# The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

Authors: Lu-Chuan Ceng; Himanshu Gupta; Ching-Feng Wen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution setΞ of a general system of variational inequalities, the solution set Γ of a split feasibility problem, and the fixed point set Fix(S) of a strictly pseudocontractive mapping S in the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element of Fix(S)∩Ξ∩Γ
under mild conditions.


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## Body

## 1. Introduction

Letℋ be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of ℋ. The projection (nearest point or metric projection) of ℋ onto C is denoted by PC. Let S:C→C be a mapping and Fix(S) be the set of fixed points of S. For a given nonlinear operator A:C→ℋ, we consider the following variational inequality problem (VIP) of finding x*∈C such that
(1)〈Ax*,x-x*〉≥0,∀x∈C.
The solution set of VIP (1) is denoted by VI(C,A). The theory of variational inequalities has been studied quite extensively and has emerged as an important tool in the study of a wide class of problems from mechanics, optimization, engineering, science, and social sciences. It is well known that the VIP is equivalent to a fixed point problem. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. In the recent past, several people have studied and proposed several iterative methods to find a solution of variational inequalities which is also a fixed point of a nonexpansive mapping or strict pseudocontractive mapping; see, for example, [1–9] and the references therein.For finding an element ofFix(S)∩VI(C,A) when C is closed and convex, S is nonexpansive, and A is α-inverse strongly monotone, Takahashi and Toyoda [10] introduced the following Mann-type iterative algorithm:
(2)xn+1=αnxn+(1-αn)SPC(1-λnAxn),∀n≥0,
where PC is the metric projection of ℋ onto C, x0=x∈C, {αn} is a sequence in (0,1), and {λn} is a sequence in (0,2α). They showed that if Fix(S)∩VI(C,A)≠∅, then the sequence {xn} converges weakly to some z∈Fix(S)∩VI(C,A). Nadezhkina and Takahashi [9] and Zeng and Yao [8] proposed extragradient methods motivated by Korpelevič [11] for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality problem. Further, these iterative methods are extended in [12] to develop a new iterative method for finding elements in Fix(S)∩VI(C,A).LetB1,B2:C→ℋ be two mappings. Recently, Ceng et al. [4] introduced and considered the following problem of finding (x*,y*)∈C×C such that
(3)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2B2x*+y*-x*,x-y*〉≥0,∀x∈C,
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 (3) is denoted by GSVI(C,B1,B2). In particular, if B1=B2, then problem (3) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [13]. Further, if x*=y*, then the NSVI reduces to VIP (1).Recently, Ceng et al. [4] transformed problem (3) into a fixed point problem in the following way.Lemma 1 (see [4]).
For givenx¯,y¯∈C, (x¯,y¯) is a solution of problem (3) if and only if x¯ is a fixed point of the mapping G:C→C defined by
(4)G(x)=PC[PC(x-μ2B2x)-μ1B1PC(x-μ2B2x)],∀x∈C,
where y¯=PC(x¯-μ2B2x¯).In particular, if the mappingBi:C→ℋ is βi-inverse strongly monotone for i=1,2, then the mapping G is nonexpansive provided μi∈(0,2βi) for i=1,2.Utilizing Lemma1, they introduced and studied a relaxed extragradient method for solving GSVI (3).Throughout this paper, unless otherwise specified, the set of fixed points of the mappingG is denoted by Ξ. Based on the relaxed extragradient method and viscosity approximation method, Yao et al. [7] proposed and analyzed an iterative algorithm for finding a common solution of GSVI (3) and fixed point problem of a strictly pseudocontractive mapping S:C→C, where C is a nonempty bounded closed convex subset of a real Hilbert space ℋ.Subsequently, Ceng at al. [14] further presented and analyzed an iterative scheme for finding a common element of the solution set of VIP (1), the solution set of GSVI (3), and fixed point set of a strictly pseudocontractive mapping S:C→C.Theorem 2 (see [14, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ. Let A:C→ℋ be α-inverse strongly monotone, and let Bi:C→ℋ be βi-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 {xn}, {yn}, and {zn} be generated iteratively by
(5)zn=PC(xn-λnAxn),yn=αnQxn+(1-αn)PC[PC(zn-μ2B2zn)+(1-αn)PC0-μ1B1PC(zn-μ2B2zn)],xn+1=βnxn+γnyn+δnSyn,∀n≥0,
where μi∈(0,2βi) for i=1,2, {λn}⊂(0,2α] and {αn}, {βn}, {γn}, {δn}⊂[0,1] such that(i)
βn+γn+δn=1 and (γn+δn)k≤γn for all n≥0;(ii)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(iv)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0;(v)
0<liminfn→∞λn≤limsupn→∞λn<2α and limn→∞|λn+1-λn|=0. Then the sequence {xn} generated by (5) converges strongly to x¯=PFix(S)∩Ξ∩VI(C,A)Qx¯ and (x¯,y¯) is a solution of GSVI (3), where y¯=PC(x¯-μ2B2x¯).On the other hand, letC and Q be nonempty closed convex subsets of real Hilbert spaces ℋ1 and ℋ2, respectively. The split feasibility problem (SFP) is to find a point x* with the following property:
(6)x*∈C,Ax*∈Q,
where A∈B(ℋ1,ℋ2) and B(ℋ1,ℋ2) denotes the family of all bounded linear operators from ℋ1 to ℋ2.In 1994, the SFP was first introduced by Censor and Elfving [15], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, for example, [16] and the references therein. Recently, it is found that the SFP can also be applied to study intensity-modulated radiation therapy; see, for example, [17–19] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see, for example, [16–26] and the references therein. A special case of the SFP is the following convex constrained linear inverse problem [27] of finding an element x such that
(7)x∈C,Ax=b.
It has been extensively investigated in the literature using the projected Landweber iterative method [28]. Comparatively, the SFP has received much less attention so far, due to the complexity resulting from the set Q. Therefore, whether various versions of the projected Landweber iterative method [28] can be extended to solve the SFP remains an interesting open topic. For example, it is yet not clear whether the dual approach to (7) of [29] can be extended to the SFP. The original algorithm given in [15] involves the computation of the inverse A-1 (assuming the existence of the inverse of A), and thus has not become popular. A seemingly more popular algorithm that solves the SFP is the CQ algorithm of Byrne [16, 21] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method [30]. The CQ algorithm only involves the computation of the projections PC and PQ onto the sets C and Q, respectively, and is therefore implementable in the case where PC and PQ have closed-form expressions; for example, C and Q are closed balls or half-spaces. However, it remains a challenge how to implement the CQ algorithm in the case where the projections PC and/or PQ fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.Very recently, Xu [20] gave a continuation of the study on the CQ algorithm and its convergence. He applied Mann's algorithm to the SFP and purposed an averaged CQ algorithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.Furthermore, Korpelevič [11] introduced the so-called extragradient method for finding a solution of a saddle point problem. He proved that the sequences generated by the proposed iterative algorithm converge to a solution of the saddle point problem.Throughout this paper, assume that the SFP is consistent; that is, the solution setΓ of the SFP is nonempty. Let f:ℋ1→R be a continuous differentiable function. The minimization problem
(8)minx∈Cf(x):=12∥Ax-PQAx∥2
is ill posed. Therefore, Xu [20] considered the following Tikhonov regularization problem:
(9)minx∈Cfα(x):=12∥Ax-PQAx∥2+12α∥x∥2,
where α>0 is the regularization parameter. The regularized minimization (9) has a unique solution which is denoted by xα. The following results are easy to prove.Proposition 3 (see [31, Proposition 3.1]).
