Some results on variational inequalities and generalized equilibrium problems with applications

Some results on variational inequalities and generalized equilibrium problems with applications

Xiaolong Qin^I; Sun Young Cho^II; Shin Min Kang^III

^IDepartment of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

^IIDepartment of Mathematics, Gyeongsang National University, Jinju 660-701, Korea

^IIIDepartment of Mathematics and the RINS, Gyeongsang National University Jinju 660-701, Korea. E-mail: [e-mail]

ABSTRACT

An iterative algorithm is considered for variational inequalities, generalized equilibrium problems and fixed point problems. Strong convergence of the proposed iterative algorithm is obtained in the framework Hilbert spaces.

Mathematical subject classification: 47H05, 47H09, 47J25, 47N10.

Keywords: generalized equilibrium problem, variational inequality, fixed point, nonexpansive mapping.

1 Introduction and preliminaries

Let H be a real Hilbert space, whose inner product and norm are denoted by ‹·, ·› and ║·║, respectively. Let C be a nonempty closed and convex subset of H and P_C be the projection of H onto C.

Let ƒ, S, A, T be nonlinear mappings. Recall the following definitions:

(1) ƒ : C → C is said to be α-contractive if there exists a constant α ∈ (0,1) such that

(2) S : C → C is said to be nonexpansive if

Throughout this paper, we use F(S) to denote the set of fixed points of the mapping S.

(3) A : C → H is said to be monotone if

(4) A : C → H is said to be inverse-strongly monotone if there exists δ > 0 such that

Such a mapping A is also called δ-inverse-strongly monotone. We know that if S : C → C is nonexpansive, then A = I – S is -inverse-strongly monotone; see [1, 21] for more details.

(5) A set-valued mapping T : H → 2^H is said to be monotone if for all x, y ∈ H, ƒ ∈ Tx and g ∈ Ty ⇒ ‹x – y, ƒ – g› > 0. A monotone mapping T : H → 2^H is maximal if the graph of G(T) of T is not properly contained in the graph of any other monotone mapping. It isknown that a monotone mapping T is maximal if and only if for (x, ƒ) ∈ H × H, ‹x – y, ƒ – g› > 0 for every (y, g) ∈ G(T) implies that ƒ ∈ Tx. Let A be a monotone mapping of C into H and let N_Cυ be the normal cone to C at υ ∈ C, i.e., N_Cυ = {w ∈ H : ‹υ – u, w› > 0, ∀u ∈ C} and define

Then T is maximal monotone and 0 ∈ Tυ if and only if ‹Aυ, u – υ› > 0, ∀u ∈ C; see [16] for more details.

Recall that the classical variational inequality is to find an x ∈ C such that

In this paper, we use VI (C, A) to denote the solution set of the variational inequality (1.1). For given z ∈ H and u ∈ C, we see that

holds if and only if u = P_Cz. It is known that projection operator P_C satisfies

One can see that the variational inequality (1.1) is equivalent to a fixed point problem. An element u ∈ C is a solution of the variational inequality (1.1) if and only if u ∈ C is a fixed point of the mapping P_C(I – λA)u, where λ > 0 is a constant and I is the identity mapping. This can be seen from the following. u ∈ C is a solution of the variational inequality (1.1), this is,

which is equivalent to

where λ > 0 is a constant. This implies from (1.2) that u = P_C(I – λA)u, that is, u is a fixed point of the mapping P_C(I – λA). This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Let A : C → H be a δ-inverse-strongly monotone mapping and F be a bifunction of C × C into , where denotes the set of real numbers. We consider the following generalized equilibrium problem:

In this paper, the set of such an x ∈ C is denoted by EP(F, A), i.e.,

Next, we give some special cases of the generalized equilibrium problem (1.3).

(I) If A ≡ 0, the zero mapping, then the problem (1.3) is reduced to the following equilibrium problem:

In this paper, the set of such an x ∈ C is denoted by EP(F), i.e.,

(II) If F ≡ 0, then the problem (1.3) is reduced to the classical variational inequality (1.1).

In 2005, Iiduka and Takahashi [8] considered the classical variational inequality (1.1) and a single nonexpansive mapping. To be more precise, they obtained the following results.

