Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings

Abstract

In this article, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

1. Introduction

Let H be a real Hilbert space with inner product 〈.,.〉 and induced norm ||·||. Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to R and let B : C → H be a nonlinear mapping, where R is the set of real numbers. Moudafi [1], Moudafi and Thera [2], Peng and Yao [3, 4], Takahashi and Takahashi [5] considered the following generalized equilibrium problem:

(1.1)

The set of solutions of (1.1) is denoted by GEP(F, B). If B = 0, the generalized equilibrium problem (1.1) becomes the equilibrium problem for F : C × C → R, which is to find x ∈ C such that

(1.2)

The set of solutions of (1.2) is denoted by EP(F).

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance [1–7].

Recall that a mapping S : C → C is nonexpansive if there holds that

We denote the set of fixed points of S by Fix(S).

Let the mapping A : C → H be monotone and k-Lipschitz-continuous. The variational inequality problem is to find x ∈ C such that

for all y ∈ C. The set of solutions of the variational inequality problem is denoted by V I(C, A).

Several algorithms have been proposed for finding the solution of problem (1.1). Moudafi [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a weak convergence theorem. Moudafi and Thera [2] introduced an auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [3, 4] introduced some iterative schemes for finding a common element of the set of solutions of problem (1.1), the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping and obtain both strong convergence theorems, and weak convergence theorems for the sequences generated by the corresponding processes in Hilbert spaces. Takahashi and Takahashi [5] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence theorem.

Some methods also have been proposed to solve the problem (1.2); see, for instance, [8–19] and the references therein. Takahashi and Takahashi [9] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a non-expansive mapping, and proved a strong convergence theorem in a Hilbert space. Su et al. [10] introduced and researched an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly monotone mapping in a Hilbert space. Tada and Takahashi [11] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and obtained both strong convergence and weak convergence theorems. Plubtieng and Punpaeng [12] introduced an iterative processes based on the extragradient method for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of variational inequality problem for an α-inverse-strongly monotone mapping. Chang et al. [13] introduced an iterative processes based on the extragradient method for finding the common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of variational inequality problem for an α-inverse-strongly monotone mapping. Yao et al. [14] and Ceng and Yao [15] introduced some iterative viscosity approximation schemes for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely nonexpansive mappings in a Hilbert space. Colao et al. [16] introduced an iterative viscosity approximation scheme for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a family of finitely nonexpansive mappings in a Hilbert space. We observe that the algorithms in [13–16] involves the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings which is an effective tool in nonlinear analysis (see [20, 21]). However, the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings is too completed to use for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely (finitely) nonexpansive mappings. It is natural to raise and to give an answer to the following question: Can one construct algorithms for finding a common element of the set of solutions of a generalized equilibrium problem (an equilibrium problem), the common set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of a variational inequality without the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings? In this article, we will give a positive answer to this question.

Recently, OHaraa et al. [22] introduced and researched an iterative approach for finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces without using the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings. Inspired by the ideas in [1–6, 8–16, 22] and the references therein, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz--continuous mapping without using the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings. We obtain both strong convergence theorems and weak convergence theorems for the sequences generated by the corresponding processes. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

2. Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. Let C be a nonempty closed convex subset of H. Let symbols → and ⇀ denote strong and weak convergences, respectively. In a real Hilbert space H, it is well known that

for all x, y ∈ H and λ ∈ [0, 1].

For any x ∈ H, there exists the unique nearest point in C, denoted by P _C (x), such that ||x - P _C (x)|| ≤ ||x - y|| for all y ∈ C. The mapping P _C is called the metric projection of H onto C. We know that P _C is a nonexpansive mapping from H onto C. It is also known that P _C x ∈ C and

(2.1)

for all x ∈ H and y ∈ C.

It is easy to see that (2.1) is equivalent to

(2.2)

for all x ∈ H and y ∈ C.

