Common Fixed Point Theorems for Totally Quasi-G-Asymptotically Nonexpansive Semigroups with the Generalized f-Projection ()
Keywords:Totally Quasi-G-Asymptotically Nonexpansive Semigroup; Generalized f-Projection Operator; Modified Halpern Type Hybrid Iterative Algorithm; Strong Convergence Theorem
1. Introduction
In this paper, we denote by 
and 
the set of real number and the set of nature number respectively. Let 
be a real Banach space with its dual 
and 
be a nonempty, closed and convex subset of
. The mapping 
is the normalized duality mapping, defined by
Recall that a mapping 
is said to be 
[1,2], if for each
,
A mapping 
is said to be 
, if there exists nonnegative real sequences 
and 
with 
as 
and a strictly increasing continuous function
with
, such that for each
,
We use 
to denote the Lyapunov function defined by
Obviously, we have
Recently, Chang et al. [3-5] and Li [6] introduced the uniformly totally quasi-
-asymptotically nonexpansive mappings and studied the strong convergence of some iterative methods for the mappings in Banach space.
Definition 1.1 [1] A countable family of mapping 
is said to be uniformly totally quasi-
-asymptotically nonexpansive, if
, and there exist nonnegative sequences
, 
with 
(as
) and a strictly increasing continuous function 
with
, such that for each
, and each
,,
(1)
More recently, Wang et al. [7] studied the strong convergence for a countable family of total quasi-
- asymptotically nonexpansive mappings by using the hybrid algorithm in 2-uniformly convex and uniformly smooth real Banach spaces. Quan et al. [8] introduced total quasi-
-asymptotically nonexpansive semigroup containing many kinds of generalized nonexpansive mappings as its special cases and used the modified Halpern-Mann iteration algorithm to prove strong convergence theorems in Banach spaces.
We use 
to denote the common fixed point set of the semigroup
, i.e.
.
Definition 1.2 [8] One-parameter family 
is said to be a quasi-
-asymptotically nonexpansive semigroup, if 
and the following conditions are satisfied:
(a) 
for each
;
(b) For each
, 
,
;
(c) For each
, the mapping 
is continuous;
(d) For each
, 
, there exists a sequences 
with 
as
, such that
(2)
One-parameter family 
is said to be a totally quasi-
-asymptotically nonexpansive semigroup, if
, the conditions (a)-(c) and the following condition are satisfied:
(e) If
, there exist sequences
, 
with 
as 
and a strictly increasing continuous function 
with
, such that
(3)
for all
,
.
On the other hand, Wu et al. [9] introduced the generalized f-projection which extends the generalized projection and always exists in a real reflexive Banach space. Li et al. [10] proved some properties of the generalized f-projection operator and studied the strong convergence theorems for the relatively nonexpansive mappings.
In 2013, by using the generalized f-projection operator, Seawan et al. [11] introduced the modified Mann type hybrid projection algorithm for a countable family of totally quasi-
-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property.
Motivated by the above researches, in this paper, we introduce a new class of the totally quasi-G-asymptotically nonexpansive mappings which contains the class of the totally quasi-
-asymptotically nonexpansive mappings and we extend from a countable family of mappings to the totally quasi-G-asymptotically nonexpansive semigroup. Then we modify the Halpern type hybrid projection algorithm by using the generalized f-projection operator for uniformly total quasi-G-asymptotically nonexpansive semigroup and prove some strong convergence theorems under some suitable conditions. The results presented in this paper extend and improve some corresponding ones by many others, such as [1,2,7,8,10,11].
2. Preliminaries
This section contains some definitions and lemmas which will be used in the proofs of our main results in the next section.
Throughout this paper, we assume that 
be a real Banach space with its dual space
. A Banach space
is said to be strictly convex, if 
for all 
with 
and
. 
is said to be uniformly convex, if 
for any two sequences
, 
in 
with 
and
. A Banach space 
is said to be smooth, if 
exists for each
with
. 
is said to be uniformly smooth, if the limit is attainted uniformly for each
.
It is well known that the normalized dual mapping 
holds the properties:
(1) If 
is a smooth Banach space, then 
is single-valued and semi-continuous;
(2) If 
is uniformly smooth Banach space, then 
is uniformly norm-to-norm continuous operator on each bounded subset of
.
A Banach space 
is said to have Kadec-Klee property, if for any sequence 
satisfies 
and
, then
. As we all know, if 
is uniformly convex, then 
has the Kadec-Klee property.
Now, we give a functional
, defined by
(4)
where
, 
, 
is a positive real number and 
is proper, convex and lower semi-continuous. From the definition of 
and
, it is easy to see the following properties:
(1) 
is convex and continuous with respect to 
when 
is fixed;
(2) 
is convex and lower semi-continuous with respect to 
when 
is fixed.
