EXTRA-GRADIENT METHODS FOR QUASI-NONEXPANSIVE OPERATORS

Abstract

In this paper, we propose an Ishikawa-type extra-gradient iterative algorithm for finding a solution of split feasibility, fixed point problems and equilibrium problems of quasi-nonexpansive mappings. It is proven that under suitable conditions, the sequences generated by the proposed iterative algorithms converge weakly to a solution of the split feasibility, fixed point problems and equilibrium problems. An example is given to illustrate the main result of this paper.

1. Introduction

Let H1 and H2 be two real Hilbert spaces and C ⊂ H1 and Q ⊂ H2 be two empty closed convex sets. The symbols ℕ and ℝ are used to denote the set of positive integers and real numbers, respectively. Let A : H1 → H2 be a bounded linear operator with its adjoint A∗. Let T : C → C be a nonlinear mapping. The set of fixed points of T is denoted by Fix(T). Let F be a bi-function from Q × Q into ℝ. The classical equilibrium problem is to find u ∈ Q such that

The symbol EP(F) is used to denote the set of all solutions of the problem (1.1), that is,

The purpose of this paper is to study the following split feasibility, fixed point problems and equilibrium problems:

We use Γ to denote the set of solutions of (1.2), that is,

In the sequel, we assume Γ ≠ ϕ. A special case of the split feasibility, fixed point problems and equilibrium problems is the split feasibility problem:

Recently, it has been found that the split feasibility problem can be applied to study intensity-modulated radiation therapy (see [1-5,9]).

In 1976, to study the saddle point problem, Korpelevich [11] introduced the so-called extra-gradient method:

where λ > 0, A is a strongly monotone and Lipschitz continuous mapping and PC is a projection operator from H1 into C.

In 2012, He et al. [8] studied the split feasibility, fixed point problems and equilibrium problems. They proposed an iterative algorithm in the following manner:

where T : C → C is a quasi-nonexpansive mapping, A∗ is the adjoint of A, is the mapping defined in Lemma 2.1, {rn} ⊂ (0,+∞) with lim infn→∞ rn > 0, , η > 0 is a constant and αn ∈ [η, 1 − η] for n ∈ ℕ. The authors proved that the sequences generated by (1.3) converge weakly to an element x ∈ Γ = {x∗ : x∗ ∈ C ∩ Fix(T), Ax∗ ∈ Q ∩ EP(F)}.

In this paper, motivated by the work of He et al. [8], we propose an Ishikawa-type extra-gradient iterative algorithm for finding a solution of the split feasibility, fixed point problems and equilibrium problems involved quasi-nonexpansive mappings. We establish weak convergence theorems for the sequences generated by the proposed iterative algorithms. Our results extend and develop the corresponding results in [8].

 

2. Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by ⟨·, ·⟩ and ║·║, respectively. Let C be a nonempty closed convex subset of H. We write xn ⇀ x to indicate that the sequence {xn} converges weakly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,

Recall that the projection from H onto C, denoted by PC, is defined in such a way that for each x ∈ H, PCx is the unique point in C with the property:

Some important properties of projections are gathered in the following proposition.

Proposition 2.1. Given x ∈ H and z ∈ C,

For all x, y ∈ H, the following conclusions hold:

and

Recall that a mapping T : C → C is said to be quasi-nonexpansive if Fix(T) ≠ ϕ and ║Tx − Tp║ ≤ ║x − p║ for all x ∈ C and p ∈ Fix(T).

A Banach space (E, ║·║) is said to satisfy Opial's condition if for each sequence {xn} in E which converges weakly to a point x ∈ E, we have

It is well known that any Hilbert space satisfies Opial's condition.

Definition 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H and T be a mapping from C into C. The mapping T is called demiclosed if for any sequence {xn} which weakly converges to and if the sequence {T(xn)} strongly converges to z, then

Let F be a bifunction of C × C into ℝ satisfying the following conditions:

Lemma 2.3 ([7]). Let C be a nonempty closed convex subset of H and let F be a bi-function of C×C into ℝ satisfying (A1)-(A4). For r > 0, define a mapping : H → C as follows:

for all x ∈ H. Then the following hold:

(i) is single-valued and Fix() = EP(F) for any r > 0 and EP(F) is closed and convex;

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

Lemma 2.4 ([10]). Let H be a real Hilbert space and let {xn} be a bounded sequence in H such that there exists a nonempty closed convex set C of H satisfying:

Lemma 2.5 ([6]). Assume that F : C × C → ℝ satisfies Assumption 2.1 and let be defined as in Lemma 2.3. Let x, y ∈ H and r1, r2 > 0. Then

 

3. Main results

Theorem 3.1. Let H1, H2 be two real Hilbert spaces and let C ⊂ H1, Q ⊂ H2 be two nonempty closed convex sets. Let A : H1 → H2 be a bounded linear operator with its adjoint A∗. Let T : C → C be a quasi-nonexpansive mapping and let F : Q×Q → ℝ be a bi-function with Γ = {x∗ : x∗ ∈ C ∩ Fix(T), Ax∗ ∈ Q ∩ EP(F)} ≠ ϕ. Suppose T − I is demiclosed at 0. For x0 ∈ H1 arbitrarily, let {xn} be a sequence defined by the following Ishikawa-type extra-gradient iterative algorithm:

where {rn} ⊂ (0,+∞), and {αn}, {βn} ⊂ (0, 1) are two real numbers satisfying 0 < b < αn, βn < a < 1.

Then the sequence {xn} generated by algorithm (3.1) converges weakly to an element of Γ.

