Open Access

A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem

  • Filomena Cianciaruso1,
  • Giuseppe Marino1Email author,
  • Luigi Muglia1 and
  • Yonghong Yao2
Fixed Point Theory and Applications20092010:383740

DOI: 10.1155/2010/383740

Received: 3 June 2009

Accepted: 16 September 2009

Published: 1 November 2009

Abstract

We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

1. Introduction

In this paper, we define an iterative method to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings generated by two sequences of firmly nonexpansive mappings and two nonlinear mappings. Let us recall from [1] that the -strict pseudocontractions in Hilbert spaces were introduced by Browder and Petryshyn in [2].

Definition 1.1.

is said to be -strict pseudocontractive if there exists such that
(1.1)

The iterative approximation problems for nonexpansive mappings, asymptotically nonexpansive mappings, and asymptotically pseudocontractive mappings were studied extensively by Browder [3], Goebel and Kirk [4], Kirk [5], Liu [6], Schu [7], and Xu [8, 9] in the setting of Hilbert spaces or uniformly convex Banach spaces. Although nonexpansive mappings are 0-strict pseudocontractions, iterative methods for -strict pseudocontractions are far less developed than those for nonexpansive mappings. The reason, probably, is that the second term appearing in the previous definition impedes the convergence analysis for iterative algorithms used to find a fixed point of the -strict pseudocontraction . However, -strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems. In the recent years the study of iterative methods like Mann's like methods and CQ-methods has been extensively studied by many authors [1, 1013] and the references therein.

If is a closed and convex subset of a Hilbert space and is a bi-function we call equilibrium problem
(1.2)

and we will indicate the set of solutions with .

If is a nonlinear mapping, we can choose , so an equilibrium point (i.e., a point of the set ) is a solution of variational inequality problem (VIP)
(1.3)

We will indicate with the set of solutions of VIP.

The equilibrium problems, in its various forms, found application in optimization problems, fixed point problems, convex minimization problems; in other words, equilibrium problems are a unified model for problems arising in physics, engineering, economics, and so on (see [10]).

As in the case of nonexpansive mappings, also in the case of -strict pseudocontraction mappings, in the recent years many papers concern the convergence of iterative methods to a solutions of variational inequality problems or equilibrium problems; see example for, [10, 1418].

Here we prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

2. Preliminaries

Let be a real Hilbert space and let be a nonempty closed convex subset of .We denote by the metric projection of onto . It is well known [19] that
(2.1)

Lemma 2.1.

(see [20]) Let be a Banach space with weakly sequentially continuous duality mapping , and suppose that converges weakly to , then for any ,
(2.2)

Moreover if is uniformly convex, equality holds in (2.2) if and only if .

Recall that a point is a solution of a VIP if and only if
(2.3)

Definition 2.2.

An operator is said to be -inverse strongly monotone operator if there exists a constant such that
(2.4)

If we say that is firmly nonexpansive. Note that every -inverse strongly monotone operator is also Lipschitz continuous (see [21]).

Lemma 2.3.

(see [2]). Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudocontractive mapping. Then with is a nonexpansive mapping with .

3. Main Theorem

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space . Let

(i) be an -inverse strongly monotone mapping of into ,

(ii) a -inverse strongly monotone mapping of into ,

(iii) and two sequences of firlmy nonexpansive mappings from to .

Let be a -strict pseudocontraction .

Set and let us define the sequence as follows:
(3.1)

where

(i) with ;

(ii) ;

(iii) .

Moreover suppose that

(i) ;

(ii) pointwise converges in to an operator and pointwise converges in to an operator ;

(iii) and .

Then strongly converges to .

Proof.

We begin to observe that the mappings and are nonexpansive for all since they are compositions of nonexpansive mappings (see [22, page 419]). As a rule, if
(3.2)

Now we divide the proof in more steps.

Step 1.

is closed and convex for each .

Indeed is the intersection of with the half space
(3.3)

where .

Step 2.

for each .

For each we have
(3.4)

So the claim immediately follows by induction.

Step 3.

exists and is asymptotically regular, that is, .

Since , and , by (2.1) choosing and , we have
(3.5)

that is, .

By and , we have
(3.6)
Then exists and is bounded. Moreover
(3.7)

and consequently .

Step 4.

and .

By , it follows
(3.8)
Moreover
(3.9)

and by boundedness of , it follows that .

Step 5.

, for each .

For , we have
(3.10)
Consequently
(3.11)

and by Step 4, the assumptions on and , we obtain the claim of Step 5.

Step 6.

.

