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It is well known that the gradient-projection algorithm (GPA) for solving constrained convex minimization problems has been proven to have only weak convergence unless the underlying Hilbert space is finite dimensional. In this paper, we introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space. It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction in a real Hilbert space.

Let

Consider the following constrained convex minimization problem:

Assume the minimization (

From the above theorem, it is known that the gradient-projection algorithm has weak convergence, in general, unless the underlying Hilbert space is finite dimensional. This gives naturally rise to a question how to appropriately modify the gradient-projection algorithm so as to have strong convergence. Xu [

Assume the minimization (

Then the sequence

On the other hand, Peng and Yao [

The problem (

If

If

If

If

The variational inequalities have been extensively studied in the literature; see [

Very recently, Peng [

Let

for each

for each

for each

Let

Furthermore, related iterative methods for solving fixed point problems, variational inequalities, equilibrium problems, and optimization problems can be found in [

In this paper, let

Compared with Xu [

Our problem of finding an element of

In our algorithm (

Here the main purpose of the reason why we use such an iteration step is to play a convenience and efficiency role in the computation of an element of

Our problem of finding an element of

Let

For every point

A set-valued mapping

Let

Recall that a mapping

In order to prove our main result in the next section, we need the following lemmas and propositions.

Let

If

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

In a real Hilbert space

Let

Recall that

Assume

If

If

If

The following lemma was proved by Suzuki [

Let

Let

Then

In order to prove our main result, we shall need the following lemma given in [

Let

We are now in a position to state and prove our main result.

Let

Then the sequences

First it is obvious that there hold the following assertions:

where

where its solution set is denoted by

We divide the proof into several steps.

Indeed, first, we can write (

Indeed, let

Indeed, set

Indeed, since

Secondly, let us show

Next, let us show

Indeed, from (

Utilizing Theorem

Let

Then,

Let

Then

Let

Then,

Let

Then,

In Theorem

Let

Let

Then,

Since

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). This research was partially supported by a Grant from NSC 101-2115-M-037-001.