We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi′s hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

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 P_C be the metric projection from H onto C. Let S : C → C be a self-mapping on C. We denote by Fix (S) the set of fixed points of S and by R the set of all real numbers. A mapping A : C → H is called monotone if

()

A mapping A : C → H is called L-Lipschitz continuous if there exists a constant L > 0, such that

()

For a given mapping A : C → H, we consider the following variational inequality (VI) of finding x^* ∈ C, such that

()

The solution set of the VI (1.3) is denoted by VI (C, A). The variational inequality was first discussed by Lions [1] and now is well known. Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, and equilibrium problems; see, for example, [2–4]. To construct a mathematical model which is as close as possible to a real complex problem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblem or the solution of one subproblem on the solution set of another subproblem. Actually, these subproblems can be given by problems of different types. For example, Antipin considered a finite-dimensional variant of the variational inequality, where the solution should satisfy some related constraint in inequality form [5] or some system of constraints in inequality and equality form [6]. Yamada [7] considered an infinite-dimensional variant of the solution of the variational inequality on the fixed-point set of some mapping.

A mapping A : C → H is called α-inverse strongly monotone if there exists a constant α > 0, such that

()

see [8, 9]. It is obvious that an α-inverse strongly monotone mapping A is monotone and Lipschitz continuous. A self-mapping S : C → C is called k-strictly pseudocontractive if there exists a constant k ∈ [0,1), such that

()

see [10]. In particular, if k = 0, then S is called a nonexpansive mapping; see [11].

A set-valued mapping M with domain D(M) and range R(M) in H is called monotone if its graph G(M) = {(x, f) ∈ H × H : x ∈ D(M), f ∈ Mx} is a monotone set in H × H; that is, M is monotone if and only if

()

A monotone set-valued mapping M is called maximal if its graph G(M) is not properly contained in the graph of any other monotone mapping in H.

Let Φ be a single-valued mapping of C into H, and let M be a multivalued mapping with D(M) = C. Consider the following variational inclusion: find x^* ∈ C, such that

()

We denote by Ω the solution set of the variational inclusion (1.7). In particular, if Φ = M = 0, then Ω = C.

In 1998, Huang [12] studied problem (1.7) in the case where M is maximal monotone, and Φ is strongly monotone and Lipschitz continuous with D(M) = C = H. Subsequently, Zeng et al. [13] further studied problem (1.7) in the case which is more general than Huang′s one [12]. Moreover, the authors [13] obtained the same strong convergence conclusion as in Huang′s result [12]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions.

In 2003, for finding an element of Fix (S)∩VI (C, A) when C ⊂ H is nonempty, closed, and convex, S : C → C is nonexpansive, and A : C → H is α-inverse strongly monotone. Takahashi and Toyoda [14] introduced the following iterative algorithm:

()

where x₀ ∈ C chosen arbitrarily, {α_n} is a sequence in (0,1), and {λ_n} is a sequence in (0,2α). They showed that, if Fix (S)∩VI (C, A) ≠ ∅, then the sequence {x_n} converges weakly to some z ∈ Fix (S)∩VI (C, A). In 2006, to solve this problem (i.e., to find an element of Fix (S)∩VI (C, A)), Nadezhkina and Takahashi [15] introduced an iterative algorithm by a hybrid method. Generally speaking, the suggested algorithm is based on two well-known types of methods, that is, on the extragradient-type method due to Korpelevich [16] for solving variational inequality and so-called hybrid or outer-approximation method due to Haugazeau (see [15]) for solving fixed point problem. It is worth emphasizing that the idea of “hybrid” or “outer-approximation” types of methods was successfully generalized and extended in many papers; see, for example, [17–23]. In addition, the idea of the extragradient iterative algorithm introduced by Korpelevich [16] was successfully generalized and extended not only in Euclidean but also in Hilbert and Banach spaces; see, for example, [24–29].

Theorem NT (see [15], Theorem 3.1.)Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be a monotone and k-Lipschitz-continuous mapping, and let S : C → C be a nonexpansive mapping such that Fix (S)∩VI (C, A) ≠ ∅. Let {x_n}, {y_n} and {z_n} be the sequences generated by

()

where x₀ ∈ C is chosen arbitrarily, {λ_n}⊂[a, b] for some a, b ∈ (0, 1/k), and {α_n}⊂[0, c] for some c ∈ [0,1). Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to P_{Fix (S)∩VI (C,A)}x₀.

