Volume 2012, Issue 1 282094

Research Article

Open Access

The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Rabian Wangkeeree,

Corresponding Author

Rabian Wangkeeree

[email protected]

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand ncetm.org.uk

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Panatda Boonman,

Panatda Boonman

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

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Rabian Wangkeeree,

Corresponding Author

Rabian Wangkeeree

[email protected]

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand ncetm.org.uk

Search for more papers by this author

Panatda Boonman,

Panatda Boonman

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand nu.ac.th

Search for more papers by this author

First published: 22 May 2012

https://doi.org/10.1155/2012/282094

Academic Editor: Rudong Chen

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Abstract

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

1. Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, let C be a nonempty closed convex subset of H, and let T be a mapping of C into H, then T : C → H is said to be nonexpansive if ∥Tx − Ty∥≤∥x − y∥ for all x, y ∈ C. A mapping T : C → H is said to be quasinonexpansive if ∥Tx − y∥≤∥x − y∥ for all x ∈ C and y ∈ F(T): = {x ∈ C : Tx = x}. It is well known that the set F(T) of fixed points of a quasi-nonexpansive mapping T is a closed and convex set [1]. A mapping T : C → C is said to be firmly nonexpansive [2] if

()

for all x, y ∈ C, and it is an important example of nonexpansive mappings in a Hilbert space.

Let φ : C → ℝ be a real-valued function, and let F : C × C → ℝ be an equilibrium bifunction, that is, F(u, u) = 0 for each u ∈ C. The mixed equilibrium problem is to find x ∈ C such that

()

Denote the set of solution of (1.2) by MEP (F, φ). In particular, if φ = 0, this problem reduces to the equilibrium problem, which is to find x ∈ C such that

()

The set of solution of (1.3) is denoted by EP (F).

The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, Min-Max problems, the Nash equilibrium problems in noncooperative games, and others; see, for example, Blum and Oettli [3] and Moudafi [4]. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3).

Let F₁, F₂ : C × C → ℝ be two monotone bifunctions and μ > 0 is constant. In 2009, Moudafi [5] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems, finding (x^′, y^′) ∈ C × C such that

()

For such mappings F₁ and F₂ and two given positive constants λ, μ > 0, Plubtieng and Sombut [6] considered the following system of mixed equilibrium problem, finding (x′, y′) ∈ C × C such that

()

In particular, if λ = μ and φ ≡ 0, then problem (SMEP) reduces to (SEP). Furthermore, Plubtieng and Sombut [6] introduced the following iterative procedure to approximate a common element of the fixed-point set of a quasi-nonexpansive mapping T and the solution set of (SMEP) in a Hilbert space H. Let {x_n}, {y_n}, and {u_n} be given by

()

where {α_n}⊂[a, b] for some a, b ∈ (0,1) and satisfying appropriate conditions. The weak convergence theorems are obtained in a real Hilbert space.

On the other hand, in 1953, Mann [7] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:

()

where the initial point x₀ is taken in C arbitrarily, and {α_n} is a sequence in [0,1].

For two nonexpansive mappings T₁, T₂ of C into itself, Moudafi [4] studied weak convergence theorems in the following iterative process:

()

for all n ∈ ℕ, where {α_n} and {β_n} are appropriate sequences in [0,1] and F(T₁)∩F(T₂) ≠ ∅. Recently, Iemoto and Takahashi [8] also considered this iterative procedure for T₁ is a nonexpansive mapping and T₂ : C → C is a nonspreading mapping. Very recently, Kim [9] studied the weak and strong convergence for the Moudafi’s iterative scheme (1.8) of two quasi-nonexpansive mappings.

In this paper, inspired and motivated by Plubtieng and Sombut [6], Moudafi [4], Iemoto and Takahashi [8], and Kim [9], we first introduce the iterative procedure to approximate a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of SMEP in a real Hilbert space. Next, we prove the weak convergence theorem for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed point of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

2. Preliminaries

Throughout this paper, let ℕ be the set of positive integers, and let ℝ be the set of real numbers. Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, respectively, and let C be a closed convex subset of H. We denote the strong convergence and the weak convergence of {x_n} to x ∈ H by x_n → x and x_n⇀x, respectively.

From [10], for each x, y ∈ H and λ ∈ [0,1], we have

()

For every point x ∈ H, there exists a unique nearest point in C, denoted by P_Cx, such that

()

P_C is called the metric projection of H onto C. It is well know that P_C is a nonexpansive mapping of H onto C and satisfies

()

Moreover, P_Cx is characterized by the following properties: P_Cx ∈ C,

()

Further, for all x ∈ H and y ∈ C, y = P_Cx if and only if 〈x − y, y − z〉≥0, for all z ∈ C.

