive mapping to a finite family of nonexpansive mappings.
Remark 2.3.
If 
is a contraction map and we replace 
by 
in the recursion formula (1.6), we obtain what some authors now call viscosity iteration
method. We note that our theorem in this paper carries over trivially to the so-called
viscosity process. Therefore, our results also include Yao et al. [16] as a special case.
Remark 2.4.
Our results partially improve Shang et al. [7] from a Hilbert space to a Banach space.
Remark 2.5.
If 
is a single nonexpansive mapping, then the strict convexity of 
may not be needed.
Acknowledgment
This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-313-C00040).
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