ABSTRACT

The aim of this paper, is to introduce and study a general iterative algorithm concerning the new mappings which the sequences generated by our proposed scheme converge strongly to a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a real Hilbert space. In addition, we obtain some applications by using this result. The results obtained in this paper generalize and refine some known results in the current literature.

1. Introduction

Letbe a real Hilbert space, whose inner product and norm are denoted by and respectively. Let be a nonempty closed convex subset of H. A mapping is called nonexpansive if for all We denote by the set of fixed points of T. A linear bounded operator A is strongly positive if there is a constant with the property for all A mapping is said to be a contraction if there exists a coefficient such that for all Let P_C be the nearest point projection of onto the convex subset (i.e., for, P_C is the only point in C such that It is known that projection operator P_C is nonexpansive. It is also known that P_C satisfies for The following characterizes the projection P_C Given and Then if and only if there holds the relations:

(1.1)

for all (see [1]). Moreover, is characterized by the properties: and for all Let be a nonlinear map. The classical variational inequality problem, denoted by is to find such that

(1.2)

for all One can see that the variational inequality problem (1.2) is equivalent to the following fixed point problem: the element is a solution of the variational inequality (1.2) if and only if satisfies the relation where is a constant. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [2-6] and the references therein. A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space:

(1.3)

where A is a linear bounded operator and b is a given point in H. In [5] (see also [6]), it is proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily,

converges strongly to the unique solution of the minimization problem (1.3) provided the sequence satisfies certain conditions. In 2006, Marino and Xu (see [3]) considered the following viscosity iterative method which was first introduced by Moudafi (see [7]):

(1.4)

They proved that the sequence generated by iterative scheme (1.4) converges strongly to the unique solution of the variational inequality , which is the optimality condition for the minimization problem

where h is a potential function for (i.e., for).

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for -cocoercive mapping, Takahashi and Toyoda (see [11]) introduced the following iterative process:

(1.5)

where B is -cocoercive, and . They showed that, if is nonempty, then the sequence generated by (1.5) converges weakly to some In 2005, Iiduka and Takahashi (see [12]) introduced the following iterative process:

(1.6)

where, and They proved that under certain appropriate conditions imposed on and the sequence generated by (1.6) converges strongly to In 2009, Qin, Kang and Shang, [13] introduced the following iterative algorithm given by

(1.7)

where, a k-strict pseudo-contraction for some, defined by A is a strongly positive linear bounded self-adjoint operator and f is a contraction. They proved that the sequence generated by the iterative algorithm (1.7) converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A.

Let be a proper extended realvalued function and F be a bifunction from to where is the set of real numbers. Ceng and Yao [14] considered the following mixed equilibrium problem: Find such that

(1.8)

for all The set of solutions of (1.8) is denoted by i.e.,

It is easy to see that x is a solution of problem (1.8) implies that Moreover, Ceng and Yao [14] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.8) and the set of common fixed points of a family of finitely nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. If then the mixed equilibrium problem (1.8) becomes the following equilibrium problem:

(1.9)

for all The set of solutions of (1.9) is denoted by i.e.,

Given a mapping let and for all Then, if and only if for all i.e., z is a solution of the variational inequality. Equilibrium problems have been studied extensively; see, for instance, [15,16]. The mixed equilibrium problem (1.8) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance, [14,16-19].

Combettes and Hirstoaga (see [15]) introduced an iterative scheme for finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. In 2007, S. Takahashi and W. Takahashi (see [20]) introduced an iterative scheme using the viscosity approximation method for finding a common element of the set of solutions of equilibrium problem (1.9) and the set of fixed points of a nonexpansive nonself-mapping in a Hilbert space. The scheme is defined as follows:

(1.10)

They proved that under certain appropriate conditions imposed on and, the sequences and generated by (1.10) converge strongly to , where In the same year, Shang et al. (see [21]) introduced the following iterative scheme:

(1.11)

for finding a common element of the set of solutions of equilibrium problem (1.9) and the set of fixed points of a nonexpansive nonself-mapping in a Hilbert space. They proved that under some sufficient suitable conditions, the sequences and generated by (1.11) converge strongly to

where

which is the unique solution of the variational inequality

for all

Let where be a finite family of nonexpansive mappings. Finding an optimal point in the intersection of the fixed points set of a finite family of nonexpansive mappings is a problem of interest in various branches of sciences; see [22-27] and also see [28] for solving the variational problems defined on the set of common fixed points of finitely many nonexpansive mappings. Atsushiba and Takahashi (see [29]), defined the mappings

