^{1}

^{2}

^{1}

^{2}

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically

Throughout this paper, we assume that

Let

Let

Since the Lipschitz continuity of the gradient

Assume that the CMP (

It is worth emphasizing that the regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem

On the other hand, consider the following variational inequality problem (VIP): find a

The VIP (

Motivated by the idea of Korpelevič's extragradient method [

Recall that a mapping

monotone if

It is obvious that if

Let

It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [

If

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [

Let

They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically

Recently, Sahu et al. [

Let

Put

Whenever

Let

Let

Subsequently, the iterative algorithms in Theorems SXY1 and SXY2 are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space; see, for example, [

On the other hand, Yao et al. [

In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (

Let

The metric (or nearest point) projection from

For given

Consequently,

If

A mapping

nonexpansive if

firmly nonexpansive if

It can be easily seen that if

A mapping

We need some facts and tools in a real Hilbert space

Let

Let

Let

Let

If

Let

Let

Let

Let

Let

Lemmas

To prove a weak convergence theorem by a modified extragradient method with regularization for the CMP (

Let

Let

Recall that a Banach space

Let

A set-valued mapping

For solving the equilibrium problem, let us assume that the bifunction

for each

Assume that

for each

Let

In this section, we prove a strong convergence theorem for a hybrid extragradient iterative algorithm with regularization for finding a common element of the set of solutions of the CMP (

Let

First of all, one can show that

We divide the proof into several steps.

Next we show that

Indeed, let

Indeed, from (

Next we show that