Proof. Let x∗ ∈ Γ. Then we have x∗ ∈ C ∩ Fix(T) and Ax∗ ∈ Q ∩ EP(F). For simplicity, we write for all n ≥ 1. For each n ≥ 1, by Lemma 2.1, we have

So, we obtain

for any n ≥ 1. By (2.2), we have

Since A is a linear operator with its adjoint A∗, we have

Again using (2.2), we obtain

From (3.2), (3.4) and (3.5), we get

Substituting (3.6) into (3.3), we deduce

Since PC is nonexpansive, we get

Since is nonexpansive, we know that I - is -inverse strongly monotone. Therefore, it is easy to see that -inverse strongly monotone, that is,

and

It follows from Proposition 2.1, (2.2), (3.9) and (3.10) that

From Proposition 2.1 and (3.9), we have

From the assumption of ε, we obtain

Similarly, we have

From (2.1), we have

From (2.1), (3.1), (3.11) and (3.13), we have

The inequality (3.14) implies that limn→∞ ║xn − x∗║2 exists. This implies that {xn} is bounded. Additionally, we get the boundedness of {yn} and {zn} from (3.8) and (3.11) immediately. Returning to (3.11), (3.12) and (3.14), we have

and

Hence

and

which imply that

and

From (3.7) and (3.8), we have

which implies that

By (3.11) and (3.14), we have

It follows that

This implies that

From (3.13), we obtain

for any n ∈ ℕ. It is from (3.18) that we get

Since the sequence {xn} is bounded, we can choose a subsequence {xni} of {xn} such that . Consequently, we derive from (3.15)-(3.17) that

Since I − T is demiclosed at 0 and (3.19), we deduce

Next we claim for any r > 0. Suppose that for any r > 0. From (3.17) and Lemma 2.3, we have

which lead to a contradiction. So, . Note that yni = PCuni ∈ C and . From (3.20), we deduce . To this end, we deduce . That is to say . This shows that ωw(xn) ⊂ Γ. Since the limn→∞ ║xn−x∗║ exists for every x∗ ∈ Γ, the weak convergence of the whole sequence {xn} follows by applying Lemma 2.2. This completes the proof. □

Furthermore, we can immediately obtain the following weak convergence result.

Corollary 3.2. Let H1 and H2 be two real Hilbert spaces and let C ⊂ H1 and Q ⊂ H2 be two nonempty closed convex sets. Let A : H1 → H2 be a bounded linear operator with its adjoint A∗. Let T : C → C be a quasi-nonexpansive mapping and let F : Q × Q → ℝ be a bi-function with Γ = {x∗ : x∗ ∈ C ∩ Fix(T), Ax∗ ∈ Q ∩ EP(F)} ≠ ϕ. Suppose T − I is demiclosed at 0. For x0 ∈ H1 arbitrarily, let {xn} be a sequence defined by the following Ishikawa-type iterative algorithm:

where {rn} ⊂ (0,+∞), {αn}, {βn} ⊂ (0, 1) such that 0 < b < αn, βn < a < 1 and is a constant. Then the sequence {xn} generated by algorithm (3.21) converges weakly to an element of Γ.

Proof. Let x∗ ∈ Γ. Then we have x∗ ∈ C ∩ Fix(T) and Ax∗ ∈ Q ∩ EP(F). For simplicity, we write for all n ≥ 1. By (3.7) and (3.8), we have

and

From (2.1), (3.21) and (3.23), we have

and

So, we obtain limn→∞ ║xn − x∗║2 exists immediately. This implies that {xn} is bounded. Additionally, we get the boundedness of {yn} and {zn} from (3.23) and (3.24) immediately. Returning to (3.22), (3.23) and (3.25), we have

Hence,

which implies that

Using the firm nonexpansiveness of PC, Proposition 2.1 and (3.22), we have

Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.1 can be obtained from Theorem 3.1 immediately. □

Remark 3.1. Theorem 3.1 extends and develops the corresponding one of He and Du [8] in the following aspects:

The corresponding iterative algorithms in [8], Theorem 3.1 and Corollary 3.1 are extended for developing our Ishikawa-type extra-gradient iterative algorithms for the split common solution problem in Theorem 3.1 and Corollary 3.1.

Example 3.3. Let H1 = H2 = ℝ with the inner product defined by ⟨x, y⟩ = xy for all x, y ∈ ℝ and the absolute valued norm |·|. Let C = [0,+∞), Q = (−∞, 0] and for all x ∈ C. Obviously, Fix(T) = {2}. It is easy to see that

for all x ∈ C. Thus, T is a continuous quasi-nonexpansive mapping.

Let {xn} be a sequence in C such that xn ⇀ z ∈ C and ║xn − Txn║ → 0 as n → ∞. Then z ∈ Fix(T) = {2}. Therefore, T is zero-demiclosed.

Define the mapping A : ℝ → ℝ and F : Q × Q → ℝ as follows:

and

Then A is a bounded linear operator with A∗ = A and ║A║ = 2, and F satisfies the conditions (A1)-(A4) with EP(F) = {−4}. Obviously, Γ = {x∗ : x∗ ∈ C ∩ Fix(T), Ax∗ ∈ Q ∩ EP(F)} = {2} = Fix(T). For rn = 1, is equivalent to

Hence, we can easily find vn = −xn − 2 ∈ Q. It is not hard to compute for all n ∈ ℕ. Hence, for , we have

for all n ∈ ℕ. So, we get

and

for all n ∈ ℕ. For every n ≥ 1, from (3.26) and (3.27), we can rewrite (3.1) as follows:

Observe that for all n ≥ 1,

Hence we have for all n ≥ 1. This implies that {xn} converges to 2.