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We introduce new implicit and explicit iterative algorithms 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 a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.

Let

A mapping

Let

The VIP (

Let

Throughout this paper, it is assumed as in [

for each

Next we list some elementary results for the MEP.

Assume that

Let

The

In 2012, combining the hybrid steepest-descent method in [

Let

Let

In 1998, Huang [

Let

Whenever

Let

Let

Since the Lipschitz continuity of the gradient

Motivated and inspired by the above facts, we in this paper introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the CMP (

Throughout this paper, we assume that

Recall that a mapping

monotone if

It is obvious that if

The metric (or nearest point) projection from

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

For given

Consequently,

If

A mapping

nonexpansive [

firmly nonexpansive if

It can be easily seen that if

A mapping

Let

Let

We need some facts and tools in a real Hilbert space

Let

Let

Let

Let

Let

The following lemma can be easily proven and, therefore, we omit the proof.

Let

Let

Recall that a set-valued mapping

Let

Assume that

Consequently,

Let

Let

Let

We now state and prove the first main result of this paper.

Let

First of all, let us show that the sequence

Put

Consider the following mapping

Note that

Now, let us show that

Next let us show that

Indeed, from (