It is easy to see that the class of α-inverse strongly monotone mappings in the above mentioned problem of Takahashi and Toyoda [14] is the quite important class of mappings in various classes of well-known mappings. It is also easy to see that while α-inverse strongly monotone mappings are tightly connected with the important class of nonexpansive mappings, α-inverse strongly monotone mappings are also tightly connected with the more general and also quite important class of strictly pseudocontractive mappings. That is, if a mapping S : C → C is nonexpansive, then the mapping I − S is (1/2-) inverse strongly monotone; moreover, Fix (S) = VI (C, I − S) (see, e.g., [14]). The construction of fixed points of nonexpansive mappings via Mann′s algorithm has extensively been investigated in the literature (see, e.g., [30, 31] and references therein). At the same time, if a mapping S : C → C is k-strictly pseudocontractive, then the mapping I − S is (1 − k)/2-inverse strongly monotone and 2/(1 − k)-Lipschitz continuous.

Let B₁, B₂ : C → H be two mappings. Recently, Ceng et al. [32] introduced and considered the following problem of finding (x^*, y^*) ∈ C × C, such that

()

which is called a general system of variational inequalities (GSVI), where μ₁ > 0 and μ₂ > 0 are two constants. The set of solutions of problem (1.10) is denoted by GSVI(C, B₁, B₂). In particular, if B₁ = B₂ = A, then problem (1.10) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [33]. Further, if x^* = y^* additionally, then the NSVI reduces to the VI (1.3).

In particular, if B₁ = A and B₂ = 0, then the GSVI (1.10) is equivalent to the VI (1.3).

Indeed, in this case, the GSVI (1.10) is equivalent to the following problem of finding (x^*, y^*) ∈ C × C, such that

()

Thus we must have x^* = y^*. As a matter of fact, if x^* ≠ y^*, then by setting x = x^* we have

()

which hence leads to a contradiction. Therefore, the GSVI (1.10) coincides with the VI (1.3).

Recently, Ceng at al. [32] transformed problem (1.10) into a fixed-point problem in the following way.

Lemma 1.1 (see [32].)For given is a solution of problem (1.10) if and only if is a fixed point of the mapping G : C → C defined by

()

where

In particular, if the mapping B_i : C → H is β_i-inverse strongly monotone for i = 1,2, then the mapping G is nonexpansive provided μ_i ∈ (0,2β_i] for i = 1,2.

Utilizing Lemma 1.1, they introduced and studied a relaxed extragradient method for solving the GSVI (1.10). Throughout this paper, the set of fixed points of the mapping G is denoted by Ξ. Based on the relaxed extragradient method and viscosity approximation method, Yao et al. [34] proposed and analyzed an iterative algorithm for finding a common solution of the GSVI (1.10) and the fixed point problem of a strictly pseudocontractive mapping S : C → C.

Subsequently, Ceng et al. [35] further presented and analyzed an iterative scheme for finding a common element of the solution set of the VI (1.3), the solution set of the GSVI (1.10), and the fixed point set of a strictly pseudo-contractive mapping S : C → C.

Theorem CGY (see [35], Theorem 3.1.)Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be α-inverse strongly monotone, and let B_i : C → H be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩VI (C, A) ≠ ∅. Let Q : C → C be a ρ-contraction with ρ ∈ [0, 1/2). For given x₀ ∈ C arbitrarily, let the sequences {x_n}, {y_n}, and {z_n} be generated iteratively by

()

where μ_i ∈ (0,2β_i) for i = 1,2, {λ_n}⊂(0,2α] and {α_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1], such that

(i)
β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n, for all n ≥ 0;
(ii)
lim _n→∞α_n = 0 and ;
(iii)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and liminf _n→∞δ_n > 0;
(iv)
lim _n→∞(γ_n+1/(1 − β_n+1) − γ_n/(1 − β_n)) = 0;
(v)
0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 2α and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequence {x_n} generated by (1.14) converges strongly to , and is a solution of the GSVI (1.10), where .

On the other hand, let A : C → H be a monotone, and let L-Lipschitz-continuous mapping, Φ : C → H be an α-inverse strongly monotone mapping. Let M be a maximal monotone mapping with D(M) = C, and let S : C → C be a nonexpansive mapping such that Fix (S)∩Ω∩VI (C, A) ≠ ∅. Motivated Nadezhkina and Takahashi′s hybrid-extragradient algorithm (1.9), Ceng et al. [36, Theorem 3.1] introduced another modified hybrid-extragradient algorithm

()

where

chosen arbitrarily, {λ_n}⊂(0, 1/L), {μ_n}⊂(0,2α], and

such that

. It was proven in [36] that under very mild conditions three sequences {x_n}, {y_n}, and {z_n} generated by (1.15) converge strongly to the same point P_{Fix (S)∩Ω∩VI (C,A)}x₀.

