1 Introduction

Let C be a closed convex subset of a real Hilbert space H, and let G : C × C → ℝ be a bifunction. We know that the equilibrium problem for a bifunction G is to find xC such that

(1.1)

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, zEP(G) if and only if 〈Tz, y - z〉 ≥ 0 for all yC, 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 uC such that

(1.2)

for all vC. The set of solutions of the variational inequality is denoted by V I(C, A). Now, we consider the following generalized equilibrium problem:

(1.3)

The set of such zC is denoted by EP(G, A), i.e.,

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, [1]-[3]).

A mapping A of C into H is called inverse-strongly monotone (see [4]), if there exists a positive real number α such that

for all x, yC.

A mapping T with domain D(T) and range R(T) is called nonexpansive if

(1.4)

for all x, yD(T) and T is said to be κ-strict pseudo-contration if there exist κ ∈ [0, 1) such that

(1.5)

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

(1.6)

T is pseudo-contractive if and only if

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 CH is bounded, closed and convex and T is a nonexpansive mapping of C into itself, then F(T) is nonempty; for instance, see [5]. Recently, Tada and Takahashi [6] and Takahashi and Takahashi [7] considered iterative methods for finding an element of EP(G) ∩ F(T). Browder and Petryshyn [8] showed that if a κ-strict pseudo-contraction T has a fixed point in C, then starting with an initial x0C, the sequence {x n } generated by the recursive formula:

(1.7)

where α is a constant such that 0 < α < 1, converges weakly to a fixed point of T. Marino and Xu [9] extended Browder and Petryshyn's above mentioned result by proving that the sequence {x n } generated by the following Manns algorithm [10]:

(1.8)

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. [11] 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:

(1.9)

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, ∀xC, S : CC is a κ-strict pseudo-contraction, and A, B : CH 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 ∈ 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. [11], 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 u, x1C, let {x n } be generated by

For i = 1, 2, ..., N, let F i : C × C → ℝ 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, ∀yC 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)

(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 xH, P C x satisfies the property:

The following characterizes the projection P C .

Lemma 2.1 [5]. Given xH and yC. Then, P C x = y if and only if there holds the inequality

Lemma 2.2 [12]. Let {s n } be a sequence of nonnegative real number satisfying

where {α n }, {β n } satisfy the conditions

Then limn→∞s n = 0.

Lemma 2.3 [13]. Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : 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 xC is well defined, nonexpansive and hold.

Lemma 2.4 [14]. 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 [15]. Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n } be a sequence in 0[1] with 0 < lim infn→∞β n ≤ lim supn→∞β n < 1.

Suppose

for all integer n ≥ 0 and

Then limn→∞||x n - z n || = 0.

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

(A 1) F(x, x) = 0 ∀xC;

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

(A 3) ∀x, y, zC,

(A 4) ∀xC, yF(x, y) is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

Lemma 2.6 [1]. Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C intosatisfying (A 1) - (A 4). Let r > 0 and xH. Then, there exists zC such that

(2.1)

for all xC.

Lemma 2.7 [16]. Assume that F : C × C → ℝ satisfies (A 1) - (A 4). For r > 0 and xH, define a mapping T r : HC as follows.

for all zH. Then, the following hold.

(1) T r is single-valued,

(2) T r is firmly nonexpansive i.e

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

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

Definition 2.1 [17]. 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.

(2.3)

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

Lemma 2.8 [17]. 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 [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 ∈ ℕ, let S n be S-mapping generated by T n , ..., T1 and α n , αn-1, ..., α1. Then, for every xC and k ∈ ℕ, limn→∞U n , k x exists.

Proof. Let xC and . Fix k ∈ ℕ, then for every n ∈ ℕ with nk,

we have

It follows that

(2.4)

where and

For any k, n, p ∈ ℕ, p > 0, nk, we have

(2.5)

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

For every k ∈ ℕ and xC, we define mapping U∞,kand S : CC as follows:

(2.6)

and

(2.7)

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

Remark 2.11. For each n ∈ ℕ, S n is nonexpansive and limn→∞supxD||S n x - Sx|| = 0 for every bounded subset D of C. To show this, let x, yC and D be a bounded subset of C. Then, we have

Then, we have that S : CC is also nonexpansive indeed, observe that for each x, yC

By (2.8), we have

This implies that for m > n and xD,

By letting m → ∞, for any xD, we have

(2.8)

It follows that

(2.9)

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 ∈ 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 ∈ ℕ, with nk, let x0F (S) and , we have

(2.10)
(2.11)
(2.12)
(2.13)

For k ∈ ℕ and (2.12), we have

(2.14)

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

Again by (2.12), we have

(2.15)

as n → ∞. Hence

(2.16)

From U∞,kx0 = x0, ∀k ∈ ℕ, and (2.15), we obtain that T k x0 = x0, ∀k ∈ ℕ. 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

(2.17)

Hence, x* = P C (I - λ i A i )x* = G i (x*), ∀xC, 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

(2.18)

Hence, 〈A i x*, x - x*〉 ≥ 0, ∀xC, 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 → ℝ 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, ∀yC 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 ∈ 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 ∈ ℕ, i = 1, 2, ..., N, let {x n } and be generated by x1, uC and

(3.1)

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, yC, we have

(3.2)

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.

Step 1. We shall show that the sequence {x n } is bounded. Since

(3.3)

we have

By Lemma 2.7, we have .

Let . Then F(z, y) + 〈y - z, A i z〉 ≥ 0 ∀yC, 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 zF(B). Setting e n = a n S n x n + b n Bx n + c n y n , ∀n ∈ ℕ, we have

(3.4)

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

(3.5)

From definition of d n , we have

(3.6)

By definition of e n , we have

(3.7)

By (3.6) and (3.7), we have

(3.8)
(3.9)

It follows that

(3.10)
(3.11)

From Remark 2.11 and conditions (i)-(iii), we have

(3.12)

From (3.5), (3.12) and Lemma 2.5, we have

(3.13)

We can rewrite (3.5) as

(3.14)

By (3.13) and (3.14), we have

(3.15)

Step. 3. Show that limn→∞||x n - e n || = 0. From (3.1), we have

It implies that

By conditions (i), (ii), and (3.15), we have

(3.16)

Step. 4. We show that lim supn→∞u - z, x n - z〉 ≤ 0, where . Let be a subsequence of {x n } such that

(3.17)

Without loss of generality, we may assume that converges weakly to some q in H. Next, we will show that

(3.18)

First, we define a mapping A : CC by

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

Next, we define Q : CC by

(3.19)

Again, by Lemma 2.3, we have

By (3.19), we have

(3.20)

By condition (iii), (3.20), and (2.11), we have

(3.21)

Since

by (3.16) and (3.21), we have

(3.22)

From, (3.22), we have

(3.23)

By Lemma 2.4, we obtain that

(3.24)

From (3.17)

(3.25)

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

By nonexpansiveness of S n and B, we can show that ||e n - z|| ≤ ||x n - z||. Then,

It follows that

(3.26)

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 → ℝ 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 ∈ 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∈ ℕ, i = 1, 2, ..., N, let {x n } and be generated by x1, uC and

(4.1)

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 = yyC. Then, we get Bx n = x n 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 [11] as follows:

Theorem 4.2. Let C be a closed convex subset of Hilbert space H and let F : C × C → ℝ 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, ∀xC. Assume that . For every n ∈ ℕ, let {x n } and {v n } be generated by x1, uC and

(4.2)

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, ∀yC. Choose , a = a n , b = b n , c = c n for all n ∈ 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 □