^{1}

^{*}

^{1}

^{*}

In this paper, we introduce some new classes of the totally quasi-G-asymptotically nonexpansive mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then, with the generalized f-projection operator, we prove some strong convergence theorems of a new modified Halpern type hybrid iterative algorithm for the totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The results presented in this paper extend and improve some corresponding ones by many others.

In this paper, we denote by

Recall that a mapping

A mapping

We use

Obviously, we have

Recently, Chang et al. [3-5] and Li [

Definition 1.1 [

(as

More recently, Wang et al. [

We use

Definition 1.2 [

(a)

(b) For each

(c) For each

(d) For each

One-parameter family

(e) If

for all

On the other hand, Wu et al. [

In 2013, by using the generalized f-projection operator, Seawan et al. [

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-

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

It is well known that the normalized dual mapping

(1) If

(2) If

A Banach space

Now, we give a functional

where

(1)

(2)

Definition 2.1 [

Lemma 2.2 [

Recall that if

And in a smooth Banach space, the definition of the generalized f-projection operator transforms into:

Definition 2.3 [

Now, we give the definition of the totally quasi-

Definition 2.4 A mapping

for any

A mapping

for all

Remark 2.5 It is easy to see that a quasi-

Definition 2.6 One-parameter family

(f) There exists a sequence

holds for all

One-parameter family

(g) if

holds for each

Remark 2.7 It is easy to see that a quasi-

When we use

The following Lemmas are necessary for proving the main results in this paper.

Lemma 2.8 [

Lemma 2.9 [

Lemma 2.10 [

for each

Lemma 2.11 [

Lemma 2.12 Let

Proof Let

Since

So,

By lemma 2.8, we have that

Submitting (15) into (14), we have

This implies that

Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space and

such that

where

Proof We divide the proof into five steps.

Step 1. Firstly, we prove that

Since

Again, by the assumption,

This shows that

Step 2. Next, we prove that

In fact,

where

Step 3. We prove that

Since

Again since

Therefore

This implies that

Step 4. Next, we prove that

By the definition of

as

Since

That is

Since

Then, by Lemma 2.8, we have

As

Since

uniformly for all

Since J is uniformly continuous, we obtain that

uniformly for all

Since

Then

as

Hence,

Step 5. Finally, we prove that

Since

Assume that

As

Just as in Remark 2.7, we use

Corollary 3.2 Let

where,

converges strongly to

In Corollary 3.2, when

Corollary 3.3 Let

and

where,

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-

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).