We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

1. Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, respectively. let C be a nonempty closed convex subset of H. Let ϕ be a bifunction of C × C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for ϕ : C × C → ℝ is to find x ∈ C such that

()

The set of solutions of (1.1) is denoted by EP (ϕ). Given a mapping T : C → H, let ϕ(x, y) = 〈Tx, y − x〉 for all x, y ∈ C. Then, z ∈ EP (ϕ) if and only if 〈Tz, y − z〉 ≥ 0 for all y ∈ C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1–6].

A mapping S of C into H is said to be nonexpansive if

()

We denote by F(S) the set of fixed points of S. The fixed point equation Tx = x is ill-posed (it may fail to have a solution, nor uniqueness of solution) in general. Regularization therefore is needed. Contractions can be used to regularize nonexpansive mappings. In fact, the following regularization has widely been implemented ([7–9]). Fixing a point u ∈ C and for each t ∈ (0,1), one defines a contraction T_t : C → C by

()

In this paper we provide an alternative regularization method. Our idea is to shrink x first and then apply T to the convex combination of the shrunk x and the anchor u (this idea appeared implicitly in [10] where iterative methods for finding zeros of maximal monotone operators were investigated). In other words, we fix an anchor u ∈ C and t ∈ (0,1) and define a contraction T_t : C → C by

()

Compared with (1.1), (1.4) looks slightly more compact in the sense that the mapping T is more directly involved in the regularization and thus may be more convenient in manipulations since the nonexpansivity of T is utilized first.

In 2000, Moudafi [11] proved the following strong convergence theorem.

Theorem 1.1 (Moudafi [11]). Let C be a nonempty closed convex subset of a Hilbert space H and let S be a nonexpansive mapping of C into itself such that F(S) is nonempty. Let f be a contraction of C into itself and let {x_n} be a sequence defined as follows: x₁ = x ∈ C and

()

for all n ∈ N, where {ε_n}⊂(0,1) satisfies

()

Then, {x_n} converges strongly to z ∈ F(S), where z = P_F(S)f(z) and P_F(S) is the metric projection of H onto F(S).

Such a method for approximation of fixed points is called the viscosity approximation method.

In 2007, S. Takahashi and W. Takahashi [5] introduced and considered the following iterative algorithm by the viscosity approximation method in the Hilbert space:

()

for all n ∈ N, where {α_n} ⊂ [0,1] and {r_n} ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they proved that {x_n} and {u_n} converge strongly to z ∈ F(S)∩EP (ϕ), where z = P_{F(S)∩EP (ϕ)}f(z).

In this paper, motivated and inspired by the above results, we introduce an iterative scheme by the general iterative method for finding a common element of the set of solutions of (1.1) and the set of fixed points of a nonexpansive mapping in Hilbert space.

Let S : C → C be a nonexpansive mapping. Starting with an arbitrary x₁, u ∈ H, define sequences {x_n} and {u_n} by

()

We will prove in Section 3 that if the sequences {α_n}, {β_n}, and {γ_n} of parameters satisfy appropriate conditions, then the sequence {x_n} and {u_n} generated by (1.8) converges strongly to the unique solution of the variational inequality

()

which is the optimality condition for the minimization problem

()

where h is a potential function for f and at the same time, the sequence {x_n} and {u_n} generated by (1.8) converges in norm to Q(u), where Q : C → Fix (T) is the sunny nonexpansive retraction.

2. Preliminaries

Throughout this paper, we consider H as the Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, respectively, C is a nonempty closed convex subset of H. Consider a subset D of C and a mapping Q : C → D. Then we say that

(i)
Q is a retraction provided Qx = x for x ∈ D;
(ii)
Q is a nonexpansive retraction provided Q is a retraction that is also nonexpansive;
(iii)
Q is a sunny nonexpansive retraction provided Q is a nonexpansive retraction with the additional property: Q(x + t(x − Qx)) = Qx whenever x + t(x − Qx) ∈ C, where x ∈ C and t ≥ 0.