(1.12)

where Such a mapping is called the W-mapping generated by and The concept of W-mappings was introduced in [30-33]. In 2008, Qin et al. (see [34]) introduced and studied the following iterative process:

(1.13)

where is defined by (1.12), is a strongly linear bounded operator and B is -Lipschitzian, relaxed -cocoercive mapping of C into H. They proved that the sequences and generated by the iterative scheme (1.13) converge strongly to

where

which is the unique solution of the variational inequality

for all

.

In the same year, Colao et al. (see [35]) introduced a new iterative scheme:

(1.14)

for approximating a common element of the set of solutions of equilibrium problem (1.9) and the set of common fixed points of a finite family of nonexpansive mappings and obtained a strong convergence theorem in a Hilbert space. In 2009, Yao et al. (see [36]) studied similar scheme as follows:

(1.15)

where, , , and is the W-mapping defined by (1.12). They proved that under certain appropriate conditions imposed on, , and , the sequences and generated by (1.15) converge strongly to

where

which is the unique solution of the variational inequality

for all.

If for some then (1.15) reduces to the iterative scheme (1.14). Very recently, Kangtunyakarn and Suantai (see [37]) defined the new mappings

(1.16)

where Such a mapping K_n is called the K-mapping generated by and Nonexpansivity of each T_i ensures the nonexpansivity of K_n Also following they defined the new mappings

(1.17)

where such that for all and Such a mapping K is called the K-mapping generated by and In [37], Lemma 2.9 and Lemma 2.10, its shown that

and for all where K_n and K are the K-mappings defined by (1.16) and (1.17), respectively. Its important tool for the proof of the main results in this paper. Moreover, Kangtunyakarn and Suantai (see [37]) introduced a new iterative scheme: and,

(1.18)

where, , , and K_n is the K-mapping defined by (1.16). They proved that under certain appropriate conditions imposed on, and , the sequences and generated by (1.18) converge strongly to

where

Motivated by the recent works, we introduce a more general iterative algorithm for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. The scheme is defined as follows: and

(1.19)

where, , , , , is a -Lipschitzian, relaxed -cocoercive mapping, f is a contraction of H into itself with a coefficient is a projection of H onto C, A is a strongly positive linear bounded operator on H, F is a mixed equilibrium bifunction, is a proper lower semicontinuous and convex function and K_n is the K-mapping generated by and We prove that the sequences and generated by the iterative scheme (1.19) converge strongly to

where

which is the unique solution of the variational inequality for all

and is also the optimality condition for the minimization problem

where h is a potential function for (i.e., for).

2. Preliminaries and Lemmas

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

A mapping B is called -strongly monotone, if each we have

for a constant v > 0, which implies that so that B is v-expansive and when v = 1, it is expansive. B is said to be v-cocoercive (see [8] and [9]), if for each we have

for a constant v > 0. Clearly, every v-cocoercive mapping B is -Lipschitz continuous. B is called relaxed u-cocoercive, if there exists a constant u > 0 such that

for all B is said to be relaxed -cocoercive, if there exist two constants u, v > 0 such that

for all for B is v-strongly monotone.

It is worth mentioning that the class of mappings which are relaxed -cocoercive more general than the class of strongly monotone mappings. It is easy to see that if B is a v-strongly monotone mapping, then it is a relaxed -cocoercive mapping (see [10]).

It is well known that for all and there holds

Recall that a space X is said to satisfy Opial’s condition (see [38]) if weakly as and for all then

A set-valued mapping is called monotone if for all, , and imply

A monotone mapping is maximal if graph of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for, for every implies Let B be a monotone mapping of C into H and let be normal cone to C at i.e.,

and define

Then T is a maximal monotone and if and only if; see [39].

In the sequel, the following lemmas are needed to prove our main results.

Lemma 2.1. (see [4,5]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that 1)

2) Then

Lemma 2.2. (see [3]). Assume A is a strong positive linear bounded operator on a Hilbert space H with coefficient and. Then.

