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The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution set

Let

For finding an element of

Let

Recently, Ceng et al. [

For given

In particular, if the mapping

Utilizing Lemma

Throughout this paper, unless otherwise specified, the set of fixed points of the mapping

Subsequently, Ceng at al. [

Let

On the other hand, let

In 1994, the SFP was first introduced by Censor and Elfving [

Very recently, Xu [

Furthermore, Korpelevič [

Throughout this paper, assume that the SFP is consistent; that is, the solution set

Given

It is clear from Proposition

The following statements hold:

Very recently, by combining the regularization method and extragradient method due to Nadezhkina and Takahashi [

Let

Motivated and inspired by the research going on this area, we propose and analyze the following Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution set of the GSVI (

Let

Under appropriate assumptions, it is proven that all the sequences

Let

Also, under mild conditions, it is shown that all the sequences

Observe that both [

Let

The metric (or nearest point) projection from

Some important properties of projections are gathered in the following proposition.

For given

A mapping

nonexpansive if

firmly nonexpansive if

Let

Given a number

Given a number

It can be easily seen that if

Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see, for example, [

A mapping

It is clear that in a real Hilbert space

This immediately implies that if

The following elementary result in the real Hilbert spaces is quite well known.

Let

Let

If

If

If

The following lemma plays a key role in proving weak convergence of the sequences generated by our algorithms.

Let

Let

Let

The following lemma is an immediate consequence of an inner product.

In a real Hilbert space

Let

The solution set of the VIP is denoted by

A set-valued mapping

It is known that in this case the mapping

In this section, we first prove the weak convergence of the sequences generated by the Mann-type extragradient iterative algorithm (

Let

First, taking into account

Now, let us show that

Indeed, it is easy to see that

Hence, it follows that

This immediately implies that

Next we divide the remainder of the proof into several steps.

Indeed, take

Indeed, utilizing Lemma

Indeed, observe that

This together with

Moreover, using the argument technique similar to the previous one, we derive

Utilizing (

Indeed, since

Let

Let

In Theorem

Next, utilizing Corollary

Let

In Corollary

Now, we are in a position to prove the weak convergence of the sequences generated by the Mann-type extragradient iterative algorithm (

Let

First, taking into account

Next we divide the remainder of the proof into several steps.

Indeed, take

Indeed, utilizing Lemma

Indeed, utilizing the Lipschitz continuity of

Indeed, since

First, it is clear from Lemma

Let

Next, utilizing Corollary

Let

In Corollary

Compared with the Ceng and Yao [

Our Theorems

Compared with the relaxed extragradient iterative algorithm in [

Because [

The problem of finding an element of

The hybrid extragradient method for finding an element of

The proof of our results are very different from that of [

Because our iterative algorithms (

In this research, the first author was partially supported by the National Science Foundation of China (11071169) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The third author was partially supported by Grant NSC 101-2115-M-037-001.