Definition 2.1 [9] 
is said to be a generalized f-projection operator, if for any
,
(5)
Lemma 2.2 [9] Let 
be a real reflexive Banach space with its dual
, 
be a nonempty closed and convex subset of
. Then 
is a nonempty closed and convex subset of 
for all
. Moreoverif 
is strictly convex, then 
is a single-valued mapping.
Recall that if 
is a smooth Banach space, then the normalized dual mapping 
is single-valued, i.e. there exists unique 
such that 
for each
. Then (4) is equivalent to
(6)
And in a smooth Banach space, the definition of the generalized f-projection operator transforms into:
Definition 2.3 [10] Let 
be a real smooth Banach space and 
be a nonempty, closed and convex subset of
. The mapping 
is called generalized f-projection operator, if for all
,
(7)
Now, we give the definition of the totally quasi-
-asymptotically nonexpansive mapping and the totally quasi-
-asymptotically nonexpansive semigroup.
Definition 2.4 A mapping 
is said to be a quasi-G-asymptotically nonexpansive, if 
and there exists a sequence 
with 
(as
), such that
(8)
for any 
and
.
A mapping 
is said to be a totally quasi-G-asymptotically nonexpansive, if 
and there exist sequences
,
with 
as 
and a strictly increasing continuous function 
with
, such that
(9)
for all 
and
.
Remark 2.5 It is easy to see that a quasi-
-asymptotically nonexpansive mapping is a quasi-G-asymptotically nonexpansive mapping with 
for all
. A totally quasi-
-asymptotically nonexpansive mapping is a totally quasi-G-asymptotically nonexpansive mapping with
. Therefore, our totally quasi-G-asymptotically nonexpansive mappings here are more widely than the totally quasi-
- asymptotically nonexpansive mappings which contain many kinds of generalized nonexpansive mappings as their special cases.
Definition 2.6 One-parameter family 
is said to be a quasi-G-asymptotically nonexpansive semigroup on
, if the conditions (a)-(c) in Definition 1.2 and the following condition are satisfied:
(f) There exists a sequence 
with 
as 
such that
(10)
holds for all
,
.
One-parameter family 
is said to be a totally quasi-G-asymptotically nonexpansive semigroup on
, if the above conditions (a)-(c) in Definition 1.2 and the following condition are satisfied:
(g) if 
and there exist sequences
,
with 
as 
and a strictly increasing continuous function 
with 
such that for all 
and
,
(11)
holds for each
.
Remark 2.7 It is easy to see that a quasi-
-asymptotically nonexpansive semigroup is a quasi-G-asymptotically nonexpansive semigroup with 
for all
. A totally quasi-
-asymptotically nonexpansive semigroup is a totally quasi-G-asymptotically nonexpansive semigroup with
.
When we use 
instead of 
in Definition 2.6 and denote 
by
, 
then a quasi-G-asymptotically nonexpansive semigroup becomes a countable family of total quasi-G-asymptotically nonexpansive mappings which contains a countable family of total quasi-
-asymptotically nonexpansive mappings (see [3,4,7]) as it’s special case. So our totally quasi-G-asymptotically nonexpansive semigroup here is the most widely family of the nonexpansive mappings so far.
The following Lemmas are necessary for proving the main results in this paper.
Lemma 2.8 [12] Let 
be a uniformly convex and smooth Banach space, and
, 
be two sequences of
. If 
and either 
or 
is bounded, then
.
Lemma 2.9 [13] If 
is a strictly convex, reflexive and smooth Banach space, then for
, 
if and only if
.
Lemma 2.10 [14] Let 
be a real Banach space and 
be a lower semicontinuous convex functional. Then there exists 
and 
such that
(12)
for each
.
Lemma 2.11 [10] Let 
be a real reflexive and smooth Banach space and 
be a nonempty, closed and convex subset of
. Let
,
. Then
(13)
Lemma 2.12 Let 
be a uniformly smooth and strictly convex Banach space, 
be a nonempty closed and convex subset of
. Let 
be a totally quasi-G-asymptotically nonexpansive mapping defined by (9). If
, then the fixed point set 
of 
is closed and convex subset of
.
Proof Let 
be a sequence in 
with 
as
, we prove that
. In fact, since 
is a quasi-G-asymptotically nonexpansive mapping, we have
Since
, it is equivalent to that
So,
By lemma 2.8, we have that 
which implies that 
is closed. Next we prove that 
is convex, i.e. for any
, 
, we prove that
. In fact,
(14)
(15)
Submitting (15) into (14), we have
This implies that 
and
. Hence we have
, i.e.
. This completes the proof of Lemma 2.12.
3. Main Results
Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space and 
be a nonempty closed and convex subset of E. Let 
be a convex and lower semicontinuous function with 
such that 
for all 
and
. Let 
be a closed and totally quasi-G-asymptotically nonexpansive semigroup defined by Definition 2.6. Assume that 
is uniformly asymptotically regular for all 
and
. Let the sequence 
be defined by
(16)
where 
and the sequence
. If 
and
then 
converges strongly to
.
