We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to ℝ, where ℝ is the set of real numbers.

The equilibrium problem for F : C × C → ℝ is to find x ∈ C such that

(1.1)

for all y ∈ C. The set of such solutions is denoted by EP (F).

A mapping S of C is said to be a κ-strict pseudocontraction if there exists a constant κ ∈ [0,1) such that

(1.2)

for all x, y ∈ C; see [1]. We denote the set of fixed points of S by F(S) (i.e., F(S) = {x ∈ C : Sx = x}).

Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mapping S on C such that

(1.3)

for all x, y ∈ C. That is, S is nonexpansive if and only if S is a 0-strict pseudocontraction.

Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem (1.1); see, for instance, [2–4]. In particular, Combettes and Hirstoaga [5] proposed several methods for solving the equilibrium problem. On the other hand, Mann [6], Shimoji and Takahashi [7] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [8] projected new iterative methods for finding a fixed point of strict pseudocontractions.

In 2006, Marino and Xu [3] introduced the general iterative method and proved that the algorithm converged strongly. Recently, Liu [2] considered a general iterative method for equilibrium problems and strict pseudocontractions. Tian [9] proposed a new general iterative algorithm combining an L-Lipschitzian and η-strong monotone operator. Very recently, Wang [10] considered a general composite iterative method for infinite family strict pseudocontractions.

In this paper, motivated by the above facts, we introduce two iterative schemes and obtain strong convergence theorems for finding a common element of the set of fixed points of a infinite family of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1).

2. Preliminaries

Throughout this paper, we always write ⇀ for weak convergence and → for strong convergence. We need some facts and tools in a real Hilbert space H which are listed as below.

Lemma 2.1. Let H be a real Hilbert space. There hold the following identities:

(i)
∥x−y∥² = ∥x∥² − ∥y∥² − 2〈x − y, y〉, ∀x, y ∈ H;
(ii)
∥tx+(1−t)y∥² = t∥x∥² + (1 − t)∥y∥² − t(1 − t)∥x−y∥², ∀ t ∈ [0,1], ∀ x, y ∈ H.

Lemma 2.2 (see [11].)Assume that {α_n} is a sequence of nonnegative real numbers such that

(2.1)

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

(i)
(ii)

Then, lim _n→∞α_n = 0.

Recall that given a nonempty closed convex subset C of a real Hilbert space H, for any x ∈ H, there exists a unique nearest point in C, denoted by P_Cx, such that

(2.2)

for all y ∈ C. Such a P_C is called the metric (or the nearest point) projection of H onto C. As known, y = P_Cx if and only if there holds the relation:

(2.3)

Lemma 2.3 (see [10].)Let A : H → H be a L-Lipschitzian and η-strongly monotone operator on a Hilbert space H with L > 0, η > 0, 0 < μ < 2η/L², and 0 < t < 1. Then, S = (I − tμA) : H → H is a contraction with contractive coefficient 1 − tτ and τ = (1/2)μ(2η − μL²).

Lemma 2.4 (see [1].)Let S : C → C be a κ-strict pseudocontraction. Define T : C → C by Tx = λx + (1 − λ)Sx for each x ∈ C. Then, as λ ∈ [κ, 1), T is a nonexpansive mapping such that F(T) = F(S).

Lemma 2.5 (see [9].)Let H be a Hilbert space and f : H → H be a contraction with coefficient 0 < α < 1, and A : H → H an L-Lipschitzian continuous operator and η-strongly monotone with L > 0, η > 0. Then for 0 < γ < μη/α:

(2.4)

That is, μA − γf is strongly monotone with coefficient μη − γα.

Let {S_n} be a sequence of κ_n-strict pseudo-contractions. Define . Then, by Lemma 2.4, is nonexpansive. In this paper, consider the mapping W_n defined by

(2.5)

where t₁, t₂, … are real numbers such that 0 ≤ t_n < 1. Such a mapping W_n is called a W-mapping generated by

and t₁, t₂, …. It is easy to see W_n is nonexpansive.

Lemma 2.6 (see [7].)Let C be a nonempty closed convex subset of a strictly convex Banach space E, let be nonexpansive mappings of C into itself such that and let t₁, t₂, … be real numbers such that 0 < t_i ≤ b < 1, for every i = 1,2, …. Then, for any x ∈ C and k ∈ N, the limit lim _n→∞U_n,kx exists.