Givenx*∈ℋ1, the following statements are equivalent: (i)
x*solves the SFP;(ii)
x*solves the fixed point equation(10)PC(I-λ∇f)x*=x*,whereλ>0,∇f=A*(I-PQ)AandA*is the adjoint ofA;(iii)
x*solves the variational inequality problem (VIP) of findingx*∈Csuch that(11)〈∇f(x*),x-x*〉≥0,∀x∈C.It is clear from Proposition3 that
(12)Γ=Fix(PC(I-λ∇f))=VI(C,∇f)
for all λ>0, where Fix(PC(I-λ∇f)) and VI(C,∇f) denote the set of fixed points of PC(I-λ∇f) and the solution set of VIP (11), respectively.Proposition 4 (see [31]).
The following statements hold:(i)
the gradient(13)∇fα=∇f+αI=A*(I-PQ)A+αIis(α+∥A∥2)-Lipschitz continuous andα-strongly monotone;(ii)
the mappingPC(I-λ∇fα)is a contraction with coefficient(14)1-λ(2α-λ(∥A∥2+α)2)(≤1-αλ≤1-12αλ),where0<λ≤α/(∥A∥2+α)2;(iii)
if the SFP is consistent, then the stronglimα→0xαexists and is the minimum-norm solution of the SFP.Very recently, by combining the regularization method and extragradient method due to Nadezhkina and Takahashi [32], Ceng et al. [31] proposed an extragradient algorithm with regularization and proved that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S)∩Γ, where S:C→C is a nonexpansive mapping.Theorem 5 (see [31, Theorem 3.1]).
LetS:C→C be a nonexpansive mapping such that Fix(S)∩Γ≠∅. Let {xn} and {yn} be the sequences in C generated by the following extragradient algorithm:
(15)x0=x∈Cchosenarbitrarily,yn=PC(xn-λn∇fαn(xn)),xn+1=βnxn+(1-βn)SPC(xn-λn∇fαn(yn)),∀n≥0,
where ∑n=0∞‍αn<∞, {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2) and {βn}⊂[c,d] for some c,d∈(0,1). Then, both sequences {xn} and {yn} converge weakly to an element x^∈Fix(S)∩Γ.Motivated and inspired by the research going on this area, we propose and analyze the following Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution set of the GSVI (3), the solution set of the SFP (6), and the fixed point set of a strictly pseudocontractive mapping S:C→C.Algorithm 6.
Letμi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {σn},{τn},{βn},{γn},{δn}⊂[0,1] such that σn+τn≤1 and βn+γn+δn=1 for all n≥0. For given x0∈C arbitrarily, let {xn}, {yn}, {zn} be the sequences generated by the Mann-type extragradient iterative scheme with regularization
(16)zn=PC(xn-λn∇fαn(xn)),yn=σnxn+τnPC(xn-λn∇fαn(zn))+(1-σn-τn)PC[PC(zn-μ2B2zn)+(1-σn-τn)PC0-μ1B1PC(zn-μ2B2zn)],xn+1=βnxn+γnyn+δnSyn,∀n≥0.
Under appropriate assumptions, it is proven that all the sequences{xn}, {yn}, {zn} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of the GSVI (3), where y¯=PC(x¯-μ2B2x¯).Algorithm 7.
Letμi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {σn},{βn},{γn},{δn}⊂[0,1] such that βn+γn+δn=1 for all n≥0. For given x0∈C arbitrarily, let {xn},{un},{u~n} be the sequences generated by the Mann-type extragradient iterative scheme with regularization
(17)un=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],u~n=PC(un-λn∇fαn(un)),yn=σnxn+(1-σn)PC(un-λn∇fαn(u~n)),xn+1=βnxn+γnyn+δnSyn,∀n≥0.
Also, under mild conditions, it is shown that all the sequences{xn}, {un}, {u~n} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of the GSVI (3), where y¯=PC(x¯-μ2B2x¯).
Observe that both [20, Theorem 5.7] and [31, Theorem 3.1] are weak convergence results for solving the SFP and so are our results as well. But our problem of finding an element of Fix(S)∩Ξ∩Γ is more general than the corresponding ones in [20, Theorem 5.7] and [31, Theorem 3.1], respectively. Hence, there is no doubt that our weak convergence results are very interesting and quite valuable. Because the Mann-type extragradient iterative schemes (16) and (17) with regularization 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 [20, Theorem 5.7] and [31, Theorem 3.1], respectively. Furthermore, the hybrid extragradient iterative scheme (5) is extended to develop the Mann-type extragradient iterative schemes (16) and (17) with regularization. In our results, the Mann-type extragradient iterative schemes (16) and (17) with regularization lack the requirement of boundedness for the domain in which various mappings are defined; see, for example, Yao et al. [7, Theorem 3.2]. Therefore, our results represent the modification, supplementation, extension, and improvement of [20, Theorem 5.7], [31, Theorem 3.1], [14, Theorem 3.1], and [7, Theorem 3.2].


## 2. Preliminaries

Letℋ be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let K be a nonempty, closed, and convex subset of ℋ. Now we present some known definitions and results which will be used in the sequel.The metric (or nearest point) projection fromℋ onto K is the mapping PK:ℋ→K which assigns to each point x∈ℋ the unique point PKx∈K satisfying the property
(18)∥x-PKx∥=infy∈K∥x-y∥=:d(x,K).Some important properties of projections are gathered in the following proposition.Proposition 8.
For givenx∈ℋ and z∈K: (i)
z=PKx⇔〈x-z,y-z〉≤0, for ally∈K;(ii)
z=PKx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2, for  ally∈K;(iii)
〈PKx-PKy,x-y〉≥∥PKx-PKy∥2, for ally∈ℋ, which hence implies thatPKis nonexpansive and monotone.Definition 9.
A mappingT:ℋ→ℋ is said to be (a)
nonexpansive if(19)∥Tx-Ty∥≤∥x-y∥,∀x,y∈ℋ;(b)
firmly nonexpansive if2T-I is nonexpansive, or equivalently,
(20)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈ℋ;
alternatively, T is firmly nonexpansive if and only if T can be expressed as
(21)T=12(I+S),
where S:ℋ→ℋ is nonexpansive; projections are firmly nonexpansive.Definition 10.
LetT be a nonlinear operator with domain D(T)⊆ℋ and range R(T)⊆ℋ. (a)
T is said to be monotone if
(22)〈x-y,Tx-Ty〉≥0,∀x,y∈D(T).(b)
Given a numberβ>0, T is said to be β-strongly monotone if
(23)〈x-y,Tx-Ty〉≥β∥x-y∥2,∀x,y∈D(T).(c)
Given a numberν>0, T is said to be ν-inverse strongly monotone (ν-ism) if
(24)〈x-y,Tx-Ty〉≥ν∥Tx-Ty∥2,∀x,y∈D(T).
It can be easily seen that ifS is nonexpansive, then I-S is monotone. It is also easy to see that a projection PK is 1-ism.
Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see, for example, [33, 34].Definition 11.
A mappingT:ℋ→ℋ is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
(25)T≡(1-α)I+αS,
where α∈(0,1) and S:ℋ→ℋ is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus, firmly nonexpansive mappings (in particular, projections) are 1/2-averaged maps.Proposition 12 (see [21]).
Let
T
:
ℋ
→
ℋ
be a given mapping.
(i)
T
is nonexpansive if and only if the complement
I
-
T
is
1
/
2
-ism.