Theorem IT. Let C be a closed convex subset of a real Hilbert space H. Let A be anα-inverse-strongly monotone mapping of C into H and S be a nonexpansive mapping of C into itself such that F(S)∩VI(C, A) ≠ . Suppose that x₁ = x ∈ C and {x_n} is given by

where {α_n} is a sequence in [0,1) and {λ_n} is a sequence in [0,2α]. If {α_n} and {λ_n} are chosen so that {λ_n} ⊂ [a, b] for some a, b with 0 < a < b < 2α,

then {x_n} converges strongly to P_{F(S)∩VI(C, A)}x.

On the other hand, we see that the problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others; see, for instance, [2, 5, 9]. Recently, many authors considered iterative methods for the problems (1.3) and (1.4), see [3-7, 11-15, 18, 20, 22, 24] for more details.

To study the equilibrium problems (1.3) and (1.4), we may assume that F satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) < 0 for all x, y ∈ C;

(A3) for each x, y, z ∈ C,

(A4) for each x ∈ C, y → F(x, y) is convex and weakly lower semi-continuous.

Put (x, y) = F(x, y) + ‹Ax, y – x› for each x, y ∈ C. It is not hard tosee that also confirms (A1)-(A4).

In 2007, Takahashi and Takahashi [20] introduced the following iterative method

where ƒ is a α-contraction, T is a nonexpansive mapping. They considered the problem of approximating a common element of the set of fixed points of a single nonexpansive mapping and the set of solutions of the equilibrium problem (1.4). Strong convergence theorems of the iterative algorithm (1.6) are established in a real Hilbert space.

Recently, Takahashi and Takahashi [22] further considered the generalized equilibrium problem (1.3). They obtained the following result in a real Hilbert space.

Theorem TT. Let C be a closed convex subset of a real Hilbert space H and F : C × C → be a bi-function satisfying (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into H and S be a non-expansive mapping of C into itself such that F(S) ∩ EP(F, A) ≠ . Let u ∈ C and x₁∈ C and let {z_n} ⊂ C and {x_n} ⊂ C be sequences generated by

where {α_n} ⊂ [0,1], {β_n} ⊂ [0,1] and {r_n} ⊂ [0,2α] satisfy

Then, {x_n} converges strongly to z = P_{F(S)∩EP(F, A)}u.

Very recently, Chang, Lee and Chan [5] introduced a new iterative method for solving equilibrium problem (1.4), variational inequality (1.1) and the fixed point problem of nonexpansive mappings in the framework of Hilbert spaces. More precisely, they proved the following theorem.

Theorem CLC. Let H be a real Hilbert space, C be a nonempty closed convex subset of H and F be a bifunction satisfying the conditions (A1)-(A4). Let A : C → H be an α-inverse-strongly monotone mapping and {S_i : C → C} be a family of infinitely nonexpansive mappings with F ∩ VI(C, A) ∩ EP(F) ≠ , where F := F(S_i) and ƒ : C → C be a ξ-contractive mapping. Let {x_n}, {y_n} {k_n} and {u_n} be sequences defined by

where {W_n : C → C} is the sequence defined by (1.9), {α_n}, {β_n} and {γ_n} are sequences in [0,1], {λ_n} is a sequence in [a, b] ⊂ (0,2α) and {r_n} is a sequence in (0, ∞). If the following conditions are satisfied:

(1) α_n + β_n+ γ_n = 1;

(2) lim_n→∞α_n = 0; α_n = ∞;

(3) 0 < lim inf_n→∞β_n< lim sup_n→∞β_n < 1;

(4) lim inf_n→∞r_n > 0; |r_n+1 – r_n| < ∞;

(5) lim_n→∞|λ_n+1 – λ_n| = 0,

then {x_n} and {u_n} converge strongly to z ∈ F ∩ VI(C, A) ∩ EP(F).

In this paper, motivated and inspired by the research going on in this direction, we introduce a general iterative method for finding a common element of the set of solutions of generalized equilibrium problems, the set of solutions of variational inequalities, and the set of common fixed points of a family ofnonexpansive mappings in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results of Ceng and Yao [3, 4], Chang Lee and Chan [5], Iiduka and Takahashi [8], Qin, Shang and Zhou [12], Su, Shang and Qin [18], Takahashi and Takahashi [20, 22], Yao and Yao [25] and many others.

In order to prove our main results, we need the following definitions and lemmas.

