- Research Article
- Open Access

# Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces

- JongKyu Kim
^{1}Email author and - Nguyen Buong
^{2}

**2010**:451916

https://doi.org/10.1155/2010/451916

© J. K. Kim and N. Buong. 2010

**Received:**1 November 2009**Accepted:**12 December 2009**Published:**22 February 2010

## Abstract

The purpose of this paper is to present a regularization variant of the inertial proximal point algorithm for finding a common element of the set of solutions for a variational inequality problem involving a hemicontinuous monotone mapping and for a finite family of -inverse strongly monotone mappings from a closed convex subset of a Hilbert space into .

## Keywords

- Hilbert Space
- Variational Inequality
- Weak Convergence
- Monotone Mapping
- Nonexpansive Mapping

## 1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a closed convex subset of . Denote the metric projection of onto by , that is

for all

Definition 1.1.

A monotone mapping in is said to be maximal monotone if it is not properly contained in any other monotone mapping on . Equivalently, a monotone mapping is maximal monotone if for all , where denotes the range of .

Definition 1.2.

Definition 1.3.

for all .

Definition 1.4.

for all .

It is easy to see that any -inverse strongly monotone mapping is monotone and Lipschitz continuous with Lipschitz constant . When , is said to be nonexpansive mapping. Note that a nonexpansive mapping in Hilbert space is -inverse strongly monotone [1].

Definition 1.5.

Clearly, when , is nonexpansive. Therefore, the class of -strictly pseudocontractive mappings includes the class of nonexpansive mappings.

Definition 1.6.

A mapping from into is said to be demiclosed at a point if whenever is a sequence in such that and , then where the symbols and denote the strong and weak convergences of any sequence, respectively.

for all . The set of solutions of the variational inequality problem (1.7) is denoted by .

Let be a finite family of -inverse strongly monotone mappings from into with the set of solutions denoted by And set

The problem which will be studied in this paper is to find an element

with assumption .

A following example shows the fact that Consider the following case:

Since for a nonexpansive mapping , the mapping is -inverse strongly monotone, the problem of finding an element of , where denotes the set of fixed points of the nonexpansive mapping , is equivalent to that of finding an element of , where denotes the set of solutions of the mapping , and contained in the class of problem (1.9).

The case, when is -inverse strongly monotone and , where is nonexpansive, is studied in [2].

Theorem 1.7 (see [2]).

For finding an element of the set , one can use the extragradient method proposed in [3] for the case of finite-dimensional spaces. In the infinite-dimensional Hilbert spaces, the weak convergence result of the extragradient method was proved [1] and it was improved to the strong convergence in [4].

On the other hand, when , (1.7) is equivalent to the operator equation

involving a maximal monotone , since the domain of is the whole space , and is hemicontinuous ([5, 6]). A zero element of (1.13) can be approximated by the inertial proximal point algorithm

where and are two sequences of positive numbers.

Note that the inertial proximal algorithm was proposed by Alvarez [7] in the context of convex minimization. Afterwards, Alvarez and Attouch [8] considered its extension to maximal monotone operators. Recently, Moudafi [9] applied this algorithm for variational inequalities; Moudafi and Elisabeth [10] studied the algorithm by using enlargement of a maximal monotone operator; Moudafi and Oliny [11] considered convergence of a splitting inertial proximal method. The main results in these papers are the weak convergence of the algorithm in Hilbert spaces.

In this paper, by introducing a regularization process we shall show that by adding the regularization term to the inertial proximal point algorithm, called regularization inertial proximal point algorithm, we obtain the strong convergence of the algorithm, and the strong convergence is proved for the general case are -inverse strongly monotone nonself mappings of into ; may not be is monotone and hemicontinuous at each point .

## 2. Main Results

Let be an equilibrium bifunction from to , that is for every In addition, assume that is convex and lower semicontinuous in the variable for each fixed .

The equilibrium problem for is to find such that

First, we recall several well-known facts in [12, 13] which are necessary in the proof of our results.

The equilibrium bifunction is said to be

We can get the following proposition from the above definitions.

- (i)
If is hemicontinuous in the first variable for each fixed and is monotone, then , where is the solution set of (2.1), is the solution set of for all , and they are closed and convex.

- (ii)
If is hemicontinuous in the first variable for each and is strongly monotone, then is a nonempty singleton.

Lemma 2.2 (see [14]).

Let be the sequences of positive numbers satisfying the conditions:

(i) ,

(ii) ,

Then,

Lemma 2.3 (see [15]).

Let be a nonempty closed convex subset of a Hilbert space and a strictly pseudocontractive mapping. Then is demiclosed at zero.