(ii)
If
T
is
ν
-ism, then for
γ
>
0
,
γ
T
is
ν
/
γ
-ism.
(iii)
T
is averaged if and only if the complement
I
-
T
is
ν
-ism for some
ν
>
1
/
2
. Indeed, for
α
∈
(
0,1
)
,
T
is
α
-averaged if and only if
I
-
T
is
1
/
2
α
-ism.Proposition 13 (see [21, 35]).
Let
S
,
T
,
V
:
ℋ
→
ℋ
be given operators.
(i)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is averaged and
V
is nonexpansive, then
T
is averaged.
(ii)
T
is firmly nonexpansive if and only if the complement
I
-
T
is firmly nonexpansive.
(iii)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0
,
1
)
and if
S
is firmly nonexpansive and
V
is nonexpansive, then
T
is averaged.
(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings
{
T
i
}
i
=
1
N
is averaged, then so is the composite
T
1
∘
T
2
∘
·
·
·
∘
T
N
. In particular, if
T
1
is
α
1
-averaged and
T
2
is
α
2
-averaged, where
α
1
,
α
2
∈
(
0
,
1
)
, then the composite
T
1
∘
T
2
is
α
-averaged, where
α
=
α
1
+
α
2
-
α
1
α
2
.
(v)
If the mappings
{
T
i
}
i
=
1
N
are averaged and have a common fixed point, then
(26)
⋂
i
=
1
N
‍
Fix
(
T
i
)
=
Fix
(
T
1
·
·
·
T
N
)
.
The notation Fix(T) denotes the set of all fixed points of the mapping T, that is, Fix(T)={x∈ℋ:Tx=x}.It is clear that in a real Hilbert spaceℋ, S:C→C is k-strictly pseudocontractive if and only if there holds the following inequality:
(27)〈Sx-Sy,x-y〉≤∥x-y∥2-1-k2∥(I-S)x-(I-S)y∥2,∀x,y∈C.This immediately implies that ifS is a k-strictly pseudocontractive mapping, then I-S is (1-k)/2-inverse strongly monotone; for further detail, we refer to [9] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.The following elementary result in the real Hilbert spaces is quite well known.Lemma 14 (see [36]).
Letℋ be a real Hilbert space. Then, for all x,y∈ℋ and λ∈[0,1],
(28)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2∥λx+(1-λ)y∥2=-λ(1-λ)∥x-y∥2.Lemma 15 (see [37, Proposition 2.1]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ and S:C→C be a mapping. (i)
IfS is a k-strict pseudocontractive mapping, then S satisfies the Lipschitz condition
(29)∥Sx-Sy∥≤1+k1-k∥x-y∥,∀x,y∈C.(ii)
IfS is a k-strict pseudocontractive mapping, then the mapping I-S is semiclosed at 0, that is, if {xn} is a sequence in C such that xn→x~ weakly and (I-S)xn→0 strongly, then (I-S)x~=0.(iii)
IfS is k-(quasi-)strict pseudocontraction, then the fixed point set Fix(S) of S is closed and convex so that the projection PFix(S) is well defined.The following lemma plays a key role in proving weak convergence of the sequences generated by our algorithms.Lemma 16 (see [38, p. 80]).
Let{an}n=0∞, {bn}n=0∞, and {δn}n=0∞ be sequences of nonnegative real numbers satisfying the inequality
(30)an+1≤(1+δn)an+bn,∀n≥0.
If ∑n=0∞‍δn<∞ and ∑n=0∞‍bn<∞, then limn→∞an exists. If, in addition, {an}n=0∞ has a subsequence which converges to zero, then limn→∞an=0.Corollary 17 (see [39, p. 303]).
Let{an}n=0∞ and {bn}n=0∞ be two sequences of nonnegative real numbers satisfying the inequality
(31)an+1≤an+bn,∀n≥0.
If ∑n=0∞‍bn converges, then limn→∞an exists.Lemma 18 (see [7]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ. Let S:C→C be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that (γ+δ)k≤γ. Then
(32)∥γ(x-y)+δ(Sx-Sy)∥≤(γ+δ)∥x-y∥,∀x,y∈C.The following lemma is an immediate consequence of an inner product.Lemma 19.
In a real Hilbert spaceℋ, there holds the inequality
(33)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈ℋ.LetK be a nonempty closed convex subset of a real Hilbert space ℋ and let F:K→ℋ be a monotone mapping. The variational inequality problem (VIP) is to find x∈K such that
(34)〈Fx,y-x〉≥0,∀y∈K.The solution set of the VIP is denoted byVI(K,F). It is well known that
(35)x∈VI(K,F)⟺x=PK(x-λFx),∀λ>0.A set-valued mappingT:ℋ→2ℋ is called monotone if for all x,y∈ℋ, f∈Tx and g∈Ty imply that 〈x-y,f-g〉≥0. A monotone set-valued mapping T:ℋ→2ℋ is called maximal if its graph Gph(T) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping T:ℋ→2ℋ is maximal if and only if for (x,f)∈ℋ×ℋ,〈x-y,f-g〉≥0 for every (y,g)∈Gph(T) implies that f∈Tx. Let F:K→ℋ be a monotone and Lipschitz continuous mapping and let NKv be the normal cone to K at v∈K, that is,
(36)NKv={w∈ℋ:〈v-u,w〉≥0,∀u∈K}.
Define
(37)Tv={Fv+NKv,ifv∈K,∅,ifv∉K.It is known that in this case the mappingT is maximal monotone, and 0∈Tv if and only if v∈VI(K,F); for further details, we refer to [40] and the references therein.


## 3. Main Results

In this section, we first prove the weak convergence of the sequences generated by the Mann-type extragradient iterative algorithm (16) with regularization.Theorem 20.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and Bi:C→ℋ1 be βi-inverse strongly monotone for i=1,2. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ξ∩Γ≠∅. For given x0∈C arbitrarily, let {xn},{yn},{zn} be the sequences generated by the Mann-type extragradient iterative algorithm (16) with regularization, where μi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {σn}, {τn}, {βn}, {γn}, {δn}⊂[0,1] such that (i)
∑n=0∞‍αn<∞;(ii)
βn+γn+δn=1 and (γn+δn)k≤γn for all n≥0;(iii)
σn+τn≤1 for all n≥0;(iv)
0<liminfn→∞τn≤limsupn→∞(σn+τn)<1;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(vi)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then all the sequences {xn}, {yn}, {zn} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of GSVI (3), where y¯=PC(x¯-μ2B2x¯).Proof.
First, taking into account0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2, without loss of generality we may assume that {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2).
Now, let us show thatPC(I-λ∇fα) is ζ-averaged for each λ∈(0,2/(α+∥A∥2)), where
(38)ζ=2+λ(α+∥A∥2)4∈(0,1).
Indeed, it is easy to see that∇f=A*(I-PQ)A is 1/∥A∥2-ism, that is,
(39)〈∇f(x)-∇f(y),x-y〉≥1∥A∥2∥∇f(x)-∇f(y)∥2.
Observe that
(40)(α+∥A∥2)〈∇fα(x)-∇fα(y),x-y〉=(α+∥A∥2)[α∥x-y∥2=(α+∥A∥2)0+〈∇f(x)-∇f(y),x-y〉∥x-y∥2]=α2∥x-y∥2+α〈∇f(x)-∇f(y),x-y〉+α∥A∥2∥x-y∥2+∥A∥2〈∇f(x)-∇f(y),x-y〉≥α2∥x-y∥2+2α〈∇f(x)-∇f(y),x-y〉+∥∇f(x)-∇f(y)∥2=∥α(x-y)+∇f(x)-∇f(y)∥2=∥∇fα(x)-∇fα(y)∥2.