A space X is said to satisfy Opial condition [10] if for each sequence {x_n} in X which converges weakly to point x ∈ X, we have

Lemma 1.1 ([2]). Let C be a nonempty closed convex subset of H and F : C × C → be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that

Lemma 1.2 ([2], [7]). Suppose that all the conditions in Lemma 1.1 are satisfied. For any give r > 0 define a mapping T_r : H → C as follows:

then the following conclusions hold:

(1) T_r is single-valued;

(2) T_r is firmly nonexpansive, i.e., for any x,y ∈ H,

(3) F(T_r) = EP(F);

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

Lemma 1.3 ([23]). Assume that {α_n} is a sequence of nonnegative real numbers such that

where {γ_n} is a sequence in (0,1) and {δ_n} is a sequence such that

(1)

γ_n = ∞;

(2) lim sup_n→∞δ_n/γ_n < 0 or|δ_n| < ∞.

Then lim_n→∞α_n = 0.

Definition 1.4 ([19]). Let {S_i : C → C} be a family of infinitely nonexpansive mappings and {γ_i} be a nonnegative real sequence with 0 < γ_i < 1, ∀i > 1. For n > 1 define a mapping W_n: C → C as follows:

Such a mapping W_n is nonexpansive from C to C and it is called a W-mapping generated by S_n, S_n-1, ..., S₁ and γ_n, γ_n-1, ..., γ₁.

Lemma 1.5 ([19]). Let C be a nonempty closed convex subset of a Hilbert space H, {S_i : C → C} be a family of infinitely nonexpansive mappings with F(S_i) ≠ and {γ_i} be a real sequence such that 0 < γ_i< l < 1, ∀i > 1. Then

(1) W_n is nonexpansive and F(W_n) = F(S_i), for each n > 1;

(2) for each x ∈ C and for each positive integer k, the limit lim_n→∞U_n,k exists.

(3) the mapping W : C → C defined by

is a nonexpansive mapping satisfying F(W) = F(S_i) and it is called the W-mapping generated by S₁, S₂, ... andγ₁, γ₂, ....

Lemma 1.6 ([5]). Let C be a nonempty closed convex subset of a Hilbert space H, {S_i : C → C} be a family of infinitely nonexpansive mappings with F(S_i) ≠ and {γ_i} be a real sequence such that 0 < γ_i< l < 1, ∀i > 1. If K is any bounded subset of C, then

Throughout this paper, we always assume that 0 < γ_i< l < 1, ∀i > 1.

Lemma 1.7 ([17]). Let {x_n} and {y_n} be bounded sequences in a Hilbert space H and {β_n} be a sequence in [0,1] with

Suppose that x_n+1 = (1 – β_n)y_n + β_nx_n for all n > 0 and

Then lim_n→∞║y_n – x_n║ = 0.

2 Main results

Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let A₁ : C → H be a δ₁-inverse-strongly monotone mapping, A₂ : C → H be a δ₂-inverse-strongly monotone mapping, A₃ : C → H be a δ₃-inverse-strongly monotone mapping and {S_i: C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F, A₃) ∩ VI ≠ , where FP = F(S_i) and VI = VI(C, A₁) ∩ VI(C, A₂). Let ƒ : C → C be an α-contraction. Let x₁∈ C and {x_n} be a sequence generated by

where {W_n : C → C} is the sequence generated in (1.9), {α_n}, {β_n} and {γ_n} are sequences in (0,1) such that α_n + β_n + γ_n = 1 for each n > 1 and {r_n}, {λ_n} and {η_n} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

(R1) 0 < a < η_n< b < 2δ₁, 0 < a' < λ_n< b' < 2δ₂, 0 < < r_n < < 2δ₃, ∀n > 1;

(R2) lim_n→∞α_n = 0 and α_n = ∞;

(R3) 0 < lim inf_n→∞β_n< lim sup_n→∞β_n < 1;

(R4) lim_n→∞(λ_n– λ_n+1) = lim_n→∞(η_n – η_n+1) = lim_n→∞(r_n – r_n+1) = 0.

Then the sequence {x_n} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:

Proof. First, we show, for each n > 1, that the mappings I – η_nA₁, I – λ_nA₂ and I – r_nA₃ are nonexpansive. Indeed, for ∀x, y ∈ C, we obtain from the restriction (R1) that

which implies that the mapping I – η_nA₁ is nonexpansive, so are I – λ_nA₂and I – r_nA₃ for each n > 1. Note that u_n can be re-written as u_n = (I – r_nA₃)x_n for each n > 1. Take x* ∈ Ω. Noticing that x* = P_C(I – η_nA₁)x* = P_C(I - λ_nA₂)x* = (I – r_nA₃)x*, we have