Inspired by the research going on this area, we propose and analyze the following hybrid extragradient iterative algorithm for finding a common element of the solution set Ξ of the GSVI (1.10), the solution set Ω of the variational inclusion (1.7), and the fixed point set Fix (S) of a strictly pseudo-contractive mapping S : C → C.

Algorithm 1.2. Assume that Fix (S)∩Ω∩Ξ ≠ ∅. Let μ_i ∈ (0,2β_i) for i = 1,2, {μ_n}⊂(0,2α], and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that β_n + γ_n + δ_n = 1, for all n ≥ 0. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by the hybrid extragradient iterative scheme

()

where

, for all n ≥ 0.

Under very appropriate assumptions, it is proven that all the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point . Furthermore, is a solution of the GSVI (1.10), where .

Let T : C → C be a k-strictly pseudocontractive mapping, let Γ : C → C be a κ-strictly pseudocontractive mapping, and let S : C → C be a nonexpansive mapping. Putting B₁ = I − T, B₂ = 0, Φ = I − Γ, M = 0, and σ_n = 0, for all n ≥ 0 in Algorithm 1.2, we consider and analyze the following hybrid extragradient iterative algorithm for finding a common fixed point of three mappings S, Γ, and T.

Algorithm 1.3. Assume that Fix (S)∩Fix (Γ)∩Fix (T) ≠ ∅. Let μ₁ ∈ (0,1 − k), {μ_n}⊂(0,1 − κ], and {β_n}, {γ_n}, {δ_n}⊂[0,1] such that β_n + γ_n + δ_n = 1, for all n ≥ 0. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by the hybrid extragradient iterative scheme

()

Under quite mild conditions, it is shown that all the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point P_{Fix (S)∩Fix (Γ)∩Fix (T)}x₀.

Observe that Ceng et al. [36, Theorem 3.1] considered the problem of finding an element of Fix (S)∩Ω∩VI (C, A) where S : C → C is nonexpansive, Nadezhkina and Takahashi [15, Theorem 3.1] studied the problem of finding an element of Fix (S)∩VI (C, A) where S : C → C is nonexpansive, and Ceng et al. [35, Theorem 3.1] investigated the problem of finding an element of Fix (S) ∩ Ξ ∩ VI (C, A) where S : C → C is strictly pseudocontractive. It is clear that every one of these three problems is very different from our problem of finding an element of Fix (S)∩Ω∩Ξ where S : C → C is strictly pseudocontractive. Hence there is no doubt that the strong convergence results for solving our problem are very interesting and quite valuable. Because our hybrid extragradient iterative algorithms involve two inverse strongly monotone mappings B₁ and B₂, a strictly pseudo-contractive self-mapping S, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [36, Theorem 3.1] and [15, Theorem 3.1], respectively. Furthermore, the relaxed extragradient iterative scheme in Yao et al. [34, Theorem 3.2] is extended to develop our hybrid extragradient iterative algorithms. In our results, the hybrid extragradient iterative algorithms drop the requirements that 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and lim _n→∞(γ_n+1/(1 − β_n+1) − γ_n/(1 − β_n)) = 0 in [34, Theorem 3.2] and [35, Theorem 3.1]. Therefore, our results represent the modification, supplementation, extension, and improvement of [36, Theorem 3.1], [15, Theorem 3.1], [34, Theorem 3.2], and [35, Theorem 3.1] to a great extent.

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 → to indicate that the sequence {x_n} converges strongly to x and ⇀ to indicate that the sequence {x_n} converges weakly to x. Moreover, we use ω_w(x_n) to denote the weak ω-limit set of the sequence {x_n}, that is,

()

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. We know that P_C is a firmly nonexpansive mapping of H onto C; that is, there holds the following relation

()

Consequently, P_C is nonexpansive and monotone. It is also known that P_C is characterized by the following properties: P_Cx ∈ C and

()

for all x ∈ H, y ∈ C; see [11, 37] for more details. Let A : C → H be a monotone mapping. In the context of the variational inequality, this implies that

()

It is also known that the norm of every Hilbert space H satisfies the weak lower semicontinuity [4]. That is, for any sequence {x_n} with x_n⇀x, the inequality

()

holds.

Recall that a set-valued mapping M : D(M) ⊂ H → 2^H is called maximal monotone if M is monotone and (I + λM)D(M) = H for each λ > 0, where I is the identity mapping of H. We denote by G(M) the graph of M. It is known that a monotone mapping M is maximal if and only if, for (x, f) ∈ H × H, 〈f − g, x − y〉≥0 for every (y, g) ∈ G(M) implies f ∈ Mx. Here the following example illustrates the concept of maximal monotone mappings in the setting of Hilbert spaces.