Using Lemma 2.6, one can define the mapping W of C into itself as follows:

(2.6)

Lemma 2.7 (see [7].)Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of C into itself such that and let t₁, t₂, … be real numbers such that 0 < t_i ≤ b < 1, for all i ≥ 1. If K is any bounded subset of C, then

(2.7)

Lemma 2.8 (see [12].)Let C be a nonempty closed convex subset of a Hilbert space be a family of infinite nonexpansive mappings with , let t₁, t₂, … be real numbers such that 0 < t_i ≤ b < 1, for every i = 1,2, …. Then .

For solving the equilibrium problem, assume that the bifunction F satisfies the following conditions:

(A1)
F(x, x) = 0 for all x ∈ C;
(A2)
F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for any x, y ∈ C;
(A3)
for each x, y, z ∈ C, limsup _t→0F(tz + (1 − t)x, y) ≤ F(x, y);
(A4)
F(x, ·) is convex and lower semicontinuous for each x ∈ C.

Recall some lemmas which will be needed in the rest of this paper.

Lemma 2.9 (see [13].)Let C be a nonempty closed convex subset of H, let F be bifunction from C × C to ℝ satisfying (A1)–(A4), and let r > 0 and x ∈ H. Then, there exists z ∈ C such that

(2.8)

Lemma 2.10 (see [5].)For r > 0, x ∈ H, define a mapping T_r : H → C as follows:

(2.9)

for all x ∈ H. Then, the following statements hold:

(i)
T_r is single-valued;
(ii)
T_r is firmly nonexpansive, that is, for any x, y ∈ H,

(2.10)

(iii)
F(T_r) = EP(F);
(iv)
EP(F) is closed and convex.

Lemma 2.11 (see [14].)Let {x_n} and {z_n} be bounded sequences in a Banach space and let {β_n} be a sequence of real numbers such that 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 for all n = 0,1, 2, …. Suppose that x_n+1 = (1 − β_n)z_n + β_nx_n for all n = 0,1, 2, … and limsup _n→∞∥z_n+1 − z_n∥−∥x_n+1 − x_n∥≤0. Then lim _n→∞∥z_n − x_n∥ = 0.

Lemma 2.12 (see [4].)Let C, H, F, and T_rx be as in Lemma 2.10. Then, the following holds:

(2.11)

for all s, t > 0 and x ∈ H.

Lemma 2.13 (see [10].)Let H be a Hilbert space and let C be a nonempty closed convex subset of H, and T : C → C a nonexpansive mapping with F(T) ≠ ∅. If {x_n} is a sequence in C weakly converging to x and if {(I − T)x_n} converges strongly to y, then (I − T)x = y.

3. Main Result

Throughout the rest of this paper, we always assume that f is a contraction of H into itself with coefficient α ∈ (0,1), and A is a L-Lipschitzian continuous operator and η-strongly monotone on H with L > 0, η > 0. Assume that 0 < μ < 2η/L² and 0 < γ < μ(η − (μL²/2))/α = τ/α.

Define a mapping

. Since both W_n and

are nonexpansive, it is easy to get V_n is also nonexpansive. Consider the following mapping G_n on H defined by

(3.1)

where α_n ∈ (0,1). By Lemmas 2.3 and 2.10, we have

(3.2)

Since 0 < 1 − α_n(τ − γα) < 1, it follows that G_n is a contraction. Therefore, by the Banach contraction principle, G_n has a unique fixed pointed

such that

(3.3)

For simplicity, we will write x_n for

provided no confusion occurs. Next we prove the sequences {x_n} converges strongly to a

which solves the variational inequality:

(3.4)

Equivalently, x^* = P_Ω(I − μA + γf)x^*.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and F a bifunction from C × C to ℝ satisfying (A1)–(A4). Let S_i : C → C be a family κ_i-strict pseudocontractions for some 0 ≤ κ_i < 1. Assume the set . Let f be a contraction of H into itself with α ∈ (0,1) and let A be a L-Lipschitzian continuous operator and η-strongly monotone with L > 0, η > 0, 0 < μ < 2η/L² and 0 < γ < μ(η − (μL²/2))/α = τ/α. For every n ∈ ℕ, let W_n be the mapping generated by and t_i as in (2.5). Let {x_n} and {u_n} be sequences generated by the following algorithm:

(3.5)

If {α_n}, {β_n}, and {r_n} satisfy the following conditions:

(i)
{α_n}⊂(0,1), lim _n→∞α_n = 0;
(ii)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(iii)
{r_n}⊂(0, ∞), liminf _n→∞r_n > 0.