On the other hand, we have

It follows from (2.2) and (2.3) that

which yields that

From the algorithm (2.1) and (2.5), we arrive at

By simple inductions, we obtain that

which gives that the sequence {x_n} is bounded, so are {y_n}, {z_n} and {u_n}. Without loss of generality, we can assume that there exists a bounded set K ⊂ C such that

Notice that u_n+1 = (I - r_n+1A₃)x_n+1 and u_n = (I – r_nA₃)x_n, wesee from Lemma 1.2 that

and

Let y = u_n in (2.7) and y = u_n+1 in (2.8). By adding up these two inequalities and using the assumption (R2), we obtain that

Hence, we have

This implies that

It follows that

where M₁ is an appropriate constant such that

From the nonexpansivity of P_C, we also have

Substituting (2.9) into (2.10), we arrive at

In a similar way, we can obtain that

Combining (2.11) with (2.12), we see that

where M₂ is an appropriate constant such that

Letting

we see that

It follows that

On the other hand, we have

where K is the bounded subset of C defined by (2.6). Substituting (2.13) into (2.16), we arrive at

which combines with (2.15) yields that

In view of the restriction (R2), (R3) and (R4), we obtain from Lemma 1.6 that

Hence, we obtain from Lemma 1.7 that

In view of (2.14), we have

Thanks to the restriction (R3), we see that

For any x* ∈ Ω, we see that

Note that

Substituting (2.19) into (2.18), we arrive at

This implies that

By virtue of the restrictions (R1) and (R2), we obtain from (2.17) that

Next, we show that

Indeed, by using (2.18), we obtain that

On the other hand, we have

Substituting (2.23) into (2.22), we arrive at

This in turn gives that

In view of the restrictions (R1) and (R2), we obtain from (2.17) that (2.21) holds.

On the other hand, we see from (2.22) that

It follows that

This implies that

In view of the restrictions (R1), (R2) and (R3), we see from (2.17) that

On the other hand, we see from Lemma 1.2 that

This in turn implies that

Combining (2.24) with (2.26), we arrive at

It follows that

Thanks to the restrictions (R2) and (R3), we see from (2.17) and (2.25) that

In view of the firm nonexpansivity of P_C, we see that

which implies that

Substituting (2.28) into (2.22), we arrive at

from which it follows that

In view of the restrictions (R2) and (R3), we obtain from (2.17) and (2.21) that

In a similar way, we can obtain that

Note that

It follows that

In view of the restrictions (R2) and (R3), we obtain from (2.17) that

Notice that

From (2.27), (2.29), (2.30) and (2.31), we arrive at

Next, we prove that

where z = P_Ωƒ(z). To see this, we choose a subsequence {} of {x_n} such that

Since {} is bounded, there exists a subsequence {} of {} which converges weakly to w. Without loss of generality, we may assume that

w. On the other hand, we have

It follows from (2.27), (2.29) and (2.30) that

Therefore, we see that

w. First, we prove that w ∈ VI(C, A₁). For the purpose, let T be the maximal monotone mapping defined by:

For any given (x, y) ∈ G(T), hence y – A₁x ∈ N_C. Since y_n ∈ C, by the definition of N_C, we have

Notice that

It follows that

and hence

From the monotonicity of A₁, we see that

Since

w and A₁ is Lipschitz continuous, we obtain from (2.30) that ‹x – w, y› > 0. Notice that T is maximal monotone, hence 0 ∈ Tw. This shows that w ∈ VI(C, A₁). It follows from (2.27) and (2.29), we also have

Therefore, we obtain

w. Similarly, we can prove w ∈ VI(C, A₂). That is, w ∈ VI = VI(C, A₂) ∩ VI(C, A₁).

Next, we show that w ∈ FP = F(S_i). Suppose the contrary, w ∉ FP, i.e., Ww ≠ w. Since

w, we see from Opial condition that

On the other hand, we have

From Lemma 1.6, we obtain from (2.32) that lim_n→∞║Wy_n – y_n║ = 0, which combines with (2.36) yields that that

This derives a contradiction. Thus, we have w ∈ FP.

Next, we show that w ∈ EP(F, A₃). It follows from (2.27) that u_n w. Since u_n = (I - r A₃)x_n, for any y ∈ C, we have

From the assumption (A2), we see that

Replacing n by n_i, we arrive at

Putting y_t = ty + (1 – t)w for any t ∈ (0,1] and y ∈ C, we see that y_t ∈ C. It follows from (2.37) that

In view of the monotonicity of A₃, (2.27) and the restriction (R1), we obtain from the assumption (A4) that

From the assumptions (A1) and (A4), we see that

from which it follows that

It follows from the assumption (A3) that w ∈ EP(F, A₃). On the other hand, we see from (2.33) that

Finally, we show that x_n → z, as n → ∞. Note that

which implies that

From the restriction (R2), we obtain from Lemma 1.3 that lim_n→∞║x_n – z║ = 0. This completes the proof.