Let A : C → H be a monotone, L-Lipschitz-continuous mapping, and let N_Cv be the normal cone to C at v ∈ C, that is,

()

Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ VI (C, A); see [38].

Assume that M : D(M) ⊂ H → 2^H is a maximal monotone mapping. Then, for λ > 0, associated with M, the resolvent operator J_M,λ can be defined as

()

In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator J_M,λ : H → H.

Lemma 2.1. J_M,λ is single valued and firmly nonexpansive, that is,

()

Consequently, J_M,λ is nonexpansive and monotone.

Lemma 2.2 (see [39].)There holds the relation:

()

for all x, y, z ∈ H and λ, μ, ν ∈ [0,1] with λ + μ + ν = 1.

Lemma 2.3 (see [36].)Let M be a maximal monotone mapping with D(M) = C. Then for any given λ > 0, x^* ∈ C is a solution of problem (1.7) if and only if x^* ∈ C satisfies

()

Lemma 2.4 (see [13].)Let M be a maximal monotone mapping with D(M) = C, and let V : C → H be a strong monotone, continuous, and single-valued mapping. Then for each z ∈ H, the equation z ∈ Vx + λMx has a unique solution x_λ for λ > 0.

Lemma 2.5 (see [36].)Let M be a maximal monotone mapping with D(M) = C, and let A : C → H be a monotone, continuous, and single-valued mapping. Then (I + λ(M + A))C = H for each λ > 0. In this case, M + A is maximal monotone.

It is clear that, in a real Hilbert space H, S : C → C is k-strictly pseudo-contractive if and only if there holds the following inequality:

()

This immediately implies that if S is a k-strictly pseudocontractive mapping, then I − S is (1 − k) /2-inverse strongly monotone; for further detail, we refer to [10] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.

Lemma 2.6 (see [10], Proposition 2.1.)Let C be a nonempty closed convex subset of a real Hilbert space H, and let S : C → C be a mapping.

(i)
If S is a k-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
()
(ii)
If S is a k-strict pseudo-contractive mapping, then the mapping I − S is semiclosed at 0; that is, if {x_n} is a sequence in C such that weakly and (I − S)x_n → 0 strongly, then .
(iii)
If S is k-quasistrict pseudo-contraction, then the fixed point set Fix (S) of S is closed and convex, so that the projection P_Fix (S) is well defined.

Lemma 2.7 (see [34].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a k-strictly pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)k ≤ γ. Then

()

The following lemma is well known to us.

Lemma 2.8 (see [11].)Every Hilbert space H has the Kadec-Klee property; that is, for given x ∈ H and {x_n} ⊂ H, we have

()

3. Main Results

In this section, we first prove the strong convergence of the sequences generated by our hybrid extragradient iterative algorithm for finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed problem of a strictly pseudocontractive self-mapping.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let B_i : C → H be β_i-inverse strongly monotone for i = 1,2, let Φ : C → H be an α-inverse strongly monotone mapping, let M be a maximal monotone mapping with D(M) = C, and let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S) ∩ Ω ∩ Ξ ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by

()

where μ_i ∈ (0,2β_i) for i = 1,2, {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α], and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that {σ_n}⊂[0, c] for some c ∈ [0,1), {δ_n}⊂[d, 1] for some d ∈ (0,1], β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n, for all n ≥ 0. Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point

if and only if

. Furthermore,

is a solution of the GSVI (1.10), where

Proof. It is obvious that C_n is closed and Q_n is closed and convex for every n = 0,1, 2, …. As

()

we also know that C_n is convex for every n = 0,1, 2, …. As

()

we have 〈x_n − z, x − x_n〉≥0, for all z ∈ Q_n, and hence

by (2.4).

First of all, assume that the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point . Then it is clear that ∥x_n − y_n∥ → 0 and ∥x_n − z_n∥ → 0. Observe that from the nonexpansiveness of the mappings P_C(I − μ₁B₁) and P_C(I − μ₂B₂) (due to μ_i ∈ (0,2β_i) for i = 1,2), we have

()

Hence, we conclude that ∥y_n − t_n∥ → 0 and

. Since

, we obtain that

and

. Thus, from the nonexpansiveness of the mapping

, we have

()

So, we deduce that

and

. Note that

()

This implies that

as n → ∞.

For the remainder of the proof, we divide it into several steps.

