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This paper presents new implicit algorithms for solving the variational inequality and shows that the proposed methods converge under certain conditions. Some special cases are also discussed.

Variational inequality theory, introduced by Stampaccia [

The projection method provides important tools for finding the approximate solution of variational inequalities. This method is due to Lions and Stampacchia [

In this paper, we use the equivalent fixed point formulation to suggest and analyze some new implicit iterative methods for solving the variational inequalities. We have shown that these new implicit methods include the unified implicit, the proximal point and the modified extra gradient methods of Noor et al. [4,5], Noor [

Let H be a real Hilbert space whose inner product and norm are denoted by and respectively. Let K be a nonempty closed convex subset in

For a given nonlinear operator, we consider the problem of finding such that

Problem (1) is called the variational inequality, introduced and studied by Stampacchia [

First we recall the following well-known results and concepts.

Lemma 1. Let K be a nonempty, closed, and convex set in. Then, for a given in, satisfies the inequality

if and only if

where is the projection of onto the closed and convex set.

It is well known that the projection operator is nonexpansive, that is

Now if is a nonempty, closed and convex subset in, then Problem (1) is equivalent to the existence of such that

where denotes the normal cone of at. Problem (2) is called the variational inclusion problem associated with the variational inequality (1).

Definition 1. An operator is said to be strongly monotone if and only if there exists a constant such that

and Lipschitz continuous if there exists a constant such that

In this section, using Lemma 1, one can easily show that the variational inequality (1) is equivalent to the existence of such that

where is constant.

Equation (3) is a fixed point problem and will be used in suggesting some new implicit methods for solving the variational inequality (1), and this is the main motivation of this paper.

Now, using the equivalent fixed point formulation (3), one can suggest the following iterative method for solving the variational inequality (1).

Algorithm 1. For a given find the approximate solution by the iterative scheme

Algorithm 1 is known as the projection iterative method.

For a given, we can rewrite (3) as

This fixed point formulation is used to suggest the following iterative method for solving variational inequality (1).

Algorithm 2. For a given, find the approximate solution by the iterative scheme

Note that Algorithm 2 is an implicit type iterative method and includes the implicit method of Noor [

In order to implement this method, we use the predictor-corrector technique. We use Algorithm 1 as the predictor and Algorithm 2 as the corrector. Consequently, we obtain the following two-step iterative method for solving the variational inequality (1).

Algorithm 3. For a given, find the approximate solution by the iterative schemes

Algorithm 3 is a new two-step implicit iterative method for solving the variational inequality (1). For, Algorithm 3 reduces to the following iterative method for solving variational inequality (1).

Algorithm 4. For a given, find the approximate solution by the iterative schemes

which is known as the modified double projection method, Noor [

For, Algorithm 3 reduces to algorithm 1 for solving variational inequality (1).

This shows that Algorithm 3 is a unified implicit method and includes the previously known implicit and predictor-corrector methods as special cases.

Now for a given and, we can rewrite (3) as

For, the fixed point formulation (7) reduces to the fixed point formulation (4).

Now we use (7) to suggest the following iterative methods for solving variational inequality (1).

Algorithm 5. For a given, find the approximate solution by the iterative scheme

Note that Algorithm 5 is an implicit type iterative method and includes the implicit method of Noor et al. [

In order to implement this method, we use the predictor-corrector technique. We use Algorithm 1 as the predictor and Algorithm 5 as the corrector. Consequently, we obtain the following iterative method for solving the variational inequality (1).

Algorithm 6. For a given, find the approximate solution by the iterative schemes

Algorithm 6 is a new two-step implicit iterative method for solving the variational inequality (1). For, Algorithm 6 reduces to the following iterative method for solving variational inequality (1).

Algorithm 7. For a given, find the approximate solution by the iterative schemes

Algorithm 7 was studied by Noor et al. [

For, Algorithm 6 reduces to the following iterative method for solving variational inequality (1), and appears to be new.

Algorithm 8. For a given, find the approximate solution by the iterative schemes

For, Algorithm 6 reduces to the following iterative method for solving variational inequality (1), and appears to be new.

Algorithm 9. For a given, find the approximate solution by the iterative schemes

For Algorithm 9 reduces to Noor [

Now one can obtains the following iterative method for solving variational inequality (1), by using the fixed point formulation (7).

Algorithm 10. For a given, find the approximate solution by the iterative scheme

In order to implement this method, we use the predictor-corrector technique. We use Algorithm 1 as the predictor and Algorithm 10 as the corrector. Consequently, we obtain the following two-step iterative method for solving the variational inequality (1).

Algorithm 11. For a given, find the approximate solution by the iterative scheme

Algorithm 11 is a new two-step implicit iterative method for solving the variational inequality (1). For, Algorithm 11 reduces to Algorithm 7 [

We now consider the convergence analysis of Algorithm 3, 6 and 11, and this is the motivation of next results.

Theorem 1. Let the operator be strongly monotone with constant and Lipschitz continuous with constant. If there exists a constant such that

then, the approximate solution obtained from Algorithm 3 converges strongly to the exact solution satisfying the variational inequality (1).

Proof. Let be a solution of (1) and be the approximate solution obtained from Algorithm 3. Then, from (3) and (5), we have

From the strongly monotonicity and Lipschitz continuity of the operator, one obtains

From (11) and (12), one obtains

where

Now from (3), (6) and (13), we have

where

From (10), it follows that. Hence, the fixed point Problem (3) has a unique solution and consequently the iterative solution obtained from Algorithm 3 converges to the exact solution and satisfying the variational inequality (1). □

Theorem 2. Let the operator T be strongly monotone with constant and Lipschitz continuous with constant. If there exists a constant such that

then, the approximate solution obtained from Algorithm 6 converges strongly to the exact solution satisfying the variational inequality (1).

Proof. Let be a solution of (1) and be the approximate solution obtained from Algorithm 6. Then, from (3), (8) and (13), we have

where

From (14), it follows that. Hence, the fixed point Problem (3) has a unique solution and consequently the iterative solution obtained from algorithm 6 converges to the exact solution of (1). □

Theorem 3. Let the operator T be strongly monotone with constant and Lipschitz continuous with constant. If there exists a constant such that

then, the approximate solution obtained from Algorithm 11 converges strongly to the exact solution and satisfying the variational inequality (1).

Proof. Let be a solution of (1) and be the approximate solution obtained from Algorithm 11. Then, from (3), (9) and (13), we have

where

From (15), it follows that. Hence, the fixed point Problem (3) has a unique solution and consequently the iterative solution obtained from algorithm 11 converges to the exact solution of (1). □

In this paper, we have used the equivalence between the variational inequality and the fixed point problem to suggest and analyze some new implicit iterative methods for solving the variational inequality. We also show that the new implicit methods includes the extra gradient method of Korpelevich [

The author would like to express her thanks to the anonymous referee for his valuable comments to improve the final version of this paper.