□

Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let A₁: C → H be a δ₁-inverse-strongly monotone mapping, A₂ : C → H be a δ₂-inverse-strongly monotone mapping and {S_i : C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F) ∩ VI ≠ , where FP = F(S_i) and VI = VI(C, A₁) ∩ VI(C, A₂). Let ƒ : C → C be an α-contraction. Let x₁∈ C and {x_n} be a sequence generated by

where {W_n : C → C} is the sequence generated in (1.9), {α_n}, {β_n} and {γ_n} are sequences in (0,1) such that α_n + β_n + γ_n = 1 for each n > 1 and {r_n}, {λ_n} and {η_n} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

(R1) 0 < a < η_n < b < 2δ₁, 0 < a' < λ_n < b' < 2δ₂, 0 < < r_n < < 2δ₃, ∀n > 1;

(R2) lim_n→∞α_n = 0 and α_n = ∞;

(R3) 0 < lim inf_n→∞β_n< lim sup_n→∞β_n < 1;

(R4) lim_n→∞(λ_n – λ_n+1) = lim_n→∞(η_n– η_n+1) = lim_n→∞(r_n – r_n+1) = 0.

Then the sequence {x_n} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:

Proof. Putting A₃≡ 0, we see that

for all δ ∈ (0, ∞). We can conclude from Theorem 2.1 the desired conclusion easily. This completes the proof.

□

Remark 2.3. If A₁≡ A₂ and λ_n≡ η_n, then Corollary 2.2 is reduced to Theorem 3.1 of Chang et al. [5]. If A₂≡ 0, ƒ(x) ≡ e ∈ C a arbitrary fixed point and S_i≡ I, the identity mapping, then Corollary 2.2 is reduced to Theorem 3.1 of Plubtieng and Punpaeng [11].

Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let A₁ : C → H be a δ₁-inverse-strongly monotone mapping, A₂ : C → H be a δ₂-inverse-strongly monotone mapping, A₃ : C→ H be a δ₃-inverse-strongly monotone mapping and {S_i : C → C} be a family of infinitely nonexpansive mappings. Assume that Ω := FP ∩ EP(F, A₃) ∩ VI ≠ , where FP = F(S_i) and VI = VI(C, A₁) ∩ VI(C, A₂). Let ƒ : C → C be an α-contraction. Let x₁∈ C and {x_n} be a sequence generated by

where {W_n : C → C} is the sequence generated in (1.9), {α_n}, {β_n} and {γ_n} are sequences in (0,1) such that α_n + β_n + γ_n = 1 for each n > 1 and {r_n}, {λ_n} and {η_n} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

(R1) 0 < a < η_n< b < 2δ₁, 0 < a' < λ_n< b' < 2δ₂, 0 < < r_n < < 2δ₃, ∀n > 1;

(R2) lim_n→∞α_n = 0 and α_n = ∞;

(R3) 0 < lim inf_n→∞β_n< lim sup_n→∞β_n < 1;

(R4) lim_n→∞(λ_n– λ_n+1) = lim_n→∞(η_n– η_n+1) = lim_n→∞(r_n – r_n+1) = 0.

Then the sequence {x_n} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:

Proof. Putting F ≡ 0, we see that

is equivalent to

This implies that

From the proof of Theorem 2.1, we can conclude the desired conclusion immediately. This completes the proof.

□

Remark 2.5. Corollary 2.4 includes Theorem 3.1 of Yao and Yao [25] as a special case, see [25] for more details.

As some applications of our main results, we can obtain the following results.

Recall that a mapping T : C → C is said to be a k-strict pseudo-contraction if there exists a constant k ∈ [0,1) such that

Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings.

Put A = I - T, where T : C → C is a k-strict pseudo-contraction. Then A is -inverse-strongly monotone; see [1] for more details.

Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C × C to which satisfies (A1)-(A4). Let T₁ : C → C be a k₁-inverse-strongly monotone mapping, T₂ : C → C be a k₂-inverse-strongly monotone mapping, T₃ : C → C be a k₃-inverse-strongly monotone mapping and {S_i: C → C} be a family of infinitely nonexpansive mappings. Assume that Ω: = FP ∩ EP(F, I – T₃) ∩ VI ≠ , where FP = F(S_i) and VI = F(T₁) ∩ F(T₂). Let ƒ : C → C be an α-contraction. Let x₁∈ C and {x_n} be a sequence generated by

where {W_n : C → C} is the sequence generated in (1.9), {α_n}, {β_n} and {γ_n} are sequences in (0,1) such that α_n+ β_n + γ_n = 1 for each n > 1 and {r_n}, {λ_n} and {η_n} are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

(R1) 0 < a < η_n< b < (1 – k₁), 0 < a' < λ_n< b' < (1 – k₂), 0 < < r_n < < (1 – k₃), ∀n > 1;

(R2) lim_n→∞α_n = 0 and α_n = ∞;

(R3) 0 < lim inf_n→∞β_n< lim sup_n→∞β_n < 1;

(R4) lim_n→∞(λ_n – λ_n+1) = lim_n→∞(η_n – η_n+1) = lim_n→∞(r_n – r_n+1) = 0.

Then the sequence {x_n} converges strongly to z ∈ Ω, which solves uniquely the following variational inequality:

and

□

3 Conclusion

The iterative process (2.1) presented in this paper which can be employed to approximate common elements in the solution set of the generalized equilibrium problem (1.3), in the solution set of the classical variational inequality (1.1) and in the common fixed point set of a family nonexpansive mappings is general.It is of interest to improve the main results presented in this paper to the framework of real Banach spaces.

Acknowledgments. The authors are extremely grateful to the editor and the referees for useful suggestions that improve the contents of the paper.

Received: 08/IV/09.

Accepted: 07/XII/09.

#CAM-90/09.

• [1] F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space J. Math. Anal. Appl., 20 (1967), 197-228.
• [2] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems Math. Stud., 63 (1994), 123-145.
• [3] L.C. Ceng and J.C. Yao, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings Appl. Math. Comput., 198 (2008), 729-741.
• [4] L.C. Ceng and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities Math. Meth. Oper. Res., 67 (2008), 375-390.
• [5] S.S. Chang, H.W.J. Lee and C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization Nonlinear Anal., 70 (2009), 3307-3319.
• [6] V. Colao, G. Marino and H.K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems J. Math. Anal. Appl., 344 (2008), 340-352.
• [7] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces J. Nonlinear Convex Anal., 6 (2005), 117-136.
• [8] H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Nonlinear Anal., 61 (2005), 341-350.
• [9] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems J. Nonlinear Convex Anal., 9 (2008), 37-43.
• [10] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings Bull. Amer. Math. Soc., 73 (1967), 595-597.
• [11] S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings Appl. Math. Comput., 197 (2008), 548-558.
• [12] X. Qin, M. Shang and H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces Appl. Math. Comput., 200 (2008), 242-253.
• [13] X. Qin, M. Shang and Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces Nonlinear Anal., 69 (2008), 3897-3909.
• [14] X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems Math. Comput. Modelling, 48 (2008), 1033-1046.
• [15] X. Qin, Y.J. Cho and S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces J. Comput. Appl. Math., 225 (2009), 20-30.
• [16] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators Trans. Amer. Math. Soc., 149 (1970), 75-88.
• [17] T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochne integrals J. Math. Anal. Appl., 305(2005), 227-239.
• [18] Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems Nonlinear Anal., 69 (2008), 2709-2719.
• [19] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications Taiwanese J. Math., 5 (2001), 387-404.
• [20] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces J. Math. Anal. Appl., 331 (2007), 506-515.
• [21] W. Takahashi, Nonlinear Functional Analysis Yokohama Publishers, Yokohama (2000).
• [22] S. Takahashi and W. Takahahsi, Strong convergence theorem of a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space Nonlinear Anal., 69 (2008), 1025-1033.
• [23] H.K. Xu, Iterative algorithms for nonlinear operators J. London Math. Soc., 66 (2002), 240-256.
• [24] Y. Yao, M.A. Noor and Y.C. Liou, On iterative methods for equilibrium problems Nonlinear Anal., 70 (2009), 497-509.
• [25] Y. Yao and J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings Appl. Math. Comput., 186 (2007), 1551-1558.

Publication Dates

• Publication in this collection
  22 Nov 2010
• Date of issue
  2010

History

• Received
  08 Apr 2009
• Accepted
  07 Dec 2009