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

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.

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 x ∈ C such that

(1.1)

The set of solutions of (1.1) is denoted by EP(G). Given a mapping T : C → H, 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 : C → H be a nonlinear mapping. The variational inequality problem is to find a u ∈ C such that

(1.2)

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:

(1.3)

The set of such z ∈ C 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, y ∈ C.

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

(1.4)

for all x, y ∈ D(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 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 [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 x₀ ∈ C, 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 : C → C 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 : C → C 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 = P_{EP(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 : C → C as follows:

This mapping is called S-mapping generated by T _n , ..., T₁ and α _n , α_n-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, x₁ ∈ C, let {x _n } be generated by

For i = 1, 2, ..., N, let F _i : C × C → ℝ be bifunction, A _i : C → H be α _i -inverse strongly monotone and let G _i : C → C 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)

(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 .

Lemma 2.1 [5]. Given x ∈ H and y ∈ C. 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 lim_n→∞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 x ∈ C 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 : C → C 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 inf_n→∞β _n ≤ lim sup_n→∞β _n < 1.

Suppose

for all integer n ≥ 0 and

Then lim_n→∞||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 ∀x ∈ C;

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

(A 3) ∀x, y, z ∈ C,

(A 4) ∀x ∈ C, y ↦ F(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 into ℝ satisfying (A 1) - (A 4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that

(2.1)

for all x ∈ C.

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

for all z ∈ H. 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 : C → C 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 : C → C 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 , α_n-1, ..., α₁. Then, for every x ∈ C and k ∈ ℕ, lim_n→∞U _n , _k x exists.

Proof. Let x ∈ C and . Fix k ∈ ℕ, then for every n ∈ ℕ with n ≥ k,

we have

It follows that

(2.4)

where and

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

(2.5)

Since a ∈ (0, 1), we have lim_n→∞aⁿ = 0. From (2.5), we have that {U _n , _k x} is a Cauchy sequence. Hence lim _n→∞U_n,kx exists. □

For every k ∈ ℕ and x ∈ C, we define mapping U_∞,kand S : C ∈ C as follows:

(2.6)

and

(2.7)

Such a mapping S is called S-mapping generated by T _n , T_n-1, ... and α _n , α_{n- 1}, ...

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

Then, we have that S : C → C is also nonexpansive indeed, observe that for each x, y ∈ C

By (2.8), we have

This implies that for m > n and x ∈ D,

By letting m → ∞, for any x ∈ D, 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 = sup_i∈κ _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 , α_n-1, ..., α₁ and T _n , T_n-1, ..., and α _n , α_n-1, ..., respectively. Then .

Proof. It is evident that . For every n, k ∈ ℕ, with n ≥ k, let x₀ ∈ F (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 : C → H be mappings and let G _i : C → C 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*), ∀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

(2.18)

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 → ℝ be a bifunction satisfying (A₁) - (A₄), let A _i : C → H be α _i -inverse strongly monotone and let G _i : C → C 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 : C → C 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 , ρ_{n - 1}, ..., ρ₁ and T _n , T_{n- 1}, ..., and ρ _n , ρ_{n - 1}, ..., respectively. Assume that . For every n ∈ ℕ, i = 1, 2, ..., N, let {x _n } and be generated by x₁, u ∈ C 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, y ∈ C, 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 ∀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 ∈ 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 lim_n→∞||x_n+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 lim_n→∞||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 sup_n→∞〈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 : C → C by

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

Next, we define Q : C → C 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 lim_n→∞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 lim_n→∞ 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 , ρ_{n - 1}, ..., ρ₁ and T _n , T_{n- 1}, ..., and ρ _n , ρ_{n- 1}, ..., respectively. Assume that . For every n∈ ℕ, i = 1, 2, ..., N, let {x _n } and be generated by x₁, u ∈ C 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 = y ∀y ∈ C. 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 : C → H 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 ∈ ℕ, let {x _n } and {v _n } be generated by x₁, u ∈ C 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, ∀y ∈ C. Choose , a = a _n , b = b _n , c = c _n for all n ∈ ℕ, and let T _κ ≡ S₁ : C → C be S-mapping generated by T₁ and ρ₁ with T₁ = T and , and then we obtain the desired result from Theorem 3.1 □