Proof We divide the proof into five steps.
Step 1. Firstly, we prove that 
and 
are closed and convex subsets in
.
Since 
is a totally quasi-G-asymptotically nonexpansive mapping, it follows the Lemma 2.12 that 
is a closed and convex subset of
. So 
is closed and convex subset of
.
Again, by the assumption, 
is closed and convex. Suppose that 
is the closed and convex subset of 
for
. In view of the definition of
, we have that
This shows that 
is closed and convex for all
.
Step 2. Next, we prove that
.
In fact,
. Suppose that
, for some
. Since 
is a totally quasi-G-asymptotically nonexpansive semigroup, for each
, we have
where
. This shows that
, which implies that 
for all
.
Step 3. We prove that 
is bounded and 
is convergent.
Since 
is a convex and lower semicontinuous function, by virtue of Lemma 2.10, we have that there exists 
and 
such that 
for each
. Then for each
, we have that
(17)
Again since 
and
, from Lemma 2.11, we have 
for any
. Hence, from (17), we have
Therefore 
and 
are bounded. As 
and
, by using Lemma 2.11, we have that
This implies that 
is bounded and nondecreasing. Hence the limit 
exists.
Step 4. Next, we prove that
.
By the definition of
, for any positive integer
, we have
. Again from Lemma 2.11, we have that
as
. It follows from Lemma 2.8 that
. Hence 
is a Cauchy sequence in
. Since 
is a nonempty closed and convex subset of Banach space
, we can assume that
. Therefore, we have
(18)
Since 
and
, it follows from the definition of 
that we have
That is
(19)
Since 
and
, from (18), (19), we can get
Then, by Lemma 2.8, we have
(20)
As 
is uniformly continuous on each bounded subset of
, we have
. Then from (20), for any
, we have
Since
, we have that
uniformly for all
.
Since J is uniformly continuous, we obtain that
(21)
uniformly for all
.
Since 
is asymptotically regular for all
, from (21), we have
Then 
as
. By virtue of the closedness of 
and 
as
, we can obtain that
, which implies 
for all
.
Hence,
.
Step 5. Finally, we prove that
.
Since 
is closed and convex, by Lemma 2.2, we know that 
is single-valued.
Assume that
. Since 
and
, we have 
for all
. As we know, 
is convex and lower semicontinuous with respect to y when x is fixed. So we have
As
, from the definition of
, we can obtain that 
and 
as
. This completes the proof of Theorem 3.1.
Just as in Remark 2.7, we use 
instead of 
in Definition 2.6 and denote 
by
, 
becomes a countable family of total quasi-G-asymptotically nonexpansive mappings. Then we get the following corollary.
Corollary 3.2 Let 
be a uniformly convex and uniformly smooth Banach space and 
be a nonempty closed and convex subset of
. Let 
be a countable family of closed and totally quasi-Gasymptotically nonexpansive mappings. Let 
be a convex and lower semicontinuous function with 
such that 
for all 
and
. Assume that 
is uniformly asymptotically regular for all 
and
. Let the sequence 
defined by
(22)
where, 
and
. If 
and
, then 
converges strongly to
.
In Corollary 3.2, when 
for all
, 
be a countable family of closed and totally quasi-
-asymptotically nonexpansive mappings. Then we can get the following theorem.
Corollary 3.3 Let 
be a uniformly convex and uniformly smooth Banach space and 
be a nonempty closed and convex subset of
. Let 
be a countable family of closed and totally quasi-
asymptotically nonexpansive mappings. Assume that 
is uniformly asymptotically regular for all 
and
. Let the sequence 
defined by
(23)
where, 
and
. If 
and
, then 
converges strongly to
.
Remark 3.4 The results in this paper improve and extend many recent corresponding main results of other authors (see, for example, [3,4,7,8,10,11,15-19]) in the following ways: (a) we introduce a new class of totally quasi-G-asymptotically nonexpansive mappings which contains the classes of the totally quasi-
-asymptotically nonexpansive mappings and many non-expansive mappings; (b) we extend from a countable family of mappings to the totally quasi-G-asymptotically nonexpansive semigroup; (c) we modify the Halpern type hybrid projection algorithm by using the generalized f-projection operator for uniformly total quasi-G-asymptotically nonexpansive semigroup. For example, Corollary 3.2 extends the main result of Seawan et al. [11] from the modified Mann type iterative algorithm to modified Halpern iterative by the generalized f-projection method. Corollary 3.3 is the main result of Chang et al.[3].
Contributions
All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the article. This study was supported by the National Natural Science Foundations of China (Grant No. 11271330) and the Natural Science Foundations of Zhejiang Province of China (Grant No. Y6110270).
NOTES
*Corresponding author.