Lemma 2.3. (see [40]). Let and be bounded sequences in a Banach space and let be a sequence in with

Suppose for all integers n ≥ 0 and

Then

Lemma 2.4. (see [37]). Let C be a nonempty closed convex set of a strictly convex Banach space. Let be a finite family of nonexpansive mappings of C into itself with and let be real numbers such that for every and Let K be the K-mapping generated by and Then.

Lemma 2.5. (see [37]). Let C be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of into itself and

be sequences in such that Moreover for every let K and be the Kmappings generated by and

and and respectively. Then for every it follows that

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and the set C:

(A1) for all

(A2) is monotone, i.e., for all

(A3) For each

(A4) For each is convex and lower semicontinuous;

(B1) For each and there exists a bounded subset and such that for any

(B2) C is a bounded set.

By a similar argument as in the proof of Lemma 2.3 in [18], we have the following result.

Lemma 2.6. Let C be a nonempty closed convex subset of a Hilbert space H and let F be a mixed equilibrium bifunction of C × C into satisfying conditions (A1)- (A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and define a mapping as follows:

for all Then is well defined and the following hold:

1) is single-valued;

2) is firmly nonexpansive, i.e., for any

3);

4) is closed and convex.

Remark 2.7. We remark that Lemma 1.6 is not a consequence of Lemma 3.1 in [14], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case

The following lemma is well known.

Lemma 2.8. In a real Hilbert space H, there holds the following inequality

for all

3. Main Results

Theorem 3.1. Let H be a real Hilbert space, C a nonempty closed convex subset of H, B a -Lipschitzian, relaxed -cocoercive mapping of C into H, F a bifunction from C × C to which satisfies (A1)-(A4), a proper lower semicontinuous and convex function and a finite family of nonexpansive mappings of C into H such that the common fixed points set

Let f be a contraction of H into itself with a coefficient and A a strongly positive linear bounded operator on H with coefficient such that

Assume that and either (B1) or (B2) holds.

Let be real numbers such that for every and and, two real sequences in (0, 1) satisfying the following conditions:

(C1) and

(C2)

(C3) and (this is weaker than the condition );

(C4)

(C5) for some a, b with

;

(C6)

Then, the sequences and generated iteratively by (1.19) converge strongly to

where

which solves the following variational inequality:

for all

Proof Since as by the condition (C1), we may assume, without loss of generality, that

for all n. We also have for all n. By using Lemma 2.2, we have

Since A is a strongly positive linear bounded operator on a Hilbert space H, we have

and

Observe that 

This shows that is positive. It follows that

Next, we will assume that First, we show is nonexpansive. Indeed, from the relaxed -cocoercive and -Lipschitzian definition on B and condition (C5), we have which implies the mapping is nonexpansive.

We shall divide our proof into 5 steps.

Step 1. We shall show that the sequence is bounded. Let

Since we have

(3.1)

Putting for all we have

Using (1.19), (3.1) and (3.2), we have

which gives that

Hence is bounded, so are ,

and

Step 2. We will show that

Observing that and we have

(3.3)

and

(3.4)

Putting in (3.3) and in (3.4), we have

and

Summing up the last two inequalities and using Lemma 2.6 (A2), we obtain

That is,

It then follows that

This implies that

where M₁ is an appropriate constant such that

Since is nonexpansive and using (3.5), we also have

where M₂ is an appropriate constant such that 

Define

for all so that

It follows that

Observe that from (3.6), we obtain

(3.7)

Next we estimate

For we have

(3.8)

and

(3.9)

where

Using (3.8) and (3.9), we have

(3.10)

Substitute (3.10) into (3.7) yields that

which implies that (noting that (C1), (C2), (C3), (C4) and (C6))

Hence by Lemma 2.3, we have

(3.11)

Using (3.11) and we have

(3.12)

Step 3. We shall show that

where

Note that

This implies

From condition (C1), (C4) and (3.12), we have

(3.13)

Next we prove that

as

Indeed, picking 

Since and T_r is firmly nonexpansive, we obtain and hence

(3.14)

Set and let be an appropriate constant such that

Therefore, from the convexity of using (3.2), (3.14) and Lemma 2.8 we have

It follows that

By using condition (C1), (C4) and (3.12), we have

(3.15)

From (3.13) and (3.15), we obtain

(3.16)

From (3.11) and (3.13), we also obtain

(3.17)

Step 4. We shall show that

where q is the unique solution of the variational inequality

Let Observe that

is a contraction. Indeed, for all, and we have

Banach’s Contraction Mapping Principle guarantees that has a unique fixed point, say That is,

by (1.1) we obtain that for all

Next, we show that

To see this, we choose a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to p. Without loss of generality, we can assume that Claim that

First, we prove.