Lemma 2.1 (see [11].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let P be the metric projection of H onto C, and let {x_n} _n∈ℕ be in H. If

()

for all u ∈ C and n ∈ ℕ. Then, {P_Cx_n} converges strongly to an element of C.

Theorem 2.2 (Opial’s theorem, [10]). Let H be a real Hilbert space, and suppose that x_n⇀x, then

()

for all y ∈ H with x ≠ y.

All Hilbert space and l^p (1 < p < ∞) satisfy Opial’s condition, while L^p with 1 < p ≠ 2 < ∞ do not.

For solving the mixed equilibrium problem for an equilibrium bifunction F : C × C → ℝ, let us assume that 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 all x, y ∈ C,
(A3)
for each y ∈ C, x ↦ F(x, y) is weakly upper semicontinuous,
(A4)
for each x ∈ C, y ↦ F(x, y) is convex and semicontinuous.

The following lemma appears implicitly in [3, 12].

Lemma 2.3 (see [3].)Let C be a nonempty closed convex subset of H, and let F : C × C → ℝ be a bifunction satisfying (A1)–(A4). Let r > 0 and x ∈ H, then there exists z ∈ C such that

()

The following lemma was also given in [12].

Lemma 2.4 (see [12].)Let C be a nonempty closed closed convex subset of H and let F : C × C → ℝ be a bifunction satisfying (A1)–(A4), then, for any r > 0 and x ∈ H, define a mapping T_rx : H → C as follows:

()

for all z ∈ H, r ∈ ℝ. Then the following hold:

(i)
T_r is single valued,
(ii)
T_r is firmly nonexpansive, that is,
()
(iii)
F(T_r) = EP (F),
(iv)
EP (F) is closed and convex.

We note that Lemma 2.4 is equivalent to the following lemma.

Lemma 2.5 (see [6].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C × C → ℝ be an equilibrium bifunction satisfying (A1)–(A4) and let φ : C → ℝ be a lower semicontinuous and convex functional. For each r > 0 and x ∈ H, define a mapping

()

Then, the following results hold:

(i)
for each x ∈ C, S_r(x) ≠ ∅,
(ii)
S_r is single valued,
(iii)
S_r is firmly nonexpansive, that is, for any x, y ∈ H,
()
(iv)
F(S_r) = MEP (F, φ),
(v)
MEP (F, φ) is closed and convex.

3. Main Results

In this section, we prove the weak convergence for approximating a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of the system of mixed equilibrium problems in a Hilbert space.

To begin with, let us state and proof the following characterizations of the solution set of GMEP.

Lemma 3.1. Let C be a closed convex subset of a real Hilbert space H. Let F₁ and F₂ be two mappings from C × C → ℝ satisfying (A1)–(A4), and let S_1,λ and S_2,μ be defined as in Lemma 2.5 associated to F₁ and F₂, respectively. For given x^′, y^′ ∈ C, (x^′, y^′) is a solution of problem (1.5) if and only if x′ is a fixed point of the mapping G : C → C defined by

()

where y^′ = S_2,μx^′.

Proof. For given x′, y′ ∈ C, we observe the following equivalency:

()

This completes the proof.

We note from Lemma 3.1 that the mapping G is nonexpansive. Moreover, if C is a closed bounded convex subset of H, then the solution of problem (1.5) always exists. Throughout this paper, we denote the set of fixed points of G by Ω.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F₁ and F₂ be two bifunctions from C × C → ℝ satisfying (A1)–(A4). Let r, λ > 0 and S_1,r and S_2,λ be defined as in Lemma 2.5 associated to F₁ and F₂, respectively. Let T_i : C → C, i = 1,2, be two quasi-nonexpansive mappings such that I − T_i are demiclosed at zero, that is, if {w_n} ⊂ C, w_n⇀w, and (I − T_i)w_n → 0, then w ∈ F(T_i), with F(T₁)∩F(T₂)∩Ω ≠ ∅. Let the sequences {x_n}, {y_n}, and {u_n}, be given by

()

where {a_n}, {b_n}⊂[a, b] for some a, b ∈ (0,1), and satisfy

()

then

and

is a solution of problem (1.5), where

Proof. Let x^* ∈ F(T₁)∩F(T₂)∩Ω, then x^* = T₁x^* = T₂x^* and x^* = S_1,r(S_2,λx^*).