Then, {x_n} converges strongly to a point x^* ∈ Ω, which solves the variational inequality (3.4).

Proof. The proof is divided into several steps.

Step 1. Show first that {x_n} is bounded.

Take any p ∈ Ω, by (3.5) and Lemma 2.3, we derive that

(3.6)

It follows that ∥x_n − p∥≤(∥γf(p) − μAp∥)/(τ − γα).

Hence, {x_n} is bounded, so are {u_n} and {y_n}. It follows from the Lipschitz continuity of A that {Ax_n} and {Au_n} are also bounded. From the nonexpansivity of f and W_n, it follows that {f(x_n)} and {W_nx_n} are also bounded.

Step 2. Show that

(3.7)

Notice that

(3.8)

By Lemma 2.10, we have

(3.9)

It follows that

(3.10)

Thus, from Lemma 2.1 and (3.10), we get

(3.11)

It follows that

(3.12)

Since α_n → 0, we have

(3.13)

From (3.8), it is easy to get

(3.14)

Step 3. Show that

(3.15)

(3.16)

This implies that

(3.17)

From condition (ii), (3.13), and (3.14), we have

(3.18)

Notice that

(3.19)

By Lemma 2.7 and (3.18), we get (3.15).

Since {u_n} is bounded, so there exists a subsequence which converges weakly to x^*.

Step 4. Show that x^* ∈ Ω.

Since C is closed and convex, C is weakly closed. So, we have x^* ∈ C.

From (3.15), we obtain . From Lemmas 2.8, 2.4, and 2.13, we have .

By , for all n ≥ 1, we have

(3.20)

It follows from (A2) that

(3.21)

Hence, we get

(3.22)

It follows from condition (iii), (3.13), and (A4) that

(3.23)

For s with 0 < s ≤ 1 and y ∈ C, let y_s = sy + (1 − s)x^*. Since y ∈ C and x^* ∈ C, we obtain y_s ∈ C and hence F(y_s, x^*) ≤ 0. So, we have

(3.24)

Dividing by s, we get

(3.25)

Letting s → 0 and from (A3), we get

(3.26)

for all y ∈ C and x^* ∈ EP (F). Hence x^* ∈ Ω.

Step 5. Show that x_n → x^*, where x^* = P_Ω(I − μA + γf)x^*:

(3.27)

Hence, we obtain

(3.28)

It follows that

(3.29)

This implies that

(3.30)

In particular,

(3.31)

Since , it follows from (3.31) that as j → ∞. Next, we show that x^* solves the variational inequality (3.4).

By the iterative algorithm (3.5), we have

(3.32)

Therefore, we have

(3.33)

that is,

(3.34)

Hence, for p ∈ Ω,

(3.35)

Since I − V_n is monotone (i.e., 〈x − y, (I − V_n)x − (I − V_n)y〉 ≥ 0, for all x, y ∈ H). This is due to the nonexpansivity of V_n.

Now replacing n in (3.35) with n_j and letting j → ∞, we obtain

(3.36)

That is, x^* ∈ Ω is a solution of (3.4). To show that the sequence {x_n} converges strongly to x^*, we assume that . By the same processing as the proof above, we derive . Moreover, it follows from the inequality (3.36) that

(3.37)

Interchanging x^* and

, we get

(3.38)

By Lemma 2.5, adding up (3.37) and (3.38) yields

(3.39)

Hence

and, therefore, x_n → x^* as n → ∞,

(3.40)

This is equivalent to the fixed point equation:

(3.41)