Since we have

for all It follows from Lemma 2.6 (A2) that

and hence

Since and together with the lower semicontinuity of and Lemma 2.6 (A4), we have for all For t with and let Since and we have and hence

So, from Lemma 2.6 (A1), (A4) and the convexity of we have

Dividing by t, we get

Letting it follows from Lemma 2.6 (A3) and the lower semicontinuity of that for all and hence Next, we prove To see this, we observe that we may assume (by passing to a further subsequence if necessary) . Let K be the K-mapping generated by and Then by Lemma 2.5, we have, for every

(3.18)

everyMoreover, from Lemma 2.4 it follows that 

Suppose for contradiction. Then. Since Hilbert space are Opial’s spaces and

from (3.17) and (3.18), we have

which derives a contradiction. Thus, we have It follows from

that

Next, we prove Put

Since B is relaxed -cocoercive and condition (C5), we have

which yields that B is monotone. Thus T is maximal monotone. Let. Since and we have

On the other hand, from and (1.1), we have

and hence

It follows that

which together with (3.16), (3.17) and B is Lipschitz continuous implies that We have and hence That is,

It follows from the variational inequality for all

that

(3.19)

Using (3.16) and (3.19), we have

(3.20)

Moreover, from (3.15) and (3.19), we have

(3.21)

Step 5. Finally, we will show that the sequences and converge strongly to q.

Since using (1.19), (3.1), (3.2) and Lemma 2.8, we have

which implies that

Since and are bounded, we can take a constant such that

for all It then follows that

(3.22)

where

By using (3.20), (3.21) and condition (C1), we get

Now applying Lemma 2.1 to (3.22) concludes that as Finally, noticing

we also conclude that as This completes the proof.

4. Applications

In this section, by Theorem 3.1, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows:

Let for all and setting and in Theorem 3.1, we obtain the following result.

Corollary 4.1. Let H be a real Hilbert space, C a nonempty closed convex subset of H, F a bifunction from to which satisfies (A1)-(A4), a proper lower semicontinuous and convex function and a finite family of nonexpansive mappings of C into H such that the common fixed points set Assume that either (B1) or (B2) holds and is an arbitrary point in C. Let and be sequences generated by and

where, , , satisfying the conditions (C1)-(C5) in Theorem 3.1. Then, and converge strongly to a point

where

Setting and for all n in Theorem 3.1, we obtain the following result.

Corollary 4.2. Let H be a real Hilbert space, C a nonempty closed convex subset of H, F a bifunction from to which satisfies (A1)-(A4), a proper lower semicontinuous and convex function and a finite family of nonexpansive mappings of C into H such that the common fixed points set Let K_n and K be the K-mappings defined by (1.16) and (1.17), respectively. Assume that either (B1) or (B2) holds and x is an arbitrary point in C. Let and be sequences generated by and

where are real numbers such that for every and

and, , satisfying the conditions (C1), (C3), (C4) and (C6) in Theorem 3.1. Then, and converge strongly to a point

where

Finally as applications, we will utilize the results presented in this paper to study the following optimization problem:

(4.1)

where C is a nonempty bounded closed convex subset of a Hilbert space and is a proper lower semicontinuous and convex function. We denote by the set of solutions in (4.1). Let for all in Corollary 4.1, then 

It follows from Corollary 4.1 that the sequence generated by and,

(4.2)

where, , and satisfying the conditions (C1)-(C5) in Theorem 3.1. Then the sequence converges strongly to a point 

where

Let for all and for all in Corollary 4.2, then It follows from Corollary 4.2 that the iterative sequence generated by and,

(4.3)

where, and satisfying the conditions (C1), (C3) and (C4) in Theorem 3.1. Then the sequence converges strongly to a point where

Remark 4.3. The algorithms (4.2) and (4.3) are variants of the proximal method for optimization problems introduced and studied by Martinet [41], Rockafellar [42], Ferris [43] and many others.

5. Acknowledgements

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The author is extremely grateful to the referees for useful suggestions that improved the contents of the paper.

REFERENCES

NOTES

^*This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.