Putting y^* = S_2,λx^*, y_n = S_1,ru_n, and u_n = S_2,λx_n, we have

()

Next, we prove that

()

Since T₁ and T₂ are quasi-nonexpansive, we obtain that

()

which gives that

()

Hence, {∥x_n − x^*∥} is a nonincreasing sequence, and hence, lim _n→∞∥x_n − x^*∥ exists. This implies that {x_n}, {y_n}, {u_n}, {T₁y_n}, and {T₂y_n} are bounded. From (3.8), we have

()

Since liminf _n→∞a_n(1 − a_n) > 0, this implies that

()

Furthermore, since 0 < a ≤ a_n ≤ b < 1, we have

()

From (3.10), we conclude that

()

From (3.7), we have

()

where M is a constant satisfying M ≥ sup _n≥1[∥x_n − x^*∥+∥b_nT₁y_n + (1 − b_n)T₂y_n − x^*∥]. Again from (3.10), we conclude that

()

Using liminf _n→∞b_n(1 − b_n) > 0, we have

()

Now, we prove that

()

We observe that

()

which gives that

()

Since 0 < a ≤ a_n ≤ b < 1, we have

()

Using (3.12) and (3.15), we conclude that

()

which gives that

()

since liminf _n→∞a_n(1 − a_n) > 0. Similarly, we have

()

which implies that

()

Thus, we have

()

Hence, lim _n→∞a_n(1 − a_n)∥x_n+1 − T₂y_n∥ = 0. Since liminf _n→∞a_n(1 − a_n) > 0, we have

()

Next, we prove that

()

Since S_1,r and S_2,λ are firmly nonexpansive, it follows that

()

which gives that

()

This implies that

()

By the convexity of ∥·∥², we have

()

Thus,

()

Since 0 < a ≤ a_n ≤ b < 1, we have

()

Since lim _n→∞∥x_n − x^*∥ exists, we have

()

Similarly, we have

()

which gives that

()

This implies that

()

By the convexity of ∥·∥², we have

()

Thus,

()

Since 0 < a ≤ a_n ≤ b < 1, we have

()

Since lim _n→∞∥x_n − x^*∥ exists, we have

()

Hence,

()

It follow from (3.10), (3.33), and (3.40) that

()

from which it follows that

()

that is,

()

Thus,

()

Similarly, we have ∥y_n − T₂y_n∥→0. Since {y_n} is bounded sequence, there exists a subsequence

of {y_n} such that

as i → ∞. Since T₁ and T₂ are demiclosed at 0, we conclude that

. Let G be a mapping which is defined as in Lemma 3.1. Thus, we have

()

and hence,

()

This together with

implies that

, if

is another subsequence of {y_n} such that

as i → ∞. Since T₁ and T₁ are demiclosed at 0, we conclude that

. From

and

, we will show that

. Assume that

. Since lim _n→∞∥x_n − x^*∥ exists for all x^* ∈ F(T₁)∩F(T₂)∩Ω, by Opial’s Theorem 2.2, we have

()

This is a contradiction. Thus, we have

. This implies that

. Since ∥x_n − y_n∥→0, we have

. Put

. Finally, we show that

. Now from (2.4) and

, we have

()

Since {∥x_n − x^*∥} is nonnegative and nonincreasing for all x^* ∈ F(T₁)∩F(T₂)∩Ω, it follows by Lemma 2.1 that {z_n} converges strongly to some

. By (3.49), we have

()

Therefore,

Setting T : = T₁ = T₂ in Theorem 3.2, we have the following result.

Corollary 3.3 (see [6].)Let C be a closed convex subset of a real Hilbert space H. Let F₁ and F₂ be two bifunctions from C × C → ℝ satisfying (A1)–(A4). Let λ, μ > 0, and let S_1,λ and S_2,μ be defined as in Lemma 2.5 associated to F₁ and F₂, respectively. Let T : C → C be a quasi-nonexpansive mapping such that I − T is demiclosed at zero and F(T)∩Ω ≠ ∅. Suppose that x₀ = x ∈ C and {x_n}, {y_n}, and {z_n} are given by

()

for all n ∈ ℕ, where {a_n}⊂[a, b] for some a, b ∈ (0,1), and satisfy liminf _n→∞a_n(1 − a_n) > 0, then {x_n} converges weakly to

and

is a solution of problem (1.5), where

Setting F₁ = F₂ ≡ 0, φ ≡ 0 in Theorem 3.2, we have the following result.

Corollary 3.4 (see [9].)Let H be a Hilbert space, let C be a nonempty, closed, and convex subset of H, and let T₁, T₂ be two quasi-nonexpansive mappings of C into itself such that I − T₁, I − T₂ are demiclosed at zero with F(T₁)∩F(T₂) ≠ ∅. For any x₁ in C, let {x_n} be defined by

()

where {a_n} and {b_n} are chosen so that

()

then x_n⇀p ∈ F(T₁)∩F(T₂).