A mapping A of C into H is called monotone if

for all x, y ∈ C. A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that

for all x, y ∈ C. A mapping A : C → H is called k-Lipschitz-continuous if there exists a positive real number k such that

for all x, y ∈ C. It is easy to see that if A is α-inverse-strongly monotone, then A is monotone and Lipschitz-continuous. The converse is not true in general. The class of α-inverse-strongly monotone mappings does not contain some important classes of mappings even in a finite-dimensional case. For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively definite, then the mapping A will be monotone and Lipschitz-continuous, but not α-inverse-strongly monotone (see [23]).

Let A be a monotone mapping of C into H. In the context of the variational inequality problem, the characterization of projection (2.1) implies the following:

and

It is also known that H satisfies the Opial's condition [24], i.e., for any sequence {x _n } ⊂ H with x _n ⇀ x, the inequality

holds for every y ∈ H with x ≠ y.

A set-valued mapping T : H → 2 ^H is called monotone if for all x, y ∈ H, f ∈ Tx and g ∈ Ty imply 〈x - y, f - g〉 ≥ 0. A monotone mapping T : H → 2 ^H is maximal if its graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f) ∈ H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) ∈ G(T) implies f ∈ Tx. Let A be a monotone, k-Lipschitz-continuous mapping of C into H and N _C v be normal cone to C at v ∈ C, i.e., N _C v = {w ∈ H : 〈v - u, w〉 ≥ 0, ∀u ∈ C}. Define

Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ V I(C, A) (see [25]).

For solving the equilibrium problem, let us assume that the bifunction F satisfies the following condition:

(A1) F(x, x) = 0 for all x ∈ C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0 for any x, y ∈ C;

(A3) for each x, y, z ∈ C,

(A4) for each x ∈ C, y ↦ F(x, y) is convex and lower semicontinuous.

We recall some lemmas which will be needed in the rest of this article.

Lemma 2.1.[7] Let C be a nonempty closed convex subset of H, let F be a bifunction from C × C to R satisfying (A1)-(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that

Lemma 2.2.[8] Let C be a nonempty closed convex subset of H, let F be a bi-function from C × C to R satisfying (A1)-(A4). For r > 0 and x ∈ H, define a mapping Tr : H → C as follows:

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

• (1) T _r is single-valued;

• (2) T _r is firmly nonexpansive, i.e., for any x, y ∈ H,

  

• (3) F(T _r ) = EP (F);

• (4) EP(F) is closed and convex.

3. The main results

We first show a strong convergence of an iterative algorithm based on extragradient and hybrid methods which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n }, {y _n } and {z _n } be sequences generated by

(3.1)

for every n = 1, 2,... where {λ _n } ⊂ [a, b] for some , {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α), and {α _n }, {β _n }, {γ _n } are three sequences in [0, 1] satisfying the conditions:

• (i) α _n + β _n ≤ 1 for all n ∈ N;

• (ii) ;

• (iii) ;

• (iv) and for all n ∈ N;

Then, {x _n }, {u _n }, {y _n } and {z _n } converge strongly to w = P_Ω(x).

Proof. It is obvious that C _n is closed, and Q _n is closed and convex for every n = 1, 2,.... Since

we also have that C _n is convex for every n = 1, 2,.... It is easy to see that 〈x _n - z, x - x _n 〉 ≥ 0 for all z ∈ Q _n and by (2.1), . Let t _n = P _C (u _n - λ _n Ay _n ) for every n = 1, 2,.... Let u ∈ Ω and let >be a sequence of mappings defined as in Lemma 2.2. Then . From and the α-inverse strongly monotonicity of B, we have

(3.2)

From (2.2), the monotonicity of A, and u ∈ V I(C, A), we have

Further, Since y _n = (1 - γ _n )u _n + γ _n P _C (u _n - λ _n Au _n ) and A is k-Lipschitz-continuous, we have

In addition, from the definition of P _C , we have

It follows from , and (3.2) that

(3.3)

In addition, from u ∈ V I(C, A) and (3.2), we have

(3.4)