Hence, it follows that∇fα=αI+A*(I-PQ)A is 1/(α+∥A∥2)-ism. Thus, λ∇fα is 1/λ(α+∥A∥2)-ism according to Proposition 12(ii). By Proposition 12(iii), the complement I-λ∇fα is λ(α+∥A∥2)/2-averaged. Therefore, noting that PC is 1/2-averaged and utilizing Proposition 13(iv), we know that for each λ∈(0,2/(α+∥A∥2)), PC(I-λ∇fα) is ζ-averaged with
(41)ζ=12+λ(α+∥A∥2)2-12·λ(α+∥A∥2)2=2+λ(α+∥A∥2)4∈(0,1).
This shows that PC(I-λ∇fα) is nonexpansive. Furthermore, for {λn}⊂[a,b] with a,b∈(0,1/∥A∥2), we have
(42)a≤infn≥0λn≤supn≥0λn≤b<1∥A∥2=limn→∞1αn+∥A∥2.
Without loss of generality, we may assume that
(43)a≤infn≥0λn≤supn≥0λn≤b<1αn+∥A∥2,∀n≥0.
Consequently, it follows that for each integer n≥0, PC(I-λn∇fαn) is ζn-averaged with
(44)ζn=12+λn(αn+∥A∥2)2-12·λn(αn+∥A∥2)2=2+λn(αn+∥A∥2)4∈(0,1).
This immediately implies thatPC(I-λn∇fαn) is nonexpansive for all n≥0.
Next we divide the remainder of the proof into several steps.
Step
1. {xn} is bounded.
Indeed, takep∈Fix(S)∩Ξ∩Γ arbitrarily. Then Sp=p, PC(I-λ∇f)p=p for λ∈(0,2/∥A∥2), and
(45)p=PC[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)].
From (16), it follows that
(46)∥zn-p∥=∥PC(I-λn∇fαn)xn-PC(I-λn∇f)p∥≤∥PC(I-λn∇fαn)xn-PC(I-λn∇fαn)p∥≤+∥PC(I-λn∇fαn)p-PC(I-λn∇f)p∥≤∥xn-p∥+∥(I-λn∇fαn)p-(I-λn∇f)p∥≤∥xn-p∥+λnαn∥p∥.
Utilizing Lemma 19, we also have
(47)∥zn-p∥2=∥PC(I-λn∇fαn)xn-PC(I-λn∇f)p∥2=∥PC(I-λn∇fαn)xn-PC(I-λn∇fαn)p=0+PC(I-λn∇fαn)p-PC(I-λn∇f)p∥2≤∥PC(I-λn∇fαn)xn-PC(I-λn∇fαn)p∥2+2〈PC(I-λn∇fαn)p+200-PC(I-λn∇f)p,zn-p(I-λn∇fαn)〉≤∥xn-p∥2+2∥PC(I-λn∇fαn)p-PC(I-λn∇f)p∥×∥zn-p∥≤∥xn-p∥2+2∥(I-λn∇fαn)p-(I-λn∇f)p∥∥zn-p∥=∥xn-p∥2+2λnαn∥p∥∥zn-p∥.
For simplicity, we write q=PC(p-μ2B2p), z~n=PC(zn-μ2B2zn),
(48)un=PC[PC(zn-μ2B2zn)-μ1B1PC(zn-μ2B2zn)],u¯n=PC(xn-λn∇fαn(zn))
for each n≥0. Then yn=σnxn+τnu¯n+(1-σn-τn)un for each n≥0. Since Bi:C→ℋ1 is βi-inverse strongly monotone and 0<μi<2βi for i=1,2, we know that for all n≥0,
(49)∥un-p∥2=∥PC[PC(zn-μ2B2zn)-PC00-μ1B1PC(zn-μ2B2zn)]-p∥2=∥PC[PC(zn-μ2B2zn)-μ1B1PC(zn-μ2B2zn)]-PC-PC[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)]∥2≤∥[PC(zn-μ2B2zn)-μ1B1PC(zn-μ2B2zn)]000-[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)]∥2=∥[PC(zn-μ2B2zn)-PC(p-μ2B2p)]000-μ1[B1PC(zn-μ2B2zn)-B1PC(p-μ2B2p)]∥2≤∥PC(zn-μ2B2zn)-PC(p-μ2B2p)∥2-μ1(2β1-μ1)∥B1PC(zn-μ2B2zn)-μ1(2β1-μ1)0-B1PC(p-μ2B2p)∥2≤∥(zn-μ2B2zn)-(p-μ2B2p)∥2-μ1(2β1-μ1)∥B1z~n-B1q∥2=∥(zn-p)-μ2(B2zn-B2p)∥2-μ1(2β1-μ1)∥B1z~n-B1q∥2≤∥zn-p∥2-μ2(2β2-μ2)∥B2zn-B2p∥2-μ1(2β1-μ1)∥B1z~n-B1q∥2≤∥zn-p∥2.
Furthermore, by Proposition 8(ii), we have
(50)∥u¯n-p∥2≤∥xn-λn∇fαn(zn)-p∥2-∥xn-λn∇fαn(zn)-u¯n∥2=∥xn-p∥2-∥xn-u¯n∥2+2λn〈∇fαn(zn),p-u¯n〉=∥xn-p∥2-∥xn-u¯n∥2+2λn(〈∇fαn(zn)-∇fαn(p),p-zn〉+〈∇fαn(p),p-zn〉+〈∇fαn(zn),zn-u¯n〉)≤∥xn-p∥2-∥xn-u¯n∥2+2λn(〈∇fαn(p),p-zn〉+〈∇fαn(zn),zn-u¯n〉)=∥xn-p∥2-∥xn-u¯n∥2+2λn[〈∇fαn(zn),〉〈(αnI+∇f)p,p-zn〉+〈∇fαn(zn),zn-u¯n〉∇f]≤∥xn-p∥2-∥xn-u¯n∥2+2λn[αn〈p,p-zn〉+〈∇fαn(zn),zn-u¯n〉]=∥xn-p∥2-∥xn-zn∥2-2〈xn-zn,zn-u¯n〉-∥zn-u¯n∥2+2λn[αn〈p,p-zn〉+〈∇fαn(zn),zn-u¯n〉]=∥xn-p∥2-∥xn-zn∥2-∥zn-u¯n∥2+2〈xn-λn∇fαn(zn)-zn,u¯n-zn〉+2λnαn〈p,p-zn〉.
Further, by Proposition 8(i), we have
(51)〈xn-λn∇fαn(zn)-zn,u¯n-zn〉=〈xn-λn∇fαn(xn)-zn,u¯n-zn〉+〈λn∇fαn(xn)-λn∇fαn(zn),u¯n-zn〉≤〈λn∇fαn(xn)-λn∇fαn(zn),u¯n-zn〉≤λn∥∇fαn(xn)-∇fαn(zn)∥∥u¯n-zn∥≤λn(αn+∥A∥2)∥xn-zn∥∥u¯n-zn∥.