Since is firmly nonexpansive, for any , we have
(3.12)
which implies
(3.13)
Consequently
(3.14)
which implies
(3.15)

By the assumptions on , Steps 4 and 6, and the boundedness of and the claim follows.

Step 7.

and .

Since is firmly nonexpansive, for each , we have
(3.16)
and consequently
(3.17)
Then, for each , we have
(3.18)
consequently
(3.19)

and by the assumptions on , Step 4 and the boundedness of and it follows that as . By Step 6 we note that also .

Finally
(3.20)

and by previous steps, it follows that as .

Step 8.

The set of weak cluster points of is contained in .

We will use three times the Opial's Lemma 2.1.

Let be a weak cluster point of and let be a subsequence of such that .

We prove that . We suppose for absurd that . By Opial's Lemma 2.1 and as , we obtain
(3.21)

which is a contradiction.

Since it is enough to prove that . Now if we note that
(3.22)

This leads to a contraddiction again. By the hypotheses and Step 7 the claim follows. By the same idea and using Step 6, we prove that .

Step 9.

.

Since and , we have
(3.23)
Let be a subsequence of such that . By Step 8, . Thus
(3.24)
Therefore we have
(3.25)

Since has the Kadec-Klee property, then as .

Moreover, by and by the uniqueness of the projection , it follows that .

Thence every subsequence converges to as and consequently , as .

Remark 3.2.

Let us observe that one can choose and as sequences of -inverse strongly monotone operators and -inverse strongly monotone operators provided for all .

The hypotheses and in the main Theorem 3.1 seem very strong but, in the sequel, we furnish two cases in which (ii) and (iii) are satisfied.

Let us remember that the metric projection on a convex closed set is a firmly nonexpansive mapping (see [19]) so we claim that have the following proposition.

Proposition 3.3.

If is such that and an -inverse strongly monotone, then realizes conditions (ii) and (iii) with .

Proof.

To prove (ii) we note that for each ,
(3.26)

Moreover, (iii) follows directly by (2.2).

Now we consider the mixed equilibrium problem
(3.27)

In the sequel we will indicate with the set of solution of our mixed equilibrium problem. If we denote with .

We notice that for and the problem is the well-known equilibrium problem [2325]. If and is an -inverse strongly monotone operator we have the equilibrium problems studied firstly in [26] and then in [18, 22, 27]. If and we refound the mixed equilibrium problem studied in [16, 28, 29].

Definition 3.4.

A bi-function is monotone if for all .

A function is upper hemicontinuous if
(3.28)

Next lemma examines the case in which .

Lemma 3.5.

Let be a convex closed subset of a Hilbert space .

Let be a bi-function such that

(f1) for all ;

(f2) is monotone and upper hemicontinuous in the first variable;

(f3) is lower semicontinuous and convex in the second variable.

Let be a bi-function such that

(h1) for all ;

(h2) is monotone and weakly upper semicontinuous in the first variable;

(h3) is convex in the second variable.

Moreover let us suppose that

()for fixed and , there exists a bounded set and such that for all , ,

for and let be a mapping defined by
(3.29)

called resolvent of and .

Then

(1) ;

(2) is a single value;

(3) is firmly nonexpansive;

(4) and it is closed and convex.

Proof.

Let . For any define
(3.30)

We will prove that, by KKM's lemma, is nonempty.

First of all we claim that is a KKM's map. In fact if there exists such that (with ) does not appartiene to for any then
(3.31)
By the convexity of and and the monotonicity of , we obtain that
(3.32)

that is absurd.

Now we prove that . We recall that, by the weak lower semicontinuity of , the relation
(3.33)

holds. Let and let be a sequence in such that .

We want to prove that
(3.34)
Since is lower semicontinuous and convex in the second variable and is weakly upper semicontinuous in the first variable, then
(3.35)
Now we observe that is weakly compact for at least a point . In fact by hypothesis (H) there exist a bounded and , such that for all it results . Then , that is, it is bounded. It follows that is weakly compact. Then by KKM's lemma is nonempty. However if then
(3.36)
As in [24, Lemma  3], since is upper hemicontinuous and convex in the first variable and monotone, we obtain that (3.36) is equivalent to claim that is such that
(3.37)
that is, . This prove (1). To prove (2) and (3) we consider and . They satisfy the relations
(3.38)
By the monotonicity of and , summing up both the terms,
(3.39)
so we conclude
(3.40)

that means simultaneously that if and is firmly nonexpansive.

To prove (4), it is enough to follow (iii) and (iv) in [25, Lemma  2.12].

