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# Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions

Fixed Point Theory and Applications20112011:23

https://doi.org/10.1186/1687-1812-2011-23

• Received: 21 February 2011
• Accepted: 29 July 2011
• Published:

## Abstract

In this article, we introduce a new mapping generated by an infinite family of κ i - strict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces.

## Keywords

• nonexpansive mappings
• strongly positive operator
• generalized equilibrium problem
• strict pseudo-contraction
• fixed point

## 1 Introduction

Let C be a closed convex subset of a real Hilbert space H, and let G : C × C be a bifunction. We know that the equilibrium problem for a bifunction G is to find x C such that
The set of solutions of (1.1) is denoted by EP(G). Given a mapping T : CH, let G(x, y) = 〈Tx, y - x〉 for all x, y . Then, z EP(G) if and only if 〈Tz, y - z〉 ≥ 0 for all y C, i.e., z is a solution of the variational inequality. Let A : CH be a nonlinear mapping. The variational inequality problem is to find a u C such that
for all v C. The set of solutions of the variational inequality is denoted by V I(C, A). Now, we consider the following generalized equilibrium problem:

In the case of A ≡ 0, EP(G, A) is denoted by EP(G). In the case of G ≡ 0, EP(G, A) is also denoted by V I(C, A). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics reduce to find a solution of (1.3) (see, for instance, -).

A mapping A of C into H is called inverse-strongly monotone (see ), if there exists a positive real number α such that

for all x, y C.

A mapping T with domain D(T) and range R(T) is called nonexpansive if
for all x, y D(T) and T is said to be κ-strict pseudo-contration if there exist κ [0, 1) such that
We know that the class of κ-strict pseudo-contractions includes class of nonexpansive mappings. If κ = 1, then T is said to be pseudo-contractive. T is strong pseudo-contraction if there exists a positive constant λ (0, 1) such that T + λI is pseudo-contractive. In a real Hilbert space H (1.5) is equivalent to
Then, T is strongly pseudo-contractive, if there exists a positive constant λ (0, 1) such that

The class of κ-strict pseudo-contractions fall into the one between classes of nonexpansive mappings and pseudo-contractions, and the class of strong pseudo-contractions is independent of the class of κ-strict pseudo-contractions.

We denote by F(T) the set of fixed points of T. If C H is bounded, closed and convex and T is a nonexpansive mapping of C into itself, then F(T) is nonempty; for instance, see . Recently, Tada and Takahashi  and Takahashi and Takahashi  considered iterative methods for finding an element of EP(G) ∩ F(T). Browder and Petryshyn  showed that if a κ-strict pseudo-contraction T has a fixed point in C, then starting with an initial x0 C, the sequence {x n } generated by the recursive formula:
where α is a constant such that 0 < α < 1, converges weakly to a fixed point of T. Marino and Xu  extended Browder and Petryshyn's above mentioned result by proving that the sequence {x n } generated by the following Manns algorithm :

converges weakly to a fixed point of T provided the control sequence satisfies the conditions that κ < α n < 1 for all n and .

Recently, in 2009, Qin et al.  introduced a general iterative method for finding a common element of EP(F, T), F(S), and F(D). They defined {x n } as follows:

where the mapping D : CC is defined by D(x) = P C (P C (x - ηBx) - λAP C (x - ηBx)), S k is the mapping defined by S k x = kx + (1 - k)Sx, x C, S : CC is a κ-strict pseudo-contraction, and A, B : C H are a-inverse-strongly monotone mapping and b-inverse-strongly monotone mappings, respectively. Under suitable conditions, they proved strong convergence of {x n } defined by (1.9) to z = PEP(F, T)∩F(S) ∩F(D)u.

Let C be a nonempty convex subset of a real Hilbert space. Let T i , i = 1, 2, ... be mappings of C into itself. For each j = 1, 2, ..., let where I = [0, 1] and . For every n , we define the mapping S n : CC as follows:

This mapping is called S-mapping generated by T n , ..., T1 and α n , αn-1, ..., α1.

Question. How can we define an iterative method for finding an element in ?

In this article, motivated by Qin et al. , by using S-mapping, we introduce a new iteration method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Our iteration scheme is define as follows.

For i = 1, 2, ..., N, let F i : C × C be bifunction, A i : CH be α i -inverse strongly monotone and let G i : CC be defined by G i (y) = P C (I - λ i A i )y, y C with (0, 1] (0, 2 α i ) such that , where B is the K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N .

We prove a strong convergence theorem of purposed iterative sequence {x n } to a point and z is a solution of (1.10)

## 2 Preliminaries

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

Let C be a closed convex subset of a real Hilbert space H, and let P C be the metric projection of H onto C i.e., so that for x H, P C x satisfies the property:

The following characterizes the projection P C .

Then limn→∞s n = 0.