So, from (46), we obtain
(52)∥u¯n-p∥2≤∥xn-p∥2-∥xn-zn∥2-∥zn-u¯n∥2+2〈xn-λn∇fαn(zn)-zn,u¯n-zn〉+2λnαn〈p,p-zn〉≤∥xn-p∥2-∥xn-zn∥2-∥zn-u¯n∥2+2λn(αn+∥A∥2)∥xn-zn∥∥u¯n-zn∥+2λnαn∥p∥∥zn-p∥≤∥xn-p∥2-∥xn-zn∥2-∥zn-u¯n∥2+λn2(αn+∥A∥2)2∥xn-zn∥2+∥zn-u¯n∥2+2λnαn∥p∥∥zn-p∥=∥xn-p∥2+2λnαn∥p∥∥zn-p∥+(λn2(αn+∥A∥2)2-1)∥xn-zn∥2≤∥xn-p∥2+2λnαn∥p∥∥zn-p∥≤∥xn-p∥2+2λnαn∥p∥[∥xn-p∥+λnαn∥p∥]≤∥xn-p∥2+4λnαn∥p∥∥xn-p∥+4λn2αn2∥p∥2=(∥xn-p∥+2λnαn∥p∥)2.
Hence, it follows from (46), (49), and (52) that
(53)∥yn-p∥=∥σn(xn-p)+τn(u¯n-p)+(1-σn-τn)(un-p)∥≤σn∥xn-p∥+τn∥u¯n-p∥+(1-σn-τn)∥un-p∥≤σn∥xn-p∥+τn∥u¯n-p∥+(1-σn-τn)∥zn-p∥≤σn∥xn-p∥+τn(∥xn-p∥+2λnαn∥p∥)+(1-σn-τn)(∥xn-p∥+λnαn∥p∥)=∥xn-p∥+[2τn+(1-σn-τn)]λnαn∥p∥≤∥xn-p∥+2λnαn∥p∥.
Since (γn+δn)k≤γn for all n≥0, utilizing Lemma 18, we obtain from (53)
(54)∥xn+1-p∥=∥βn(xn-p)+γn(yn-p)+δn(Syn-p)∥≤βn∥xn-p∥+∥γn(yn-p)+δn(Syn-p)∥≤βn∥xn-p∥+(γn+δn)∥yn-p∥≤βn∥xn-p∥+(γn+δn)×[∥xn-p∥+2λnαn∥p∥]≤∥xn-p∥+2b∥p∥αn.
Since ∑n=0∞‍αn<∞, it is clear that ∑n=0∞‍2b∥p∥αn<∞. Thus, by Corollary 17, we conclude that
(55)limn→∞∥xn-p∥existsforeachp∈Fix(S)∩Ξ∩Γ,
and the sequence {xn} is bounded. Taking into account that PC, ∇fαn, B1 and B2 are Lipschitz continuous, we can easily see that {zn}, {un}, {u¯n}, {yn}, and {z~n} are bounded, where z~n=PC(zn-μ2B2zn) for all n≥0.
Step
2. Consider limn→∞∥B2zn-B2p∥=0, limn→∞∥B1z~n-B1q∥=0 and limn→∞∥xn-zn∥=0, where q=PC(p-μ2B2p).
Indeed, utilizing Lemma18 and the convexity of ∥·∥2, we obtain from (16) and (47)–(52) that
(56)∥xn+1-p∥2=∥βn(xn-p)+γn(yn-p)+δn(Syn-p)∥2≤βn∥xn-p∥2+(γn+δn)00×∥1γn+δn[γn(yn-p)+δn(Syn-p)]∥2≤βn∥xn-p∥2+(γn+δn)∥yn-p∥2≤βn∥xn-p∥2+(γn+δn)00×[σn∥xn-p∥2+τn∥u¯n-p∥2≤×+(1-σn-τn)∥un-p∥2]≤βn∥xn-p∥2+(γn+δn)≤×{σn∥xn-p∥20000+τn[∥xn-p∥2+2λnαn∥p∥∥zn-p∥000000000+(λn2(αn+∥A∥2)2-1)∥xn-zn∥2]0000+(1-σn-τn)0000×[∥zn-p∥2-μ2(2β2-μ2)∥B2zn-B2p∥20000000000-μ1(2β1-μ1)∥B1z~n-B1q∥2]}≤βn∥xn-p∥2+(γn+δn)×{σn∥xn-p∥2000+τn[(αn+∥A∥2)2∥xn-p∥2+2λnαn∥p∥∥zn-p∥0000000+(λn2(αn+∥A∥2)2-1)∥xn-zn∥2(αn+∥A∥2)2]0000000+(1-σn-τn)0000000×[∥xn-p∥2+2λnαn∥p∥∥zn-p∥0000000000-μ2(2β2-μ2)∥B2zn-B2p∥20000000000-μ1(2β1-μ1)∥B1z~n-B1q∥2]}=∥xn-p∥2+(γn+δn)0000×{∥p∥[μ2(2β2-μ2)∥B2zn-B2p∥2(1-σn)2λnαn∥p∥∥zn-p∥=×+τn(λn2(αn+∥A∥2)2-1)∥xn-zn∥2=×-(1-σn-τn)=××[μ2(2β2-μ2)∥B2zn-B2p∥2=×00+μ1(2β1-μ1)∥B1z~n-B1q∥2]}≤∥xn-p∥2+2λnαn∥p∥∥zn-p∥≤-(γn+δn){τn(1-λn2(αn+∥A∥2)2)∥xn-zn∥2≤-(γn+δn)0+(1-σn-τn)≤-(γn+δn)0×[μ2(2β2-μ2)∥B2zn-B2p∥2≤-(γn+δn)0000+μ1(2β1-μ1)∥B1z~n-B1q∥2](1-λn2(αn+∥A∥2)2)∥xn-zn∥2}.
Therefore,
(57)(γn+δn){τn(1-λn2(αn+∥A∥2)2)∥xn-zn∥2+δn0+(1-σn-τn)+δn0×[μ2(2β2-μ2)∥B2zn-B2p∥2+μ1(2β1-μ1)∥B1z~n-B1q∥2](1-λn2(αn+∥A∥2)2)∥xn-zn∥2}≤∥xn-p∥2-∥xn+1-p∥2+2λnαn∥p∥∥zn-p∥.
Since αn→0,limn→∞∥xn-p∥ exists, liminfn→∞(γn+δn)>0, {λn}⊂[a,b] and 0<liminfn→∞τn≤limsupn→∞(σn+τn)<1, it follows that
(58)limn→∞∥xn-zn∥=0,limn→∞∥B1z~n-B1q∥=0,limn→∞∥B2zn-B2p∥=0.
Step
3. Consider  limn→∞∥Syn-yn∥=0.
Indeed, observe that(59)∥u¯n-zn∥=∥PC(xn-λn∇fαn(zn))-PC(xn-λn∇fαn(xn))∥≤∥(xn-λn∇fαn(zn))-(xn-λn∇fαn(xn))∥=λn∥∇fαn(zn)-∇fαn(xn)∥≤λn(αn+∥A∥2)∥zn-xn∥.
This together with∥zn-xn∥→0 implies that limn→∞∥u¯n-zn∥=0 and hence limn→∞∥u¯n-xn∥=0. By firm nonexpansiveness of PC, we have
(60)∥z~n-q∥2=∥PC(zn-μ2B2zn)-PC(p-μ2B2p)∥2≤〈(zn-μ2B2zn)-(p-μ2B2p),z~n-q〉=12[∥zn-p-μ2(B2zn-B2p)∥2+∥z~n-q∥2-∥(zn-p)-μ2(B2zn-B2p)-(z~n-q)∥2]≤12[∥zn-p∥2∥zn-p∥2+∥z~n-q∥2-∥(zn-z~n)-μ2(B2zn-B2p)-(p-q)∥2∥zn-p∥2]=12[∥zn-p∥2∥zn-p∥2+∥z~n-q∥2-∥zn-z~n-(p-q)∥2+2μ2〈zn-z~n-(p-q),B2zn-B2p〉-μ22∥B2zn-B2p∥2∥zn-p∥2]≤12[∥zn-p∥2∥zn-p∥2+∥z~n-q∥2-∥zn-z~n-(p-q)∥2+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥∥zn-p∥2∥z~n-q∥2],
that is,
(61)∥z~n-q∥2≤∥zn-p∥2-∥zn-z~n-(p-q)∥2+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥.