Let now T : C → C be a nonexpansive mapping. For a fixed anchor u ∈ C and each t ∈ (0,1) recall that z_t ∈ C is the unique fixed point of the contraction C∋x ↦ T(tu + (1 − t)x). Namely, z_t ∈ C is the unique solution in C to the fixed point equation

()

In the Hilbert space (either uniformly smooth or reflexive with a weakly continuous duality map), then z_t is strongly convergent should it is bounded as t → 0⁺.

We also know that for any sequence {x_n} ⊂ H with x_n⇀x, the inequality

()

holds for every y ∈ H with x ≠ y, (we usually call it Opial’s condition); see [12, 13] for more details.

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

A₁ ϕ(x, x) = 0, for all x ∈ C;
A₂ ϕ is monotone, that is, ϕ(x, y) + ϕ(y, x) ≤ 0, for all x, y ∈ C;
A₃ For each x, y, z ∈ C, lim _t↓0ϕ(tz + (1 − t)x, y) ≤ ϕ(x, y);
A₄ For each x ∈ C, the function y ↦ ϕ(x, y) is convex and lower semicontinuous.

The following lemma appeared implicitly in [14].

Lemma 2.1 (see [14].)Let C be a nonempty closed convex subset of H and let ϕ : C × C → ℝ be a bifunction satisfying (A₁)–(A₄). Let r > 0 and x ∈ H, then, there exists z ∈ C such that

()

Lemma 2.2 (see [6].)Assume that ϕ : C × C → ℝ satisfies (A₁)–(A₄). 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, that is, for any x, y ∈ H,

()

(3) F(T_r) = EP (ϕ);
(4) EP (ϕ) is closed and convex.

Lemma 2.3 (see [15].)Let {a_n}⊂[0, ∞), {b_n}⊂[0, ∞) and {c_n}⊂[0,1) be sequences of real numbers such that

()

then, lim _n→∞a_n = 0.

Lemma 2.4 (see [9].)Suppose that X is a smooth Banach space. Then a retraction Q : C → D is sunny nonexpansive if and only if

()

Lemma 2.5. Let X be a uniformly smooth Banach space, C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping. Let z_t be defined by (2.1). Then (z_t) remains bounded as t → 0 if and only if Fix (T)≠. Moreover, if Fix (T)≠, then (z_t) converges in norm, as t → 0⁺, to a fixed point of T; and if one sets

()

then Q defines the unique sunny nonexpansive retraction from C onto Fix (T).

Lemma 2.6. In the Hilbert space, the following inequalities always hold

(i)
∥x+y∥² ≤ ∥x∥² + 2〈y, x + y〉;
(ii)
∥tx+(1−t)y∥² ≤ t∥x∥² + (1 − t)∥y∥².

3. Main Results

Theorem 3.1. Let C be a nonempty closed convex subset of H, ϕ : C × C → ∞ be a bifunction satisfying (A₁)–(A₄) and T : C → C be a nonexpansive mapping of C into H such that F(T)∩EP (ϕ) ≠ ∅. Let f be a contraction of H into itself with α ∈ (0,1), initially give an arbitrary element x₁ ∈ H and let {x_n} and {u_n} be sequences generated by (1.8), where {α_n}⊂[0,1] and {r_n}⊂(0, ∞) satisfy the following conditions:

(I)
(II)
(III)
(IV)
lim _n→∞(α_n/β_n) = 0.

Then, the sequences {x_n} and {u_n} converge strongly to z ∈ F(T)∩EP (ϕ), where z = P_{F(T)∩EP (ϕ)}f(z) and converge in norm to Q(u), where Q : C → Fix (T) is the sunny nonexpansive retraction.

Proof. Let Q = P_{F(S)∩EP (ϕ)}. Then Qf is a contraction of H into itself. In fact, there exists a ∈ [0,1) such that ∥f(x) − f(y)∥≤a∥x − y∥ for all x, y ∈ H. So, we have that

()

for all x, y ∈ H. So, Qf is a contraction of H into itself. Since H is complete, there exists a unique element z ∈ H such that z = Qf(z), such a z ∈ H is an element of C. We divide the proof into serval steps.

