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,,

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

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

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].

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

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,

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

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,

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

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

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

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,

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

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

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,

Submitting (15) into (14), we have

This implies that and. Hence we have, i.e.. This completes the proof of Lemma 2.12.

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

where and the sequence. If andthen 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

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

Since and, it follows from the definition of that we have

That is

Since and, from (18), (19), we can get

Then, by Lemma 2.8, we have

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

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

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

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].

All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.

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).