Therefore, from (3.2) to (3.4) and z _n = (1 - α _n - β _n )x _n + α _n y _n + β _n S _n t _n and u = S _n u, we have

(3.5)

for every n = 1, 2,... and hence u ∈ C _n . So, Ω ⊂ C _n for every n = 1, 2,.... Next, let us show by mathematical induction that x _n is well defined and Ω ⊂ C _n ∩ Q _n for every n = 1, 2,.... For n = 1 we have x₁ = x ∈ C and Q₁ = C. Hence, we obtain Ω ⊂C₁ ∩ Q₁. Suppose that x _k is given and Ω ⊂ C _k ∩ Q _k for some k ∈ N. Since Ω is nonempty, C _k ∩ Q _k is a nonempty closed convex subset of H. Hence, there exists a unique element x_k+1∈ C _k ∩ Q _k such that . It is also obvious that there holds 〈x_k+1- z, x - x_k+1〉 ≥ 0 for every z ∈ C _k ∩ Q _k . Since Ω ⊂ C _k ∩ Q _k , we have 〈x_k+1- z, x - x_k+1〉 ≥ 0 for every z ∈ Ω and hence Ω ⊂ Q_k+1. Therefore, we obtain Ω ⊂ C_k+1∩ Q_k+1.

Let l₀ = P_Ωx. From and l₀ v Ω ⊂ C _n ∩ Q _n , we have

(3.6)

for every n = 1, 2,.... Therefore, {x _n } is bounded. From (3.2) to (3.5) and the lipschitz continuity of A, we also obtain that {u _n }, {y _n }, {Au _n }, {t _n } and {z _n } are bounded. Since x_n+1∈ C _n ∩ Q _n ⊂ C _n and , we have

for every n = 1, 2,.... It follows from (3.6) that lim_n→∞||x _n - x|| exists.

Since and x_n+1∈ Q _n , using (2.2), we have

for every n = 1, 2,.... This implies that

Since x_n+1∈ C _n , we have ||z _n - x_n+1||² ≤ ||x _n - x_n+1||² + (3 - 3γ _n + α _n )b²||Au _n ||² and hence it follows from lim_n→∞γ _n = 1 and lim_n→∞α _n = 0 that lim_n→∞||z _n - x_n+1|| = 0. Since

for every n = 1, 2,..., we have ||x _n - z _n || → 0.

For u ∈ Ω, from (3.5), we obtain

Since lim_n→∞γ _n = 1 and lim_n→∞α _n = 0, {x _n }, {y _n }, {Au _n }, and {z _n } are bounded, we have

By lim inf_n→∞β _n > 0, we get

From (3.3) and u = S _n u, we have

Thus, lim_n→∞||t _n - u||²- ||x _n - u||² = 0.

From (3.3) and (3.2), we have

It follows that

The assumptions on γ _n and λ _n imply that and . Consequently, lim_n→∞||u _n - y _n || = lim_n→∞||t _n - y _n || = 0. Since A is Lipschitz-continuous, we have lim_n→∞||At _n - Ay _n || = 0. It follows from ||u _n - t _n || ≤ ||u _n - y _n || + ||t _n - y _n || that lim_n→∞||u _n - t _n || = 0.

We rewrite the definition of z _n as

From lim_n→∞||z _n - x _n || = 0, lim_n→∞α _n = 0, the boundedness of {x _n }, {y _n } and lim inf_n→∞β _n > 0 we infer that lim_n→∞||S _n t _n - x _n || = 0.

By (3.2)-(3.5), we have

(3.7)

Hence, we have

Since lim_n→∞α _n = 1, lim inf_n→∞β _n > 0, lim_n→∞γ _n = 1, ||x _n - z _n || → 0 and the sequences {x _n } and {z _n } are bounded, we obtain ||Bx _n - B _u || → 0.