We construct a regularization solution for (1.9) by solving the following variational inequality problem: find such that

where , is the regularization parameter.

We have the following result.

Theorem 2.4.

Let be a nonempty closed convex subset of . Let be a -inverse strongly monotone mapping of into , and let be a monotone hemicontinuous mapping of into such that . Then, we have

(i)For each , the problem (2.5) has a unique solution ;

where is a positive constant.

- (ii)
Now we prove that

Letting , we obtain for any By virtue of Proposition 2.1, we have

Now we show that for all From (2.8), for any , and the monotone property of , it implies that

(iii) From (2.8) and the properties of , for each , it follows

For each is bounded since the operator is Lipschitzian with Lipschitz constants . Using (2.10), the boundedness of and the Lagrange's mean-value theorem for the function , , , on if or if , we have conclusion (iii). This completes the proof.

Remark 2.5.

Obviously, if , where is the solution of (2.8) with , as , then

Now, we consider the regularization inertial proximal point algorithm

Clearly,

is a bifunction. Moreover, it is strongly monotone with By Proposition 2.1, there exists a unique element satisfying (2.20).

Theorem 2.6.

Let be a nonempty closed convex subset of a Hilbert space . Let be a -inverse strongly monotone mapping of into , and let be a monotone hemicontinuous mapping of into such that . Assume that the parameters , and are chosen such that

(i) , ,

(ii) ,

Then the sequence defined by (2.20) converges strongly to the element as .

Proof.

Since the seris in (iii) is convergent, Consequently, From Lemma 2.2, it follows that as .

On the other hand, as . Therefore, we have as This completes the proof.

Remark 2.7.

with satisfy all conditions in Theorem 2.6.

## Declarations

### Acknowledgment

This work was supported by the Kyungnam University Research Fund 2009.

## Authors’ Affiliations

## References

- Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2006, 128(1):191–201. 10.1007/s10957-005-7564-zMATHMathSciNetView ArticleGoogle Scholar - Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2003, 118(2):417–428. 10.1023/A:1025407607560MATHMathSciNetView ArticleGoogle Scholar - Korpelevič GM: An extragradient method for finding saddle points and for other problems.
*Èkonomika i Matematicheskie Metody*1976, 12(4):747–756.Google Scholar - Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems.
*Taiwanese Journal of Mathematics*2006, 10(5):1293–1303.MATHMathSciNetGoogle Scholar - Barbu V:
*Nonlinear Semigroups and Differential Equations in Banach Spaces*. Editura Academiei Republicii Socialiste România, Bucharest, Romania; 1976:352.MATHView ArticleGoogle Scholar - Brezis H: Opératuer maximaux monotones. In
*Mathematics Studies*.*Volume 5*. North-Holland, Amsterdam, The Netherlands; 1973.Google Scholar - Alvarez F: On the minimizing property of a second order dissipative system in Hilbert spaces.
*SIAM Journal on Control and Optimization*2000, 38(4):1102–1119. 10.1137/S0363012998335802MATHMathSciNetView ArticleGoogle Scholar - Alvarez F, Attouch H: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping.
*Set-Valued Analysis*2001, 9(1–2):3–11.MATHMathSciNetView ArticleGoogle Scholar - Moudafi A: A hybrid inertial projection-proximal method for variational inequalities. Journal of Inequalities in Pure and Applied Mathematics 2004., 5(3, article 63):Google Scholar
- Moudafi A, Elisabeth E: An approximate inertial proximal method using the enlargement of a maximal monotone operator.
*International Journal of Pure and Applied Mathematics*2003, 5(3):283–299.MATHMathSciNetGoogle Scholar - Moudafi A, Oliny M: Convergence of a splitting inertial proximal method for monotone operators.
*Journal of Computational and Applied Mathematics*2003, 155(2):447–454.MATHMathSciNetView ArticleGoogle Scholar - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*Mathematics Students*1994, 63: 123–145.MATHMathSciNetGoogle Scholar - Oettli W: A remark on vector-valued equilibria and generalized monotonicity.
*Acta Mathematica Vietnamica*1997, 22(1):213–221.MATHMathSciNetGoogle Scholar - Vasin VV, Ageev AL:
*Incorrect Problems with Priori Information*. Nauka, Ekaterinburg, Russia; 1993:263.Google Scholar - Osilike MO, Udomene A: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type.
*Journal of Mathematical Analysis and Applications*2001, 256(2):431–445. 10.1006/jmaa.2000.7257MATHMathSciNetView ArticleGoogle Scholar

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