Moreover, using the argument technique similar to the previous one, we derive(62)∥un-p∥2=∥PC(z~n-μ1B1z~n)-PC(q-μ1B1q)∥2≤〈(z~n-μ1B1z~n)-(q-μ1B1q),un-p〉=12[∥z~n-q-μ1(B1z~n-B1q)∥2+∥un-p∥2-∥(z~n-q)-μ1(B1z~n-B1q)-(un-p)∥2]≤12[∥zn-p∥2∥z~n-q∥2+∥un-p∥2-∥(z~n-un)-μ1(B1z~n-B1q)+(p-q)∥2∥zn-p∥2]=12[∥zn-p∥2∥z~n-q∥2+∥un-p∥20-∥z~n-un+(p-q)∥20+2μ1〈z~n-un+(p-q),B1z~n-B1q〉0-μ12∥B1z~n-B1q∥2∥zn-p∥2]≤12[∥z~n-q∥2+∥un-p∥2-∥z~n-un+(p-q)∥2+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥],
that is,
(63)∥un-p∥2≤∥z~n-q∥2-∥z~n-un+(p-q)∥2+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥.
Utilizing (47), (52), (61), and (63), we have
(64)∥yn-p∥2=∥σn(xn-p)+τn(u¯n-p)0+(1-σn-τn)(un-p)∥2≤σn∥xn-p∥2+τn∥u¯n-p∥2+(1-σn-τn)∥un-p∥2≤σn∥xn-p∥2+τn(∥xn-p∥2+2λnαn∥p∥∥zn-p∥)+(1-σn-τn)×[∥z~n-q∥2-∥z~n-un+(p-q)∥20000+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥∥z~n-un+(p-q)∥2]≤σn∥xn-p∥2+τn(∥xn-p∥2+2λnαn∥p∥∥zn-p∥)00+(1-σn-τn)00×{∥zn-p∥2-∥zn-z~n-(p-q)∥20+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥0-∥z~n-un+(p-q)∥20+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥∥zn-p∥2}≤σn∥xn-p∥2+τn(∥xn-p∥2+2λnαn∥p∥∥zn-p∥)+(1-σn-τn)×{∥xn-p∥2+2λnαn∥p∥∥zn-p∥0-∥zn-z~n-(p-q)∥2+2μ2∥zn-z~n-(p-q)∥0×∥B2zn-B2p∥-∥z~n-un+(p-q)∥20+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥∥xn-p∥2}≤∥xn-p∥2+2λnαn∥p∥∥zn-p∥+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥-(1-σn-τn)(∥zn-z~n-(p-q)∥20000000000000+∥z~n-un+(p-q)∥2).
Thus, utilizing Lemma 14, from (16) and (64) it follows that
(65)∥xn+1-p∥2=∥βn(xn-p)+γn(yn-p)+δn(Syn-p)∥2=∥βn(xn-p)+(1-βn)·11-βn0×[γn(yn-p)+δn(Syn-p)](1-βn)·11-βn∥2=βn∥xn-p∥2+(1-βn)×∥11-βn[γn(yn-p)+δn(Syn-p)]∥2-βn(1-βn)0000×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2-βn(1-βn)×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤βn∥xn-p∥2+(1-βn)×{∥xn-p∥2+2λnαn∥p∥∥zn-p∥0+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥0+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥0-(1-σn-τn)(∥zn-z~n-(p-q)∥2-(1-σn-τn)+∥z~n-un+(p-q)∥2)}-βn(1-βn)0000×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤∥xn-p∥2+2λnαn∥p∥∥zn-p∥+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥-(1-βn)(1-σn-τn)(∥zn-z~n-(p-q)∥2(-βn)(1-σn-τn)+∥z~n-un+(p-q)∥2)-βn(1-βn)×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2,
which hence implies that
(66)(1-βn)(1-σn-τn)×(∥zn-z~n-(p-q)∥2+∥z~n-un+(p-q)∥2)+βn(1-βn)∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤∥xn-p∥2-∥xn+1-p∥2+2λnαn∥p∥∥zn-p∥+2μ2∥zn-z~n-(p-q)∥∥B2zn-B2p∥+2μ1∥z~n-un+(p-q)∥∥B1z~n-B1q∥.
Since 0<liminfn→∞βn≤limsupn→∞βn<1, limsupn→∞(σn+τn)<1, {λn}⊂[a,b], αn→0, ∥B2zn-B2p∥→0, ∥B1z~n-B1q∥→0 and limn→∞∥xn-p∥ exists, it follows from the boundedness of {un}, {zn} and {z~n} that limn→∞∥zn-z~n-(p-q)∥=0,
(67)limn→∞∥z~n-un+(p-q)∥=0,limn→∞∥γn(xn-yn)+δn(xn-Syn)∥=0.
Consequently, it immediately follows that
(68)limn→∞∥zn-un∥=0,limn→∞∥un-u¯n∥=0.
Also, note that
(69)∥yn-u¯n∥≤σn∥xn-u¯n∥+(1-σn-τn)∥un-u¯n∥→0.
This together with ∥xn-u¯n∥→0 implies that
(70)limn→∞∥xn-yn∥=0.
Since
(71)∥δn(Syn-xn)∥=∥γn(xn-yn)+δn(xn-Syn)+γn(yn-xn)∥≤∥γn(xn-yn)+δn(xn-Syn)∥+γn∥xn-yn∥,
we have
(72)limn→∞∥Syn-xn∥=0,limn→∞∥Syn-yn∥=0.
Step
4. {xn}, {yn},and {zn} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ.
Indeed, since{xn} is bounded, there exists a subsequence {xni} of {xn} that converges weakly to some x¯∈C. We obtain that x¯∈Fix(S)∩Ξ∩Γ. Taking into account that ∥xn-yn∥→0 and ∥xn-zn∥→0 as n→∞, we deduce that yni→x¯ weakly and zni→x¯ weakly. First, it is clear from Lemma 15 and ∥Syn-yn∥→0 that x¯∈Fix(S). Now let us show that x¯∈Ξ. Note that
(73)∥zn-G(zn)∥=∥zn-PC[PC(zn-μ2B2zn)-μ1B1PC(zn-μ2B2zn)]∥=∥zn-un∥→0,
as n→∞ where G:C→C is defined as that in Lemma 1. According to Lemma 15, we get x¯∈Ξ. Further, let us show that x¯∈Γ. As a matter of fact, define
(74)Tv={∇f(v)+NCv,ifv∈C,∅,ifv∉C,
where NCv={w∈ℋ1:〈v-u,w〉≥0,∀u∈C}. Then, T is maximal monotone and 0∈Tv if and only if v∈VI(C,∇f); see [40] for more details. Let (v,w)∈Gph(T). Then, we have
(75)w∈Tv=∇f(v)+NCv,
and hence
(76)w-∇f(v)∈NCv.
So, we have
(77)〈v-u,w-∇f(v)〉≥0,∀u∈C.