For u ∈ Ω, we have, from Lemma 2.2,

Hence,

Then, by (3.5), we have

Hence,

Since lim_n→∞α _n = 0, lim inf_n→∞β _n > 0, lim_n→∞γ _n = 1, ||x _n - z _n || → 0, ||Bx _n - Bu|| → 0 and the sequences {x _n }, {u _n } and {z _n } are bounded, we obtain ||x _n - u _n || → 0. From ||z _n - t _n || ≤ ||z _n - x _n ||+||x _n - u _n ||+||u _n - t _n ||, we have ||z _n - t _n || → 0.

From ||t _n - x _n || ≤ ||t _n - u _n || + ||x _n - u _n ||, we also have ||t _n - x _n || → 0.

Since z _n = (1 - α _n - β _n )x _n + α _n y _n + β _n S _n t _n , we have β _n (S _n t _n - t _n ) = (1 - α _n - β _n )(t _n - x _n ) + α _n (t _n - y _n ) + (z _n - t _n ). Then

and hence ||S _n t _n - t _n || → 0. At the same time, observe that for all i ∈ {1, 2,...},

It follows from (3.8) and the condition (*) that for all i ∈ {1, 2,...},

(3.9)

As {x _n } is bounded, there exists a subsequence of {x _n } such that x _ni ⇀ w. From ||x _n - u _n || → 0, we obtain that u _ni ⇀ w. From ||u _n - t _n || → 0, we also obtain that t _ni ⇀ w. Since {u _ni } ⊂ C and C is closed and convex, we obtain w ∈ C.

First, we show w ∈ GEP(F, B). By , we know that

It follows from (A2) that

Hence,

(3.10)

For t with 0 < t ≤ 1 and y ∈ C, let y _t = t _y + (1 - t)w. Since y ∈ C and w ∈ C, we obtain y _t ∈ C. So, from (3.10) we have

Since , we have . Further, from the inverse-strongly monotonicity of B, we have . Hence, from (A4), and , we have

(3.11)

as i → ∞. From (A1), (A4) and (3.11), we also have

and hence

Letting t → 0, we have, for each y ∈ C,

This implies that w ∈ GEP(F, B).

We next show that . Assume . Since and for some i₀ ∈ {1, 2,...} from the Opial condition, we have

This is a contradiction. Hence, we get .

Finally we show w ∈ V I(C, A). Let

where N _C v is the normal cone to C at v ∈ C. We have already mentioned that in this case the mapping T is maximal monotone, and 0 ∈ Tv if and only if v ∈ V I(C, A). Let (v, g) ∈ G(T). Then Tv = Av + N _C v and hence g - Av ∈ N _C v.

Hence, we have 〈v - t, g - Av〉 ≥ 0 for all t ∈ C. On the other hand, from t _n = P _C (u _n - λ _n Ay _n ) and v ∈ C, we have

and hence

Therefore, we have

Hence, we obtain 〈v - w, g〉 ≥ 0 as i → ∞. Since T is maximal monotone, we have w ∈ T^-10 and hence w ∈ V I(C, A). This implies that w ∈ Ω.

From l₀ = P_Ωx, w ∈ Ω and (3.6), we have

Hence, we obtain

From , we have , and hence . Since and l₀ ∈ Ω ⊂ C _n ∩ Q _n ⊂ Q _n , we have

As i → ∞, we obtain - ||l₀- w||² ≥ 〈l₀- w, x - l₀〉 ≥ 0 by l₀ = P_Ωx and w ∈ Ω. Hence, we have w = l₀. This implies that x _n → l₀. It is easy to see u _n → l₀, y _n → l₀ and z _n → l₀. The proof is now complete.

By combining the arguments in the proof of Theorem 3.1 and those in the proof of Theorem 3.1 in [3], we can easily obtain the following weak convergence theorem for an iterative algorithm based on the extragradient method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone, and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n } and {y _n } be the sequences generated by

(3.12)

for every n = 1, 2,.... If {λ _n } ⊂ [a, b] for some , {β _n } ⊂ [δ, ε] for some δ, ε ∈ (0, 1) and {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α). Then, {x _n }, {u _n } and {y _n } converge weakly to w ∈ Ω, where w = lim_n→∞P_Ωx _n .