Step 1. We claim that Fix (S)∩Ω∩Ξ ⊂ C_n∩Q_n for every n = 0,1, 2, ….

Indeed, take a fixed p ∈ Fix (S)∩Ω∩Ξ arbitrarily. Then , for all n ≥ 0, and

()

For simplicity, we write

, and

()

for each n ≥ 0. Since B_i : C → H is β_i-inverse strongly monotone, and 0 < μ_i < 2β_i for i = 1,2, we know that for all n ≥ 0,

()

Repeating the same argument, we can obtain that for all n ≥ 0,

()

Furthermore, by Lemma 2.1 we derive from (3.9) and (3.10)

()

Since (γ_n + δ_n)k ≤ γ_n, for all n ≥ 0, utilizing Lemmas 2.2 and 2.7, we get from (3.11)

()

for every n = 0,1, 2, …, and hence p ∈ C_n. So, Fix (S) ∩ Ω ∩ Ξ ⊂ C_n for every n = 0,1, 2, …. Next, let us show by mathematical induction that {x_n} is well defined and Fix (S) ∩ Ω ∩ Ξ ⊂ C_n∩Q_n for every n = 0,1, 2, …. For n = 0, we have Q₀ = C. Hence we obtain Fix (S)∩Ω∩Ξ ⊂ C₀∩Q₀. Suppose that x_n is given and Fix (S)∩Ω∩Ξ ⊂ C_n∩Q_n for some integer n ≥ 0. Since Fix (S)∩Ω∩Ξ is nonempty, C_n∩Q_n is a nonempty closed convex subset of C. So, there exists a unique element x_n+1 ∈ C_n∩Q_n such that

. It is also obvious that there holds 〈x_n+1 − z, x₀ − x_n+1〉≥0 for z ∈ Fix (S)∩Ω∩Ξ, and hence Fix (S)∩Ω∩Ξ ⊂ Q_n+1. Therefore, we derive Fix (S)∩Ω∩Ξ∩C_n+1∩Q_n+1.

Step 2. We claim that

()

Indeed, let l₀ = P_{Fix (S)∩Ω∩Ξ}x₀. From , and l₀ ∈ Fix (S)∩Ω∩Ξ ⊂ C_n∩Q_n, we have

()

for every n = 0,1, 2, …. Therefore, {x_n} is bounded. From (3.9)–(3.12), we also obtain that

, {y_n},

, {t_n},

, and {z_n} all are bounded. Since x_n+1 ∈ C_n ∩ Q_n ⊂ Q_n and

, we have

()

for every n = 0,1, 2, …. Therefore, there exists lim _n→∞∥x_n − x₀∥. Since

and x_n+1 ∈ Q_n, utilizing (2.5), we have

()

for every n = 0,1, 2, …. This implies that

()

Since x_n+1 ∈ C_n, we have ∥z_n − x_n+1∥ ≤ ∥x_n − x_n+1∥, and hence

()

for every n = 0,1, 2, …. From ∥x_n+1 − x_n∥ → 0 it follows that

()

Step 3. We claim that

()

Indeed, for p ∈ Fix (S)∩Ω∩Ξ, we obtain from (3.12)

()

Therefore, we have

()

Since {δ_n}⊂[d, 1] for some d ∈ (0,1], ∥x_n − z_n∥ → 0, and the sequences {x_n} and {z_n} are bounded, we deduce that

()

On the other hand, by firm nonexpansiveness of P_C, we have

()

that is,

()

Repeating the same argument, we can also obtain

()

Moreover, using the argument technique similar to the above one, we derive

()

that is,

()

Repeating the same argument, we can also obtain

()

Utilizing (3.11), (3.25)–(3.29), we have

()

which hence implies that

()

Since {δ_n}⊂[d, 1] for some d ∈ (0,1], ∥x_n − z_n∥ → 0, and {x_n}, {y_n}, {z_n},

, and {t_n} all are bounded, it follows from (3.23) that

()

Consequently, it immediately follows that

()

This shows that

()

Also, note that

()

Thus we have

()

This together with ∥z_n − x_n∥ → 0 and

implies that

()

Consequently, from (3.34) we immediately derive

()

Step 4. We claim that ω_w(x_n) ⊂ Fix (S)∩Ω∩Ξ.