On the other hand, from
(78)zn=PC(xn-λn∇fαn(xn)),v∈C,
we have
(79)〈xn-λn∇fαn(xn)-zn,zn-v〉≥0,
and hence,
(80)〈v-zn,zn-xnλn+∇fαn(xn)〉≥0.
Therefore, from
(81)w-∇f(v)∈NCv,zni∈C,
we have
(82)〈v-zni,w〉≥〈v-zni,∇f(v)〉≥〈v-zni,∇f(v)〉-〈v-zni,zni-xniλni+∇fαni(xni)〉=〈v-zni,∇f(v)〉-〈v-zni,zni-xniλni+∇f(xni)〉-αni〈v-zni,xni〉=〈v-zni,∇f(v)-∇f(zni)〉+〈v-zni,∇f(zni)-∇f(xni)〉-〈v-zni,zni-xniλni〉-αni〈v-zni,xni〉≥〈v-zni,∇f(zni)-∇f(xni)〉-〈v-zni,zni-xniλni〉-αni〈v-zni,xni〉.
Hence, we get
(83)〈v-x¯,w〉≥0,asi→∞.
Since T is maximal monotone, we have x¯∈T-10, and hence, x¯∈VI(C,∇f). Thus, it is clear that x¯∈Γ. Therefore, x¯∈Fix(S)∩Ξ∩Γ.
Let{xnj} be another subsequence of {xn} such that {xnj} converges weakly to x^∈C. Then, x^∈Fix(S)∩Ξ∩Γ. Let us show that x¯=x^. Assume that x¯≠x^. From the Opial condition [41], we have
(84)limn→∞∥xn-x¯∥=liminfi→∞∥xni-x¯∥<liminfi→∞∥xni-x^∥=liminfn→∞∥xn-x^∥=liminfj→∞∥xnj-x^∥<liminfj→∞∥xnj-x¯∥=limn→∞∥xn-x¯∥.
This leads to a contradiction. Consequently, we have x¯=x^. This implies that {xn} converges weakly to x¯∈Fix(S)∩Ξ∩Γ. Further, from ∥xn-yn∥→0 and ∥xn-zn∥→0, it follows that both {yn} and {zn} converge weakly to x¯. This completes the proof.Corollary 21.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and Bi:C→ℋ1 be βi-inverse strongly monotone for i=1,2. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ξ∩Γ≠∅. For given x0∈C arbitrarily, let the sequences {xn},{yn},{zn} be generated iteratively by
(85)zn=PC(xn-λn∇fαn(xn)),yn=τnPC(xn-λn∇fαn(zn))00+(1-τn)PC[PC(zn-μ2B2zn)00+(1-τn)PC0-μ1B1PC(zn-μ2B2zn)],xn+1=βnxn+γnyn+δnSyn,∀n≥0,
where μi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {τn}, {βn}, {γn}, {δn}⊂[0,1] such that (i)
∑n=0∞‍αn<∞;(ii)
βn+γn+δn=1 and (γn+δn)k≤γn for all n≥0;(iii)
0<liminfn→∞τn≤limsupn→∞τn<1;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(v)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then all the sequences {xn}, {yn}, {zn} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of the GSVI (3), where y¯=PC(x¯-μ2B2x¯).Proof.
In Theorem20, put σn=0 for all n≥0. Then, in this case, Theorem 20 reduces to Corollary 21.Next, utilizing Corollary21, we give the following result.Corollary 22.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and S:C→C be a nonexpansive mapping such that Fix(S)∩Γ≠∅. For given x0∈C arbitrarily, let the sequences {xn}, {yn}, {zn} be generated iteratively by
(86)zn=PC(xn-λn∇fαn(xn)),yn=(1-τn)zn+τnPC(xn-λn∇fαn(zn)),xn+1=βnxn+(1-βn)Syn,∀n≥0,
where {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {τn},{βn}⊂[0,1] such that (i)
∑n=0∞‍αn<∞;(ii)
0<liminfn→∞τn≤limsupn→∞τn<1;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iv)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then all the sequences {xn}, {yn}, {zn} converge weakly to an element x¯∈Fix(S)∩Γ.Proof.
In Corollary21, put B1=B2=0 and γn=0. Then, Ξ=C, βn+δn=1 for all n≥0, and the iterative scheme (85) is equivalent to
(87)zn=PC(xn-λn∇fαn(xn)),yn=τnPC(xn-λn∇fαn(zn))+(1-τn)zn,xn+1=βnxn+δnSyn,∀n≥0.
This is equivalent to (86). Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k=0. In this case, it is easy to see that all the conditions (i)–(v) in Corollary 21 are satisfied. Therefore, in terms of Corollary 21, we obtain the desired result.Now, we are in a position to prove the weak convergence of the sequences generated by the Mann-type extragradient iterative algorithm (17) with regularization.Theorem 23.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and Bi:C→ℋ1 be βi-inverse strongly monotone for i=1,2. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ξ∩Γ≠∅. For given x0∈C arbitrarily, let {xn},{un},{u~n} be the sequences generated by the Mann-type extragradient iterative algorithm (17) with regularization, where μi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {σn},{βn},{γn},{δn}⊂[0,1] such that (i)
∑n=0∞‍αn<∞;(ii)
βn+γn+δn=1 and (γn+δn)k≤γn for all n≥0;(iii)
limsupn→∞σn<1;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(v)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then the sequences {xn}, {un}, {u~n} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of the GSVI (3), where y¯=PC(x¯-μ2B2x¯).Proof.
First, taking into account0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2, without loss of generality, we may assume that {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2). Repeating the same argument as that in the proof of Theorem 20, we can show that PC(I-λ∇fα) is ζ-averaged for each λ∈(0,1/(α+∥A∥2)), where ζ=(2+λ(α+∥A∥2))/4. Further, repeating the same argument as that in the proof of Theorem 20, we can also show that for each integer n≥0, PC(I-λn∇fαn) is ζn-averaged with ζn=(2+λn(αn+∥A∥2))/4∈(0,1).
Next we divide the remainder of the proof into several steps.
Step
1.  {xn} is bounded.
Indeed, takep∈Fix(S)∩Ξ∩Γ arbitrarily. Then Sp=p, PC(I-λ∇f)p=p for λ∈(0,2/∥A∥2), and
(88)p=PC[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)].
For simplicity, we write
(89)q=PC(p-μ2B2p),x~n=PC(xn-μ2B2xn),u¯n=PC(un-λn∇fαn(u~n)),
for each n≥0. Then yn=σnxn+(1-σn)u¯n for each n≥0. Utilizing the arguments similar to those of (46) and (47) in the proof of Theorem 20, from (17) we can obtain
(90)∥u~n-p∥≤∥un-p∥+λnαn∥p∥,(91)∥u~n-p∥2≤∥un-p∥2+2λnαn∥p∥∥u~n-p∥.
Since Bi:C→ℋ1 is βi-inverse strongly monotone and 0<μi<2βi for i=1,2, utilizing the argument similar to that of (49) in the proof of Theorem 20, we can obtain that for all n≥0,
(92)∥un-p∥2≤∥xn-p∥2-μ2(2β2-μ2)∥B2xn-B2p∥2-μ1(2β1-μ1)∥B1x~n-B1q∥2≤∥xn-p∥2.
Utilizing the argument similar to that of (52) in the proof of Theorem 20, from (90) we can obtain
(93)∥u¯n-p∥2≤∥un-p∥2+2λnαn∥p∥∥u~n-p∥+(λn2(αn+∥A∥2)2-1)∥un-u~n∥2≤(∥un-p∥+2λnαn∥p∥)2.