Step 1. {x_n} and {u_n} are all bounded. Let p ∈ F(T)∩EP (ϕ), Then from , we have

()

for all n ∈ N. Put y_n = α_nu + (1 − α_n)u_n, so {x_n+1} can be rewritten as

()

Therefore, from (3.2) we get

()

If ∥x_n − p∥≤∥u − p∥, then {x_n} is bounded. So, we assume that ∥x_n − p∥≥∥u − p∥.

Therefore ∥y_n − p∥≤∥x_n − p∥,

()

So, by induction, we have

()

hence {x_n} is bounded. we also obtain that {u_n}, {Tu_n}, {Tx_n}, {f(x_n)} and {y_n} are bounded.

Step 2. ∥x_n+1 − x_n∥→0 as n → ∞,

()

On the other hand, from and , we have

()

Putting y = u_n+1 in (3.9) and y = u_n in (3.10), we have

()

So, from (A₂) we have

()

and hence

()

Without loss of generality, let us assume that there exists a real number b such that r_n > b > 0 for all n ∈ N. Then, we have

()

and hence

()

where L = sup {∥u_n − x_n∥ : n ∈ N}. Then we obtain

()

So, put (3.8) and (3.16) into (3.7) we have

()

where K₁ : = sup {∥u − u_n∥, ∀ n ≥ 1} is a constant; K₂ : = sup {∥f(x_n−1)∥ + ∥Ty_n−1∥, ∀ n ≥ 1} is a constant.

Using Lemma 2.3 and conditions (I), (II), (III) we have

()

From (3.15) and |r_n+1 − r_n| → 0, we have

()

Since y_n = α_nu + (1 − α_n)u_n, x_n = β_nf(x_n−1) + (1 − β_n)Ty_n−1, we have

()

For p ∈ F(T)∩EP (ϕ), we have

()

Therefore, we have

()

By the above of what we have and the condition of lim _n→∞β_n = 0, we get lim _n→∞∥x_n − u_n∥ = 0. Since ∥Tu_n − u_n∥ ≤ ∥Tu_n − x_n∥ + ∥x_n − u_n∥, it follows that ∥Tu_n − u_n∥ → 0.

Step 3. we show that

()

where z = P_{F(S)∩EP (ϕ)}f(z). To show this inequality, we choose a subsequence

of {u_n} such that

()

Since

is bounded, there exists a subsequence

which converges weakly to w. Without loss of generality, we can assume that

. From ∥Tu_n − u_n∥→0, we obtain

. Let us show w ∈ EP (ϕ). By

, we have

()

From (A₂), we also have

()

and hence

()

Since

and

, from (A₄) we have 0 ≥ ϕ(y, w) for all y ∈ C. For t with 0 < t < 1 and y ∈ C, let y_t = ty + (1 − t)w. Since y ∈ C and w ∈ C, we have y_t ∈ C and hence ϕ(y_t, w) ≤ 0. So, from (A₁) and A₄ we have

()

and hence 0 ≤ ϕ(y_t, y). From (A₃), we have 0 ≤ ϕ(w, y) for all y ∈ C, and hence w ∈ EP (ϕ). We will show that w ∈ F(T). Assume that w ∉ F(T). Since

and w ≠ Tw, from Opial’s theorem we have

()

This is a contradiction. So, we get w ∈ F(T). Therefore, w ∈ F(T)∩EP (ϕ). Since z = P_{F(T)∩EP (ϕ)}f(z), we have

()

From x_n+1 − z = (1 − β_n)(Ty_n − z) + β_n(f(x_n) − z), we have

()

where

, δ_n = 2(1 − α)β_n/1 − αβ_n and ζ_n : = β_n/2(1 − α)M + (1 − β_n) ²α_n/2(1 − α)β_n∥u−z∥² + 1/1 − α〈f(z) − z, x_n+1 − z〉. It is easy to see that

and limsup _n→∞ζ_n/δ_n ≤ 0 by (3.31) and the conditions. Hence, by Lemma 2.3, the sequence {x_n} converges strongly to z.