Indeed, as {x_n} is bounded, there is a subsequence of {x_n} such that converges weakly to some u ∈ ω_w(x_n). We can obtain that u ∈ Fix (S)∩Ω∩Ξ. First, it is clear from Lemma 2.6(ii) that u ∈ Fix (S). Now let us show that u ∈ Ξ. We note that

()

where G : C → C is defined as that in Lemma 1.1. According to Lemma 2.6(ii) we obtain u ∈ Ξ. Further, let us show that u ∈ Ω. As a matter of fact, since Φ is α-inverse strongly monotone, and M is maximal monotone, by Lemma 2.5 we know that M + Φ is maximal monotone. Take a fixed (y, g) ∈ G(M + Φ) arbitrarily. Then we have g ∈ My + Φ(y). So, we have g − Φ(y) ∈ My. Since

()

implies

()

where

, we have

()

which hence yields

()

Observe that

()

From

, it follows that

()

Since

, and

, we derive

, and hence by letting i → ∞ we get from (3.43)

()

This shows that 0 ∈ Φ(u) + Mu. Thus, u ∈ Ω. Therefore, u ∈ Fix (S)∩Ω∩Ξ.

Step 5. We claim that

()

where l₀ = P_{Fix (S)∩Ω∩Ξ}x₀.

Indeed, Since l₀ = P_{Fix (S)∩Ω∩Ξ}x₀, and u ∈ Fix (S)∩Ω∩Ξ, from (3.14) we have

()

So, we obtain

()

From

, we have

(due to the Kadec-Klee property of Hilbert spaces [37]), and hence

. Since

, and l₀ ∈ Fix (S)∩Ω∩Ξ ⊂ C_n∩Q_n ⊂ Q_n, we have

()

As i → ∞, we obtain

by l₀ = P_{Fix (S)∩Ω∩Ξ}x₀, and u ∈ Fix (S)∩Ω∩Ξ. Hence we have u = l₀. This implies that x_n → l₀. It is easy to see that y_n → l₀ and z_n → l₀. This completes the proof.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let B_i : C → H be β_i-inverse strongly monotone for i = 1,2, let Φ : C → H be an α-inverse strongly monotone mapping, let M be a maximal monotone mapping with D(M) = C, and let S : C → C be a nonexpansive mapping such that Fix (S)∩Ω∩Ξ ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by

()

where μ_i ∈ (0,2β_i) for i = 1,2, {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α], and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that {σ_n}⊂[0, c] for some c ∈ [0,1), {γ_n}, {δ_n}⊂[d, 1] for some d ∈ (0,1], and β_n + γ_n + δ_n = 1 for all n ≥ 0. Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point

. Furthermore,

is a solution of the GSVI (1.10), where

Proof. Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k = 0. Take a fixed p ∈ Fix (S)∩Ω∩Ξ arbitrarily. Note that in Step 1 for the proof of Theorem 3.1, we have obtained that {x_n} is bounded and the relation holds

()

(due to (3.11)). Moreover, in Step 2 for the proof of Theorem 3.1, we have proven that

()

Now, utilizing Lemma 2.2, from the nonexpansiveness of S we deduce that

()

This together with {γ_n}, {δ_n}⊂[d, 1] implies that

()

So, we immediately derive

()

It is easy to see that all the conditions of Theorem 3.1 are satisfied. Therefore, in terms of Theorem 3.1 we obtain the desired result.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let B_i : C → H be β_i-inverse strongly monotone for i = 1,2, and let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by

()

where μ_i ∈ (0,2β_i) for i = 1,2, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that {δ_n}⊂[d, 1] for some d ∈ (0,1], β_n + γ_n + δ_n = 1, and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0. Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point

. Furthermore,

is a solution of the GSVI (1.10), where

Proof. Putting Φ = M = 0 in Theorem 3.1, we have Ω = C and Fix (S)∩Ω∩Ξ = Fix (S)∩Ξ. Let α be any positive number in the interval (0, ∞), and take any sequence {σ_n}⊂[0, c] for some c ∈ [0,1) and any sequence {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α]. Then Φ is α-inverse strongly monotone, and we have

()

which is just equivalent to (3.57). In this case, we have

()

Note that in Steps 2 and 3 for the proof of Theorem 3.1, we have proven that

()

respectively. Thus, we have

()

Consequently, it follows from {δ_n}⊂[d, 1] that ∥St_n − x_n∥ → 0, and hence ∥St_n − t_n∥ → 0. This shows that

. Utilizing Theorem 3.1, we obtain the desired result.

Remark 3.4. Our Theorems 3.1 improves, extends, and develops [36, Theorem 3.1], [15, Theorem 3.1], [34, Theorem 3.2], and [35, Theorem 3.1] in the following aspects.