Lemma 2.3 . Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n } be a sequence of nonexpansive mappings on C. Suppose is nonempty. Let {λ n } be a sequence of positive numbers with . Then, a mapping S on C defined by

for x C is well defined, nonexpansive and hold.

Lemma 2.4 . Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC be a nonexpansive mapping. Then, I - S is demi-closed at zero.

Lemma 2.5 . Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n } be a sequence in 0 with 0 < lim infn→∞β n ≤ lim supn→∞β n < 1.

Then limn→∞||x n - z n || = 0.

For solving the equilibrium problem for a bifunction F : C × C, let us assume that F satisfies the following conditions:

(A 1) F(x, x) = 0 x C;

(A 2) F is monotone, i.e. F(x, y) + F(y, x) ≤ 0, x, y C;

(A 4) x C, y F(x, y) is convex and lower semicontinuous.

The following lemma appears implicitly in .

Lemma 2.6 . Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into satisfying (A 1) - (A 4). Let r > 0 and x H. Then, there exists z C such that

for all x C.

Lemma 2.7 . Assume that F : C × C satisfies (A 1) - (A 4). For r > 0 and x H, define a mapping T r : HC as follows.

for all z H. Then, the following hold.

(1) T r is single-valued,

(3) F(T r ) = EP (F );

(4) EP(F) is closed and convex.

Definition 2.1 . Let C be a nonempty convex subset of real Banach space. Let be a finite family of nonexpanxive mappings of C into itself, and let λ1, ..., λ N be real numbers such that 0 ≤ λ i ≤ 1 for every i = 1, ..., N . We define a mapping K : CC as follows.

Such a mapping K is called the K-mapping generated by T1, ..., T N and λ1, ..., λ N .

Lemma 2.8 . Let C be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpanxive mappings of C into itself with and let λ1, ..., λ N be real numbers such that 0 < λ i < 1 for every i = 1, ..., N - 1 and 0 < λ N ≤ 1. Let K be the K-mapping generated by T1, ..., T N and λ1, ..., λ N . Then .

Lemma 2.9 . Let C be a nonempty closed convex subset of a real Hilbert space H and S : CC be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition.

Lemma 2.10. Let C be a nonempty closed convex subset of a real Hilbert space. Let be κ i -strict pseudo-contraction mappings of C into itself with and κ = sup i κ i and let , where I = [0, 1], , , and for all j = 1, 2, .... For every n , let S n be S-mapping generated by T n , ..., T1 and α n , αn-1, ..., α1. Then, for every x C and k , limn→∞U n , k x exists.

Proof. Let x C and . Fix k , then for every n with nk,

where and Since a (0, 1), we have limn→∞a n = 0. From (2.5), we have that {U n , k x} is a Cauchy sequence. Hence lim n→∞Un,kx exists. □

Such a mapping S is called S-mapping generated by T n , Tn-1, ... and α n , αn- 1, ...

Lemma 2.12. Let C be a nonempty closed convex subset of a real Hilbert space. Let be κ i -strict pseudo-contraction mappings of C into itself with and κ = supiκ i and let , where I = [0, 1], , and for all j = 1, .... For every n , let S n and S be S-mappings generated by T n , ..., T1 and α n , αn-1, ..., α1 and T n , Tn-1, ..., and α n , αn-1, ..., respectively. Then .

Proof. It is evident that . For every n, k , with nk, let x0 F (S) and , we have

as n → ∞. This implies that U, k x0 = x0, k .

From U∞,kx0 = x0, k , and (2.15), we obtain that T k x0 = x0, k . This implies that . □

Lemma 2.13. Let C be a closed convex subset of Hilbert space H. Let A i : CH be mappings and let G i : CC be defined by G i (y) = P C (I - λ i A i )y with λ i > 0, i = 1, 2, ... N. Then if and only if .

Proof. For given , we have x* VI(C, A i ), i = 1, 2, ..., N. Since 〈A i x*, x - x*〉 ≥ 0, we have 〈λ i A i x*, x - x*〉 ≥ 0, λ i > 0, i = 1, 2, ..., N. It follows that
Hence, x* = P C (I - λ i A i )x* = G i (x*), x C, i = 1, 2, ..., N. Therefore, we have . For the converse, let ; then, we have for every i = 1, ..., N, x* = G i (x*) = P C (I - λ i A i )x*, λ i > 0, i = 1, 2, ..., N. It implies that

Hence, 〈A i x*, x - x*〉 ≥ 0, x C, so x* VI(C, A i ), i = 1, 2, ..., N. Hence, .