If z_t is definition as (2.1), then, from Lemma 2.5, we have ∥z_t − q∥→0 as t → 0, and if we set Q(u): = lim _t→0z_t, then Q defines the unique sunny nonexpansive retraction from C onto Fix (T). So, if we replace t with α_n, the corollary still holds. and it is that z_n = T(α_nu + (1 − α_n)z_n) is a fixed point sequence and ∥z_n − q∥→0 as n → ∞, and if we set Q(u) : = lim _n→∞z_n, then Q defines the unique sunny nonexpansive retraction from C onto Fix (T). In the iterative algorithm of Theorem 3.1, we can take z_n to replace Ty_n in particular. Then, we have x_n+1 = β_nf(x_n)+(1 − β_n)z_n, so ∥x_n+1 − z_n∥ = β_n∥f(x_n) − z_n∥→0 as n → ∞. By the uniqueness of limit, we have z = q, that is, z = Q(u), where Q defines the unique sunny nonexpansive retraction from C onto Fix (T).

Remark 3. We notice that has not influence on x_n, u_n → z = P_{F(T)∩EP (ϕ)}f(z).

As direct consequences of Theorem 3.1, we obtain corollary.

Corollary 3.2. Let C be a nonempty closed convex subset of H, S : C → C be a nonexpansive mapping of C into H such that F(S) ≠ ∅. Let f be a contraction of H into itself and let {x_n} and {u_n} be sequences generated initially by an arbitrary elements x₁ ∈ H and then by

()

for all n ∈ N, where {α_n}⊂(0, ∞) satisfies the following conditions:

(I)
(II)
(III)
lim _n→∞α_n/β_n = 0.

Then, the sequences {x_n} converge strongly to z ∈ F(S), where z = P_F(S)f(z).

Proof. Put ϕ(x, y) = 0, for all x, y ∈ C and r_n = 1, for all n ∈ ℕ in Theorem 3.1.

Then we have u_n = P_Cx_n. So, from Theorem 3.1, the sequence x_n generated by x₁ ∈ H and

()

for all n ∈ ℕ converges strongly to z ∈ F(S), where z = P_F(S)f(z).

4. Application for Zeros of Maximal Monotone Operators

We adapt in this section the iterative algorithm (3.1) to find zeros of maximal monotone operators and find EP (ϕ). Let us recall that an operator A with domain D(A) and range R(A) in a real Hilbert space H with inner product 〈·, ·〉 and norm ∥·∥ is said to be monotone if the graph of A,

()

is a monotone set. Namely,

()

A monotone operator A is said to be maximal monotone of the graph G(T) is not properly contained in the graph of any other monotone defined in H. See Brezis [16] for more details on maximal monotone operators.

In this section we always assume that A is maximal monotone and the set of zeros of A, N(A) = {x ∈ D(A) : 0 ∈ Ax}, is nonempty so that the metric projection P_N(A) from H onto N(A) is well-defined.

One of the major problems in the theory of maximal operators is to find a point in the zero set N(A) because various problems arising from economics, convex programming, and other applied areas can be formulated as finding a zero of maximal monotone operators. The proximal point algorithm (PPA) of Rockafellar [17] is commonly recognized as the most powerful algorithm in finding a zero of maximal monotone operators. This (PPA) generates, starting with any initial guess x₀ ∈ H, a sequence {x_n} according to the inclusion:

()

where {e_n} is a sequence of errors and {c_n} is a sequence of positive regularization parameters. Equivalently, we can write

()

where for

denotes the resolvent of A,

()

with I being the identity operator on the space H.

Rockafellar [17] proved the weak convergence of his algorithm (4.4) provided the regularization sequence {c_n} remains bounded away from zero and the error sequence {e_n} satisfies the condition

()

The aim of this section is to combine algorithm (3.1) with algorithm (4.4). Our algorithm generates a sequence {x_n} and {u_n} be sequences generated initially by an arbitrary elements x₁ ∈ H and then by

()

where α_n and c_n are sequences of positive real numbers. Furthermore, we prove that {x_n} and {u_n} converge strongly to z ∈ N(A)∩EP (ϕ), where z = P_{N(A)∩EP (ϕ)}f(z).

