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We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

Let

Let

Very recently, Cai and Bu [

Recently, Ceng et al. [

For given

In particular, if the mappings

Let

We list some lemmas that will be used in the sequel. Lemma

Let

In a smooth Banach space

Let

Let

Let

It is well known that if

Given a number

Let

Let

Let

Let

In this section, we introduce our implicit iterative schemes and show the strong convergence theorems. We will use the following useful lemmas in the sequel.

Let

Let

Let

By Lemma

We now state and prove our first result on the implicit iterative scheme.

Let

Take a fixed

Let us show that

Next we show that

For simplicity, put

Now, we claim that

Finally, let us show that

Let

Further, we illustrate Theorem

Let

In Corollary

Theorem

The problem of finding a point

The iterative scheme in [

The iterative scheme (

The proof in [

The iterative scheme in [

In this section, we introduce our explicit iterative schemes and show the strong convergence theorems. First, we give several useful lemmas.

Let

Taking into account the

Let

According to Lemma

Let

We can rewrite GSVI (

By Lemma

We are now in a position to state and prove our result on the explicit iterative scheme.

Let

Take a fixed

Let us show that

Now, we claim that

Finally, let us show that

Let

Further, we illustrate Theorem

Let

Utilizing the arguments similar to those in the proof of Corollary

As previous, we emphasize that our composite iterative algorithms (i.e., the iterative schemes (

Theorem

The problem of finding a point

The iterative scheme in [

The iterative scheme (

The proof in [

The assumption of the uniformly convex and 2-uniformly smooth Banach space

The iterative scheme in [

Finally, we observe that related results can be found in recent papers, for example, [

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 101-2115-M-037-001.