Remark 3.6.

We note that if , our lemma reduces to [25, Lemma  2.12]. The coercivity condition (H) is fulfilled.

Moreover our lemma is more general than [16, Lemma  2.2]. In fact

(i)our hypotheses on are weaker ( weak upper semicontinuous implies upper hemicontinuous);

(ii)if satisfies the condition in Lemma  2.2, choosing one has that is concave and upper semicontinuous in the first variable and convex and lower semicontinous in the second variable;

(iii)the coercivity condition (H) by the equivalence of (3.36) and (3.37) is the same.

Lemma 3.7.

Let us suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Let , . Then
(3.41)

Proof.

By Lemma 3.5, defining and , we know that
(3.42)
In particular,
(3.43)
Hence, summing up this two inequalities and using the monotonicity of and ,
(3.44)
We derive from (3.44) that
(3.45)
and so
(3.46)
Then,
(3.47)

and thus the claim holds.

Proposition 3.8.

Let us suppose that and are two bi-functions satisfying the hypotheses of Lemma 3.5. Let be the resolvent of and . Let be an -inverse strongly monotone operator. Let us suppose that is such that . Then realize (ii) and (iii) in Theorem 3.1.

Proof.

Let be in a bounded closed convex subset of . To prove (i) it is enough to observe that by Lemma 3.7
(3.48)

When , by boundedness of the terms that do not depend on , we obtain (ii).

To prove (iii) let the pointwise limit of . It is necessary to prove only that . Let . We want to prove that . Let . Thus, by definition of , is the unique point such that
(3.49)
By monotonicity of and this implies
(3.50)
Passing to the limit on , by (f3) and (h2) we obtain
(3.51)
Let now with . Then by the convexity of and
(3.52)
Passing we obtain by (f1) and (h1)
(3.53)

That is, . At this point we observe that from the definitions of and , one has .

By Propositions 3.3 and 3.8 we can exhibit iterative methods to approximate fixed points of the -strict pseudo contraction that are also

(1)solution of a system of two variational inequalities VI(C,A) and VI(C,B) ( );

(2)solution of a system of two mixed equilibrium problems ( and );

(3)solution of a mixed equilibrium problem and a variational inequality ( and ).

However when the properties of the mapping and are well known, one can prove convergence theorems like Theorem 3.1 without use of Opial's lemma.

In next theorem our purpose is to prove a strong convergence theorem to approximate a fixed point of that is also a solution of a mixed equilibrium problem and a solution of a variational inequality . One can note that we relax the hypotheses on the convergence of the sequences and .

Theorem 3.9.

Let be a closed convex subset of a real Hilbert space , let be two bi-functions satisfying (f1)–(f3),(h1)–(h3), and (H). Let be a -strict pseudocontraction.

Let be an -inverse strongly monotone mapping of into and let be a -inverse strongly monotone mapping of into .

Let us suppose that .

Set , one defines the sequence as follows:
(3.54)

where

(i) with ;

(ii) ;

(iii) .

Then strongly converges to .

Proof.

First of all we observe that by Lemma 3.5 we have that . We can follow the proof of Theorem 3.1 from Steps 1–7. We prove only the following.

Step 10.

The set of weak cluster points of is contained in .

Let be a cluster point of ; we begin to prove that . We know that
(3.55)
and by (f2)
(3.56)
Let be a subsequence of weakly convergent to , then by Step 7 as . Let . Then by (3.56)
(3.57)

Since is Lipschitz continuous and as , we have as .

By condition , for fixed, the function is lower semicontinuos and convex, and thus weakly lower semicontinuous [30].

Since , as and by the assumption on we obtain . Then we obtain by (h2)
(3.58)
Using (f1), (f3), (h1), (h3) we obtain
(3.59)
Consequently
(3.60)

by (f2) and (h2), as , we obtain .

Now we prove that .

We define the maximal monotone operator
(3.61)
where is the normal cone to at , that is,
(3.62)
Since , by the definition of we have
(3.63)
But , then
(3.64)
and hence
(3.65)
By (3.63), (3.65), and by the -inverse monotonicity of , we obtain
(3.66)
By as (immediately consequence of Steps 6 and 7), it follows that as . Then
(3.67)

moreover, since is a maximal operator, , that is, .

Finally, to prove that we follow Step 8 as in Theorem 3.1.

Since also Step 9 can be followed as in Theorem 3.1, we obtain the claim.

Authors’ Affiliations

(1)
Dipartimento di Matematica, Universitá della Calabria
(2)
Department of Mathematics, Tianjin Polytechnic University

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