Before stating the convergence theorem of the algorithm (4.7), we list some properties of maximal monotone operators.

Proposition 4.1. Let A be a maximal monotone operator in H and let denote the resolvent, where c > 0,

(a)
is nonexpansive for all c > 0;
(b)
N(A) = F(J_c) for all c > 0;
(c)
For ;
(d)
(The Resolvent Identity) For λ, μ > 0, there holds the identity:

()

Theorem 4.2. Let C be a nonempty closed convex subset of H, ϕ : C × C → ∞ be a bifunction satisfying (A₁)–(A₄) and A be a maximal monotone operator such that N(A)∩EP (ϕ) ≠ ∅. Let f be a contraction of H into itself and let {x_n} and {u_n} be sequences generated initially by an arbitrary elements x₀ ∈ H and then by

()

for all n ∈ N, where {α_n}⊂[0,1] and {r_n}⊂(0, ∞) satisfy the following conditions:

(I)
lim _n→∞α_n = 0, and ;
(II)
liminf _nr_n > 0 and ;
(III)
lim _n→∞(c_n+1/c_n) = 1;
(IV)
lim _n→∞β_n = 0, , , and lim _n→∞(α_n/β_n) = 0.

Then, the sequences {x_n} and {u_n} converge strongly to z ∈ N(A)∩EP (ϕ), where z = P_{N(A)∩EP (ϕ)}f(z).

Proof. Below we write for simplicity. Setting

()

we rewrite x_n+1 of (4.7) as

()

Because the proof is similar to Theorem 3.1, here we just give the main steps as follows:

(1)
{x_n} is bounded;
(2)
∥x_n+1 − x_n∥ → 0, as n → 0;
(3)
, as n → 0;
(4)
∥x_n − u_n∥ → 0, as n → 0;
(5)
limsup _n→∞〈f(z) − z, x_n − z〉 ≤ 0;
(6)
x_n, u_n → z, as n → z.

5. Application for Optimization Problem

In this section, we study a kind of optimization problem by using the result of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions

()

where h(x) is a convex and lower semicontinuous functional defined on a closed convex subset C of a Hilbert space H. We denoted by B the set of solutions of (5.1). Let ϕ be a bifunction from C × C to R defined by ϕ(x, y) = h(y) − h(x). We consider the following equilibrium problem, that is, to find x ∈ C such that

()

It is obvious that EP (ϕ) = B, where EP (ϕ) denotes the set of solutions of equilibrium problem (5.2). In addition, it is easy to see that ϕ(x, y) satisfies the conditions (A₁)–(A₄) in the Section 2. Therefore, from Theorem 3.1, we know that the following iterative algorithm:

()

for any initial guess x₁, converges strongly to a solution z = P_Bf(z) of optimization problem (5.1), where {α_n}⊂[0,1], {β_n}⊂[0,1], and {r_n}⊂[0, ∞) satisfy

()

For a special case, we pick f(x) = η, for all η ∈ H, and r_n = 1, β_n = 1/2 and α_n = 0 for all n ≥ 1, then x_n+1 = (1/2)Tu_n + (1/2)η, from (5.3), we get the special iterative algorithm as follows:

()

Then {u_n} converges strongly to a solution z = P_Bη of optimization problem (5.1).

In fact, z is the minimum norm point from η onto the B, furthermore, if η = 0, then z is the minimum norm point on the B.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants (11071270), and (10771050).

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An Alternative Regularization Method for Equilibrium Problems and Fixed Point of Nonexpansive Mappings

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Application for Zeros of Maximal Monotone Operators

5. Application for Optimization Problem

Acknowledgment

References

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An Alternative Regularization Method for Equilibrium Problems and Fixed Point of Nonexpansive Mappings

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Application for Zeros of Maximal Monotone Operators

5. Application for Optimization Problem

Acknowledgment

References

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