(i)
Compared with the relaxed extragradient iterative algorithm in [34, Theorem 3.2] and the hybrid extragradient iterative algorithm in [35, Theorem 3.1], our hybrid extragradient iterative algorithms remove the requirements that 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and lim _n→∞(γ_n+1/(1 − β_n+1) − γ_n/(1 − β_n)) = 0.
(ii)
The problem of finding an element of Fix (S)∩Ω∩Ξ in our Theorem 3.1 is more general than the corresponding ones in [36, Theorem 3.1], [15, Theorem 3.1], and [34, Theorem 3.2] to a great extent. Thus, beyond question our results are very interesting and quite valuable.
(iii)
The relaxed extragradient method for finding an element of Fix (S)∩Ξ in [34, Theorem 3.2] is extended to develop our hybrid extragradient iterative algorithms for finding an element of Fix (S)∩Ω∩Ξ.
(iv)
The proof of our results are very different from that of [15, Theorem 3.1] because our argument technique depends on two inverse strongly monotone mappings B₁ and B₂, the property of strict pseudocontractions (see Lemmas 2.6 and 2.7), and the properties of the resolvent J_M,λ to a great extent.
(v)
Because our iterative algorithms involve two inverse strongly monotone mappings B₁ and B₂, a k-strictly pseudocontractive self-mapping S, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [36, Theorem 3.1], [15, Theorem 3.1], and [34, Theorem 3.2], respectively.

4. Applications

Utilizing Theorem 3.1, we prove some strong convergence theorems in a real Hilbert space.

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let B_i : C → H be β_i-inverse strongly monotone for i = 1,2, let Φ : C → H be an α-inverse strongly monotone mapping, and let M be a maximal monotone mapping with D(M) = C such that Ω∩Ξ ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by

()

where μ_i ∈ (0,2β_i) for i = 1,2, {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α], and {σ_n}, {β_n}⊂[0,1] such that {σ_n}⊂[0, c] for some c ∈ [0,1), and {β_n}⊂[0, d] for some d ∈ [0,1). Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to the same point

. Furthermore,

is a solution of the GSVI (1.10), where

Proof. In Corollary 3.2, putting S = I, we have

()

which is just equivalent to (4.1). In this case, we know that Fix (S)∩Ω∩Ξ = Ω∩Ξ. Therefore, by Corollary 3.2 we obtain desired result.

Theorem 4.2 (see [15], Theorem 4.2.)Let C be a nonempty closed convex subset of a real Hilbert space H, and let S : C → C be a nonexpansive mapping such that Fix (S) is nonempty. For given x₀ ∈ C arbitrarily, let {x_n} and {z_n} be the sequences generated by

()

where {δ_n}⊂[d, 1] for some d ∈ (0,1]. Then the sequences {x_n} and {z_n} converge strongly to P_Fix (S)x₀.

Proof. Putting B₁ = B₂ = Φ = M = 0 in Corollary 3.2, we let β₁, β₂, and α be any positive numbers in the interval (0, ∞), and take any numbers μ_i ∈ (0,2β_i) for i = 1,2 and any sequence {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α]. Then B_i : C → H is β_i-inverse strongly monotone for i = 1,2, and Φ : C → H is α-inverse strongly monotone. In this case, we know that Fix (S)∩Ω∩Ξ = Fix (S) and

()

which is just equivalent to (4.3). Therefore, by Corollary 3.2 we obtain the desired result.

Remark 4.3. Originally Theorem 4.2 is the result of Nakajo and Takahashi [22].

Theorem 4.4. Let H be a real Hilbert space. Let A : H → H be a λ-inverse strongly monotone mapping, let Φ : H → H be an α-inverse strongly monotone mapping, let M : H → 2^H be a maximal monotone mapping, and let S : H → H be a nonexpansive mapping such that Fix (S)∩Ω∩A⁻¹0 ≠ ∅. For given x₀ ∈ H arbitrarily, let {x_n} and {z_n} be the sequences generated by

()

where μ ∈ (0,2λ), {μ_n}⊂[ϵ, 2α] for some ϵ ∈ (0,2α], and {β_n}, {γ_n}, {δ_n}⊂[0,1] such that {γ_n}, {δ_n}⊂[d, 1] for some d ∈ (0,1], and β_n + γ_n + δ_n = 1 for all n ≥ 0. Then the sequences {x_n} and {z_n} converge strongly to

Proof. Putting C = H, B₁ = A, B₂ = 0, μ₁ = μ, and σ_n = 0, for all n ≥ 0 in Corollary 3.2, we know that P_C = P_H = I and the GSVI (1.10) coincides with the VI (1.3). Hence we have A⁻¹0 = VI (H, A) = Ξ. In this case, we conclude that Fix (S)∩Ω∩Ξ = Fix (S)∩Ω∩A⁻¹0 and

()

Therefore, by Corollary 3.2 we obtain the desired result.