4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting convergence theorems in a real Hilbert space. We give some examples as follows:

Let A = 0, by Theorems 3.1 and 3.2, respectively, we obtain the following results.

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n } {y _n }, and {z _n } be the sequences generated by

for every n = 1, 2,.... where {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α), and {α _n }, {β _n } are sequences in [0, 1] satisfying the conditions:

• (i) α _n + β _n ≤ 1 for all n ∈ N;

• (ii) ;

• (iii) for all n ∈ N;

Then, {x _n }, {u _n }, and {z _n } converge strongly to w = P_∑(x).

Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n } and {u _n } be sequences generated by

for every n = 1, 2,.... If {β _n } ⊂ [δ, ε] for some δ, ε ∈ (0, 1) and {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α). Then, {x _n } and {u _n } converge weakly to w ∈ ∑, where w = lim_n→∞P_∑x _n .

Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n }, {y _n }, and {z _n } be sequences generated by

for every n = 1, 2,... where {λ _n } ⊂ [a, b] for some , {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α), and {β _n } is a sequence in [0, 1] satisfying . Then, {x _n }, {u _n }, {y _n }, and {z _n } converge strongly to w = P_Ω(x).

Proof. Putting γ _n = 1 and α _n = 0, by Theorem 3.1, we obtain the desired result.

Let B = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results.

Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...}, and for any bounded subset K of C, there holds

Let {x _n }, {u _n }, {y _n }, and {z _n } be the sequences generated by

for every n = 1, 2,.... where {λ _n } ⊂ [a, b] for some , {r _n } ⊂ [d, +∞) for some d > 0, and {α _n }, {β _n }, {γ _n } are three sequences in [0, 1] satisfying the following conditions:

• (i) α _n + β _n ≤ 1 for all n ∈ N;

• (ii) ;

• (iii) ;

• (iv) and for all n ∈ N;

Then, {x _n }, {u _n }, {y _n } and {z _n } converge strongly to w = P_Λ(x).

Theorem 4.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n }, and {y _n } be the sequences generated by

for every n = 1, 2,.... If {λ _n } ⊂ [a, b] for some ,{β _n } ⊂ [δ, ε], for some δ, ε ∈ (0, 1) and {r _n } ⊂ [d, +∞] for some d > 0, then {x _n }, {u _n } and {y _n } converge weakly to w ∈ Λ, where w = lim_n→∞P_Λx _n .

Theorem 4.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n } {y _n }, and {z _n } be the sequences generated by

for every n = 1, 2,.... where {λ _n } ⊂ [a, b] for some , {r _n } ⊂ [d, +∞) and for some d > 0, and {β _n } is a sequence in [0, 1] satisfying . Then, {x _n }, {u _n }, {y _n }, and {z _n } converge strongly to w = P_Λ(x).

Let B = 0 and F(x, y) = 0 for x, y ∈ C, by Theorems 3.1 and 4.3, we obtain the following results.

Theorem 4.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {y _n }, and {z _n } be the sequences generated by

for every n = 1, 2,.... where {λ _n } ⊂ [a, b] for some , and {α _n }, {β _n }, {γ _n }are three sequences in [0, 1] satisfying the following conditions:

• (i) α _n + β _n ≤ 1 for all n ∈ N;

• (ii) ;

• (iii) ;

• (iv) and for all n ∈ N;

Then, {x _n }, {y _n }, and {z _n } converge strongly to w = P_Γ(x).

Theorem 4.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {y _n }, and {z _n } be the sequences generated by

for every n = 1, 2,.... where {λ _n } ⊂ [a, b] for some , and {β _n } is a sequence in [0, 1] satisfying . Then, {x _n }, {y _n }, and {z _n } converge strongly to w = P_Γ(x).

Let F(x, y) = 0 for x, y ∈ C, then by Theorem 3.2 and the proof of Theorem 4.7 in [3], we obtain the following result.