4. Applications

In this section, we apply our results to approximate a common element of the set of common fixed points of an asymptotic nonspreading mapping and an asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space. We recall the following definitions. A mapping T : C → C is called nonspreading [13] if

()

Furthermore, Takahashi and Yao [14] also introduced two nonlinear mappings in Hilbert spaces. A mapping T : C → C is called a TJ − 1 mapping [14] if

()

for all x, y ∈ C. A mapping T : C → C is called a TJ − 2 [14] mapping if

()

for all x, y ∈ C. For these two nonlinear mappings, TJ − 1 and TJ − 2 mappings, Takahashi and Yao [14] studied the existence results of fixed points in Hilbert spaces. Very recently, Lin et al. [15] introduced the following definitions of new mappings.

Definition 4.1. Let C be a nonempty closed convex subset of a Hilbert space H. We say that T : C → C is an asymptotic nonspreading mapping if there exist two functions α : C → [0,2) and β : C → [0, k], k < 2, such that

(A1)
2∥Tx − Ty∥² ≤ α(x)∥Tx − y∥² + β(x)∥Ty − x∥² for all x, y ∈ C,
(A2)
0 < α(x) + β(x) ≤ 2 for all x ∈ C.

Remark 4.2. The class of asymptotic nonspreading mappings contains the class of nonspreading mappings and the class of TJ − 2 mappings in a Hilbert space. Indeed, in Definition 4.1, we know that

(i)
if α(x) = β(x) = 1 for all x ∈ C, then T is a nonspreading mapping,
(ii)
if α(x) = 4/3 and β(x) = 2/3 for all x ∈ C, then T is a TJ − 2 mapping.

Definition 4.3. Let C be a nonempty closed convex subset of a Hilbert space H. We say T : C → C is an asymptotic TJ mapping if there exists two functions α : C → [0,2] and β : C → [0, k], k < 2, such that

(B1)
2∥Tx − Ty∥² ≤ α(x)∥x − y∥² + β(x)∥Tx − y∥² for all x, y ∈ C,
(B2)
α(x) + β(x) ≤ 2 for all x ∈ C.

Remark 4.4. The class of asymptotic TJ mappings contains the class of TJ − 1 mappings and the class of nonexpansive mappings in a Hilbert space. Indeed, in Definition 4.3, we know that

(i)
if α(x) = 2 and β(x) = 0 for each x ∈ C, then T is a nonexpansive mapping,
(ii)
if α(x) = β(x) = 1 for each x ∈ C, then T is a TJ − 1 mapping.

It is well known that the set F(T) of fixed points of a quasi-nonexpansive mapping T is a closed and convex set [1]. Hence, if T : C → C is an asymptotic nonspreading mapping (resp., asymptotic TJ mapping) with F(T) ≠ ∅, then T is a quasi-nonexpansive mapping, and this implies that F(T) is a nonempty closed convex subset of C.

Theorem 4.5 (see [15].)Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be an asymptotic nonspreading mapping, then I − T is demiclosed at 0.

Theorem 4.6 (see [15].)Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be an asymptotic TJ mapping, then I − T is demiclosed at 0.

Applying the above results, we have the following theorem.

Theorem 4.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F₁ and F₂ be two bifunctions from C × C → ℝ satisfying (A1)–(A4). Let r, λ > 0 and S_1,r and S_2,λ be defined as in Lemma 2.5 associated to F₁ and F₂, respectively. Let T_i : C → C, i = 1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping such that F(T₁)∩F(T₂)∩Ω ≠ ∅. Let {x_n}, {y_n}, and {u_n} be given by

()

where {a_n}, {b_n}⊂[a, b] for some a, b ∈ (0,1) and satisfying

()

then

, and

is a solution of problem (1.5), where

Setting F₁ = F₂ : = F and φ ≡ 0 in the above theorem, we have the following result.

Corollary 4.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be the bifunctions from C × C → ℝ satisfying (A1)–(A4). Let T_i : C → C, i = 1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping such that ℱ : = F(T₁)∩F(T₂)∩EP (F) ≠ ∅. For given u ∈ C and r > 0, let the sequences {x_n} and {u_n} be defined by

()

where {a_n}, {b_n} are two sequences in (0,1) satisfying

()

then x_n⇀w for some w ∈ ℱ.

Setting F₁ = F₂ ≡ 0 and φ ≡ 0 in Theorem 4.7, we have the following result.

Corollary 4.9 (see [15].)Let C be a nonempty closed convex subset of a real Hilbert space H, and let T_i : C → C, i = 1,2, be any one of asymptotic nonspreading mapping and asymptotic TJ mapping. Let ℱ : = F(T₁)∩F(T₂) ≠ ∅. Let {a_n} and {b_n} be two sequences in (0,1). Let {x_n} be defined by

()

Assume that liminf _n→∞ a_n(1 − a_n) > 0 and liminf _n→∞b_n(1 − b_n) > 0, then x_n⇀w for some w ∈ ℱ.

Acknowledgment

This paper is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.

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The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

Acknowledgment

References

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The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

Acknowledgment

References

References

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