Let B : H → 2^H be a maximal monotone mapping. Then, for any x ∈ H and r > 0, consider J_B,rx = (I + rB) ⁻¹x. It is known that such a J_B,r is the resolvent of B.

Theorem 4.5. Let H be a real Hilbert space. Let A : H → H be a λ-inverse strongly monotone mapping, let Φ : H → H be an α-inverse strongly monotone mapping, and let B, M : H → 2^H be two maximal monotone mappings such that A⁻¹0∩B⁻¹0∩Ω ≠ ∅. Let J_B,r be the resolvent of B for each r > 0. For given x₀ ∈ H arbitrarily, let {x_n} and {z_n} be the sequences generated by

()

Proof. Putting S = J_B,r in Theorem 4.4, we know that Fix (S) = Fix (J_B,r) = B⁻¹0. In this case, (4.5) is coincident with (4.7). Therefore, by Theorem 4.4 we obtain the desired result.

It is well known that a mapping T : C → C is called pseudocontractive if ∥Tx−Ty∥² ≤ ∥x−y∥² + ∥(I−T)x−(I−T)y∥², for all x, y ∈ C. It is easy to see that this definition is equivalent to the one that a mapping T : C → C is called pseudocontractive if 〈Tx − Ty, x − y〉≤∥x−y∥², for all x, y ∈ C; see [8]. In the meantime, we also know one more definition of a k-strictly pseudocontractive mapping, which is equivalent to the definition given in the introduction. A mapping T : C → C is called k-strictly pseudocontractive if there exists a constant k ∈ [0,1), such that

()

for all x, y ∈ C. It is clear that in this case the mapping I − T is (1 − k)/2-inverse strongly monotone. From [10], we know that if T is a k-strictly pseudocontractive mapping, then T is Lipschitz continuous with constant (1 + k)/(1 − k), such that Fix (T) = VI (C, I − T) (see, e.g., the proof of Theorem 4.6). It is obvious that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and the class of pseudocontractions strictly includes the class of strict pseudocontractions.

In the following theorem we introduce an iterative algorithm that converges strongly to a common fixed point of three mappings: one of which is nonexpansive, and the other two ones are strictly pseudocontractive mappings.

Theorem 4.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a k-strictly pseudocontractive mapping, let Γ : C → C be a κ-strictly pseudocontractive mapping, and let S : C → C be a nonexpansive mapping such that Fix (T)∩Fix (S)∩Fix (Γ) ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, and {z_n} be the sequences generated by

()

where μ₁ ∈ (0,1 − k), {μ_n}⊂[ϵ, 1 − κ] for some ϵ ∈ (0,1 − κ], and {β_n}, {γ_n}, {δ_n}⊂[0,1] such that {γ_n}, {δ_n}⊂[d, 1] for some d ∈ (0,1], and β_n + γ_n + δ_n = 1 for all n ≥ 0. Then the sequences {x_n}, {y_n}, and {z_n} converge strongly to P_{Fix (T)∩Fix (S)∩Fix (Γ)}x₀.

Proof. Putting B₁ = I − T, B₂ = 0, Φ = I − Γ, M = 0, and σ_n = 0, for all n ≥ 0 in Corollary 3.2, we know that B₁ is β₁-inverse strongly monotone with β₁ = (1 − k)/2 and Φ is α-inverse strongly monotone with α = (1 − κ)/2. Moreover, we have Ξ = VI (C, B₁) = VI (C, I − T). Noticing μ₁ ∈ (0,1 − k) and k ∈ [0,1), we know that μ₁ ∈ (0,1), and hence (1 − μ₁)x_n + μ₁Tx_n ∈ C. Also, noticing {μ_n}⊂[ϵ, 1 − κ]⊂(0,1 − κ], we know that {μ_n}⊂(0,1], and hence (1 − μ_n)t_n + μ_nΓt_n ∈ C. This implies that

()

Now let us show Fix (T) = VI (C, B₁). In fact, we have, for λ > 0,

()

Next let us show Ω = Fix (Γ). In fact, noticing that M = 0 and Φ = I − Γ, we have

()

Consequently,

()

Therefore, by Theorem 3.1 we obtain the desired result.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Ph.D. Program Foundation of Ministry of Education of China (20123127110002), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by the Grant NSC 101-2115-M-037-001.

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All articles

Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed-Point Problems

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

Acknowledgments

References

References

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Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed-Point Problems

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

Acknowledgments

References

References

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