Theorem 4.9. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H. Let S₁, S₂,... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i ∈ {1, 2,...} and for any bounded subset K of C, thenthere holds

Let {x _n }, {u _n }, and {y _n } be the sequences generated by

for every n = 1, 2,.... if {λ _n } ⊂ [a, b] for some , {β _n } ⊂ [δ, ε] for some δ, ε ∈ (0, 1) and {r _n } ⊂ [d, e] for some d, e ∈ (0, 2α). Then, {x _n } and {u _n } converge weakly to w ∈ Ξ, where w = lim_n→∞P_Ξx _n .

Remark 4.1.

• (i) For all n ≥ 1, let S _n = S be a nonexpansive mapping, by Theorems 3.2, 4.2, 4.7, 4.8, and 4.9 we recover Theorem 3.1 in [5], Theorem 3.1 in [1], Theorem 5 in [26], Theorem 3.1 in [23], and Theorem 4.7 in [3]. In addition, let A = 0, by Theorems 4.6 and 4.5, respectively, we recover Theorems 3.1 and 4.1 in [11].

• (ii) For all n ≥ 1, let S _n = S be a nonexpansive mapping, by Theorems 3.1, 4.3, and 4.4, respectively, we recover Theorems 4.3, 4.4, and 4.7 in [4] with some modified conditions on F.

• (iii) Theorems 3.1, 3.2, 4.3-4.7 also improve the main results in [10, 12, 13] because the inverse strongly monotonicity of A has been replaced by the monotonicity and Lipschitz continuity of A.

The following result illustrates that there are the nonexpansive mappings S₁, S₂ ,... satisfying the condition (*).

Lemma 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that Fix(T) ≠ ∅. If we define for n ∈ {1, 2,...}, and x ∈ C, then the following results hold:

• (a) For any bounded subset K of C, there holds

  

• (b) .

• (c) for all i ∈ {1, 2,...} and for any bounded subset K of C, there holds

  

Proof.

• (a) It is due to Bruck [27, 28] (please also see Lemma 3.1 in [22]).

• (b) It follows from (a) that .

  Moreover, it is obvious that . Hence, .

• (c) It can be proved by mathematical induction. In fact, it is clear that this conclusion holds for i = 1. Assume that the conclusion holds for i = m, that is, for any bounded subset K of C, there holds

  

  (4.1)

We now prove that the conclusion also holds for i = m + 1. In fact, we observe that

(4.2)

It is easy to verify that S₁, S₂,... are nonexpansive mappings. It follows from (4.1) and (4.2) that for any bounded subset K of C, there holds

From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain the following results.

Theorem 4.10. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H. Let T be a nonexpansive mapping of C into itself such that Θ = Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅. Let {λ _n } ⊂ [a, b] for some , {r _n } ⊂ [d, e] and for some d, e ∈ (0, 2α), and {α _n }, {β _n }, and {γ _n } be three sequences in [0, 1] satisfying the following conditions:

• (i) α _n + β _n ≤ 1 for all n ∈ N;

• (ii) ;

• (iii) ;

• (iv) and for all n ∈ N; If we define for n ∈ {1, 2,...}, and x ∈ C, then the sequences {x _n }, {u _n }, {y _n }, and {z _n } generated by algorithm (3.1) converge strongly to w = P _Θ(x).

Theorem 4.11. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an α-inverse-strongly monotone mapping of C into H, and T be a nonexpansive mapping of C into itself such that Θ = Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅. Assume that {λ _n } ⊂ [a, b] for some {β _n } ⊂ [δ, ε] for some δ, ε ∈ (0, 1), and {r _n } ⊂ [d, e] some d, e ∈ (0, 2α). If we define for n ∈ {1, 2,...} and x ∈ C, then the sequences {x _n }, {u _n }, and {y _n } generated by algorithm (3.12) converge weakly to w ∈ Θ, where w = lim_n→∞P_Θx _n .