## 3 Main results

Theorem 3.1. Let C be a closed convex subset of Hilbert space H. For every i = 1, 2, ..., N, let F i : C × C be a bifunction satisfying (A1) - (A4), let A i : CH be α i -inverse strongly monotone and let G i : CC be defined by G i (y) = P C (I - λ i A i )y, y C with λ i (0, 1] (0, 2α i ). Let B : CC be the K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N where β i (0, 1), i = 1, 2, 3, ..., N - 1, β N (0, 1] and let be κ i -strict pseudo-contraction mappings of C into itself with κ = sup i κ i and let , where I = [0, 1], , , and for all j = 1, 2, ... . For every n , let S n and S are S-mapping generated by T n , ..., T1 and ρ n , ρn - 1, ..., ρ1 and T n , Tn- 1, ..., and ρ n , ρn - 1, ..., respectively. Assume that . For every n , i = 1, 2, ..., N, let {x n } and be generated by x1, u C and

where {α n }, {β n }, {γ n }, {a n }, {b n }, {c n } (0, 1), , and , satisfy the following conditions:

(i) and ,

(ii) ,

(iii) , , , with a, b, c (0, 1).

Then, the sequence {x n }, {y n }, , i = 1, 2, ..., N, converge strongly to and z is a solution of (1.10).

Proof. First, we show that (I - λ i A i ) is nonexpansive mapping for every i = 1, 2, ..., N. For x, y C, we have

Thus, (I - λ i A i ) is nonexpansive, and so are B and G i , for all i = 1, 2, ..., N.

Now, we shall divide our proof into five steps.

By Lemma 2.7, we have .

Let . Then F(z, y) + 〈y - z, A i z〉 ≥ 0 y C, so we have
Again by Lemma 2.7, we have , i = 1, 2, ..., N. Since B is K-mapping generated by G1, G2, ..., G N and β1, β2, ..., β N and . By Lemma 2.8, we have . Since , we have z F(B). Setting e n = a n S n x n + b n Bx n + c n y n , n , we have

By induction, we can prove that {x n } is bounded, and so are , {y n }, {Bx n } {S n x n }, {e n }.

Step 2. We will show that limn→∞||xn+1- x n || = 0. Let , and then we have
Step. 4. We show that lim supn→∞u - z, x n - z〉 ≤ 0, where . Let be a subsequence of {x n } such that
Without loss of generality, we may assume that converges weakly to some q in H. Next, we will show that

Since , we have . By Lemma 2.3, we have .

Step. 5. Finally, we show that limn→∞x n = z, where .

From Step 4, (3.26), and Lemma 2.2, we have limn→∞ x n = z, where . The proof is complete. □

## 4 Applications

From Theorem 3.1, we obtain the following strong convergence theorems in a real

Hilbert space:

Theorem 4.1. Let C be a closed convex subset of Hilbert space H. For every i = 1, 2, ..., N, let F i : C × C be a bifunction satisfying (A1) - (A4) and let be κ i -strict pseudo-contraction mappings of C into itself with κ = sup i κ i and let , where I = [0, 1], , , and for all j = 2, ... .. For every n , let S n and S are S-mappings generated by T n , ..., T1 and ρ n , ρn - 1, ..., ρ1 and T n , Tn- 1, ..., and ρ n , ρn- 1, ..., respectively. Assume that . For every n , i = 1, 2, ..., N, let {x n } and be generated by x1, u C and

where {α n }, {β n }, {γ n }, {a n }, {b n }, {c n } (0, 1), , and , satisfy the following conditions:

(i) and ,

(ii) ,

(iii) , , , with a, b, c (0, 1),

Then, the sequence {x n }, {y n }, , i = 1, 2, ..., N, converge strongly to , and z is solution of (1.10)

Proof. From Theorem 3.1, let A i ≡ 0; then we have G i (y) = P Cy = y y C. Then, we get Bx n = x n n . Then, from Theorem 3.1, we obtain the desired conclusion. □

Next theorem is derived from Theorem 3.1, and we modify the result of  as follows:

Theorem 4.2. Let C be a closed convex subset of Hilbert space H and let F : C × C be a bifunction satisfying (A1)-(A4), let A : CH be α-inverse strongly monotone mapping, and let T be κ-strict pseudo-contraction mappings of C into itself. Define a mapping T κ by T κ x = κx + (1 - κ)Tx, x C. Assume that . For every n , let {x n } and {v n } be generated by x1, u C and

where {α n }, {β n }, {γ n }, {a, b, c} (0, 1), α n + β n + γ n = a + b + c = 1, and {r, λ} (ς, τ) (0, 2α) satisfy the following conditions:

(i) and ,

(ii) ,

Then, the sequence {x n } and {v n } converge strongly to .

Proof. From Theorem 3.1, choose N = 1 and let A1 = A, λ1 = λ. Then, we have B(y) = G1(y) = P C (I - λA)y, y C. Choose , a = a n , b = b n , c = c n for all n , and let T κ S1 : CC be S-mapping generated by T1 and ρ1 with T1 = T and , and then we obtain the desired result from Theorem 3.1 □

## Declarations

### Acknowledgements

The authors would like to thank Professor Dr. Suthep Suantai for his valuable suggestion in the preparation and improvement of this article.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok, 10520, Thailand

## References

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