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

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 .

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

(1.1)

for all

Definition 1.1.

A set is called monotone on if has the following property:

(1.2)

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.

A mapping is called hemicontinuous at a point in if

(1.3)

Definition 1.3.

A mapping of into is called -inverse strongly monotone if there exists a positive number such that

(1.4)

for all .

Definition 1.4.

A mapping of into is called Lipschitz continuous on if there exists a positive number , named Lipschitz constant, such that

(1.5)

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.

A mapping is said to be strictly pseudocontractive, if there exists a constant such that

(1.6)

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.

The variational inequality problem is to find such that

(1.7)

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

(1.8)

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

(1.9)

with assumption .

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

(1.10)

, where denote the metric projections of onto , and the matrix has the elements , , and . Then, is -inverse strongly monotone. Clearly, if and only if . It means that . Consequently, . It is easy to see that is monotone, and for , if and only if . Therefore, .

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]).

Let be a nonempty closed convex subset of a Hilbert space . Let be a -inverse strongly monotone mapping of into for , and let be a nonexpansive mapping of into itself such that . Let be a sequence generated by

(1.11)

where for some and for some . Then converges weakly to , where

(1.12)

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

(1.13)

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

(1.14)

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

(2.1)

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

(i)monotone, if for all , we have

(2.2)

(ii)strongly monotone with constant , if, for all , we have

(2.3)

(iii)hemicontinuous in the variable for each fixed , if

(2.4)

We can get the following proposition from the above definitions.

Proposition 2.1.

• (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

(2.5)

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 ;

(ii)If   then and for all

• (iii)
  

  (2.6)

where is a positive constant.

Proof.

• (i) Let

  

  (2.7)

Then, problem (2.5) has the following form: find such that

(2.8)

where

(2.9)

It is not difficult to verify that are the monotone bifunctions, and for each fixed , they are hemicontinuous in the variable . Therefore, also is monotone hemicontinuous in the variable for each fixed . Moreover, it is strongly monotone with constant . Hence, (2.8) (consequently (2.5)) has a unique solution for each .

• (ii) Now we prove that

(2.10)

Since and , and

(2.11)

By adding the last inequality to (2.8) in which is replaced by and using the properties of , we obtain

(2.12)

that implies (2.10). It means that is bounded. Let , as . Since is closed and convex, is weakly closed. Hence . We prove that . From the monotone property of and (2.8), it follows

(2.13)

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

(2.14)

On the base of -inverse strongly monotone property of , the monotone property of , , , for all , . From the last inequality, we have

(2.15)

Tending in the last inequality, we obtain

(2.16)

Since is -inverse strongly monotone, the mapping satisfies (1.6), where Because , we have . When , this inequality will not be changed if is replaced by . Thus, is strictly pseudocontractive. Applying Lemma 2.2, we can conclude that It means that It is well known that the sets are closed and convex. Therefore, is also closed and convex. Then, from (2.10) it implies that is the unique element in having a minimal norm. Consequently, we have

(2.17)

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

(2.18)

or

(2.19)

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

(2.20)

Clearly,

(2.21)

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),

(iii)

• (iv)
  

  (2.22)

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

Proof.

From (2.20) it follows

(2.23)

By the similar argument, from (2.5) it implies

(2.24)

where is the solution of (2.5) when is replaced by . By setting in (2.23) and in (2.24) and adding one obtained result to the other, we have,

(2.25)

Therefore, from the monotone property of the mappings , , it follows

(2.26)

From (2.23), (2.5) with and , we have

(2.27)

where

(2.28)

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.

The sequences and which are defined by

(2.29)

with satisfy all conditions in Theorem 2.6.