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and F a bifunction from C × C to ℝ satisfying (A1)–(A4). Let S_i : C → C be a family κ_i-strict pseudocontractions for some 0 ≤ κ_i < 1. Assume the set . Let f be a contraction of H into itself with α ∈ (0,1) and let A be a L-Lipschitzian continuous operator and η-strongly monotone with L > 0, η > 0,0 < μ < 2η/L², and 0 < γ < μ(η − (μL²/2))/α = τ/α. For every n ∈ ℕ, let W_n be the mapping generated by and 0 < t_i ≤ b < 1. Given x₁ ∈ H, let {x_n} and {u_n} be sequences generated by the following algorithm:

(3.42)

If {α_n}, {β_n} and {r_n} satisfy the following conditions:

(i)
{α_n}⊂(0,1), lim _n→∞α_n = 0 and ;
(ii)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(iii)
{r_n}⊂(0, ∞), liminf _n→∞r_n > 0 and lim _n→∞ | r_n+1 − r_n | = 0.

Then, {x_n} converges strongly to x^* ∈ Ω, which solves the variational inequality (3.4).

Proof. The proof is divided into several steps.

Step 1. Show first that {x_n} is bounded.

Taking any p ∈ Ω, we have

(3.43)

By induction, we obtain ∥x_n − p∥≤max {∥x₁ − p∥, ∥γf(p) − μAp)∥/(τ − αγ)}, n ≥ 1. Hence, {x_n} is bounded, so are {u_n} and {y_n}. It follows from the Lipschitz continuity of A that {Ax_n} and {Au_n} are also bounded. From the nonexpansivity of f and W_n, it follows that {f(x_n)} and {W_nx_n} are also bounded.

Step 2. Show that

(3.44)

Observe that

(3.45)

and from (2.5), we have

(3.46)

where M₁ = sup _n{∥U_n+1,n+1u_n − U_n,n+1u_n∥}.

Suppose x_n+1 = β_nx_n + (1 − β_n)z_n, then z_n = (x_n+1 − β_nx_n)/(1 − β_n) = (α_nγf(x_n)+(I − μα_nA)y_n − β_nx_n)/(1 − β_n).

Hence, we have

(3.47)

It follows from (3.45), (3.46), and the above result that

(3.48)

where M₂ = sup _n{∥γf(x_n)∥ + ∥μAy_n∥}. Hence, we get

(3.49)

From condition (i), (iii), 0 < t_n ≤ b < 1, and Lemma 2.12, we obtain

(3.50)

By Lemma 2.11,we have lim _n→∞∥z_n − x_n∥ = 0. Thus,

(3.51)

By Lemma 2.12, (3.45) and (3.44), we obtain

(3.52)

Step 3. Show that

(3.53)

Observe that

(3.54)

From condition (i) and (3.5),we can obtain

(3.55)

By Lemma 2.10, we get

(3.56)

This implies that

(3.57)

By nonexpansivity of W_n, we have

(3.58)

It follows from (3.42) that

(3.59)

This implies that

(3.60)

From condition (i), (ii), and (3.44), we have

(3.61)

Further we have ∥x_n − W_nu_n∥→0. Thus we get

(3.62)

On the other hand, we have

(3.63)

Combining (3.62), the last inequality, and Lemma 2.7, we obtain (3.53).

Step 4. Show that

(3.64)

where x^* = P_Ω(I − μA + γf)x^* is a unique solution of the variational inequality (3.4). Indeed, take a subsequence

of {x_n} such that

(3.65)

Since is bounded, there exists a subsequence of which converges weakly to q. Without loss of generality, we can assume . From (3.53), we obtain .

By the same argument as in the proof of Theorem 3.1, we have q ∈ Ω. Since x^* = P_Ω(I − μA + γf)x^*, it follows that

(3.66)

Step 5. Show that

(3.67)

Since

(3.68)

It follows from (3.44) and (3.66) that

(3.69)

This implies that

(3.70)

where M₃ = sup _n∥x_n − x^*∥², n ≥ 1. It is easily to see that γ_n = 2α_n(τ − αγ)/(1 − α_nαγ). Hence, by Lemma 2.2, the sequence {x_n} converges strongly to x^*.

Remark 3.3. If F ≡ 0, then Theorem 3.2 reduces to Theorem 3.1 of Wang [10].

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve the manuscript NSFC Tianyuan Youth Foundation of Mathematics of China (no. 11126136), and the Fundamental Research Funds for the Central Universities (GRANT: ZXH2011C002).

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General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

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General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

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1. Introduction

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Acknowledgments

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