Hence, it follows from (92) and (93) that
(94)∥yn-p∥=∥σn(xn-p)+(1-σn)(u¯n-p)∥≤σn∥xn-p∥+(1-σn)∥u¯n-p∥≤σn∥xn-p∥+(1-σn)×(∥xn-p∥+2λnαn∥p∥)≤∥xn-p∥+2λnαn∥p∥.
Since (γn+δn)k≤γn for all n≥0, by Lemma 18 we can readily see from (94) that
(95)∥xn+1-p∥≤βn∥xn-p∥+(γn+δn)∥yn-p∥≤∥xn-p∥+2b∥p∥αn.
Since ∑n=0∞‍αn<∞, it is clear that ∑n=0∞‍2b∥p∥αn<∞. Thus, by Corollary 17 we conclude that
(96)limn→∞∥xn-p∥existsforeachp∈Fix(S)∩Ξ∩Γ,
and the sequence {xn} is bounded. Since PC, ∇fαn, B1 and B2 are Lipschitz continuous, it is easy to see that {un}, {u~n}, {u¯n}, {yn} and {x~n} are bounded, where x~n=PC(xn-μ2B2xn) for all n≥0.
Step
2. Consider limn→∞∥B2xn-B2p∥=0, limn→∞∥B1x~n-B1q∥=0 and limn→∞∥un-u~n∥=0, where q=PC(p-μ2B2p).
Indeed, utilizing Lemma18 and the convexity of ∥·∥2, we obtain from (17), (92), and (93) that
(97)∥xn+1-p∥2≤βn∥xn-p∥2+(γn+δn)∥yn-p∥2≤βn∥xn-p∥2+(γn+δn)×[σn∥xn-p∥2+(1-σn)∥u¯n-p∥2]≤βn∥xn-p∥2+(γn+δn)≤×{σn∥xn-p∥2+(1-σn)00×[∥un-p∥2(λn2(αn+∥A∥2)2-1)∥un-p∥2+2λnαn∥p∥∥u~n-p∥0000s0+(λn2(αn+∥A∥2)2-1)∥un-u~n∥2(λn2(αn+∥A∥2)2-1)}≤βn∥xn-p∥2+(γn+δn)×{σn∥xn-p∥2+(1-σn)0000000×[∥xn-p∥2-μ2(2β2-μ2)∥B2xn-B2p∥20000-μ1(2β1-μ1)∥B1x~n-B1q∥2+2λnαn∥p∥∥u~n-p∥0000+(λn2(αn+∥A∥2)2-1)∥un-u~n∥2]}≤∥xn-p∥2+2λnαn∥p∥∥u~n-p∥-(γn+δn)(1-σn)×{(1-λn2(αn+∥A∥2)2)μ2(2β2-μ2)∥B2xn-B2p∥2+μ1(2β1-μ1)∥B1x~n-B1q∥2+(1-λn2(αn+∥A∥2)2)∥un-u~n∥2(1-λn2(αn+∥A∥2)2)}.
Therefore,
(98)(γn+δn)(1-σn){(2β2-μ2)∥B2xn-B2p∥2(1-λn2(αn+∥A∥2)2)μ2(2β2-μ2)∥B2xn-B2p∥20+μ1(2β1-μ1)∥B1x~n-B1q∥20+(1-λn2(αn+∥A∥2)2)∥un-u~n∥2}≤∥xn-p∥2-∥xn+1-p∥2+2λnαn∥p∥∥u~n-p∥.
Since αn→0, limsupn→∞σn<1, limn→∞∥xn-p∥ exists, liminfn→∞(γn+δn)>0 and {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2), it follows from the boundedness of {u~n} that
(99)limn→∞∥un-u~n∥=0,limn→∞∥B1x~n-B1q∥=0,limn→∞∥B2xn-B2p∥=0.
Step
3. Consider limn→∞∥Syn-yn∥=0.
Indeed, utilizing the Lipschitz continuity of∇fαn, we have
(100)∥u¯n-u~n∥=∥PC(un-λn∇fαn(u~n))-PC(un-λn∇fαn(un))∥≤λn(αn+∥A∥2)∥u~n-un∥.
This together with ∥u~n-un∥→0 implies that limn→∞∥u¯n-u~n∥=0 and hence limn→∞∥u¯n-un∥=0. Utilizing the arguments similar to those of (61) and (63) in the proof of Theorem 20, we get
(101)∥x~n-q∥2≤∥xn-p∥2-∥xn-x~n-(p-q)∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥,∥un-p∥2≤∥x~n-q∥2-∥x~n-un+(p-q)∥2+2μ1∥x~n-un+(p-q)∥∥B1x~n-B1q∥.
Utilizing (91) and (101), we have
(102)∥u¯n-p∥2=∥u~n-p+u¯n-u~n∥2≤∥u~n-p∥2+2〈u¯n-u~n,u¯n-p〉≤∥u~n-p∥2+2∥u¯n-u~n∥∥u¯n-p∥≤∥un-p∥2+2λnαn∥p∥∥u~n-p∥+2∥u¯n-u~n∥∥u¯n-p∥≤∥x~n-q∥2-∥x~n-un+(p-q)∥2+2μ1∥x~n-un+(p-q)∥∥B1x~n-B1q∥+2λnαn∥p∥∥u~n-p∥+2∥u¯n-u~n∥∥u¯n-p∥≤∥xn-p∥2-∥xn-x~n-(p-q)∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥-∥x~n-un+(p-q)∥2+2μ1∥x~n-un+(p-q)∥∥B1x~n-B1q∥+2λnαn∥p∥∥u~n-p∥+2∥u¯n-u~n∥∥u¯n-p∥.
Thus, utilizing Lemma 14, from (17) and (102), it follows that
(103)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2-βn(1-βn)×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤βn∥xn-p∥2+(1-βn)×[σn∥xn-p∥2+(1-σn)∥u¯n-p∥2]-βn(1-βn)∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤βn∥xn-p∥2+(1-βn)000×{σn∥xn-p∥2000000+(1-σn)[∥xn-p∥2-∥xn-x~n-(p-q)∥2000000000000000+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥000000000000000-∥x~n-un+(p-q)∥2000000000000000+2μ1∥x~n-un+(p-q)∥000000000000000×∥B1x~n-B1q∥+2λnαn∥p∥∥u~n-p∥000000000000000+2∥u¯n-u~n∥∥u¯n-p∥∥xn-x~n-(p-q)∥2]∥xn-x~n-(p-q)∥2}000-βn(1-βn)000×∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤∥xn-p∥2-(1-βn)(1-σn)×(∥xn-x~n-(p-q)∥2+∥x~n-un+(p-q)∥2)+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+2μ1∥x~n-un+(p-q)∥∥B1x~n-B1q∥+2λnαn∥p∥∥u~n-p∥+2∥u¯n-u~n∥∥u¯n-p∥-βn(1-βn)∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2,
which hence implies that
(104)(1-βn)(1-σn)×(∥xn-x~n-(p-q)∥2+∥x~n-un+(p-q)∥2)+βn(1-βn)∥11-βn[γn(xn-yn)+δn(xn-Syn)]∥2≤∥xn-p∥2-∥xn+1-p∥2+2λnαn∥p∥∥u~n-p∥+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+2μ1∥x~n-un+(p-q)∥∥B1x~n-B1q∥+2∥u¯n-u~n∥∥u¯n-p∥.
Since 0<liminfn→∞βn≤limsupn→∞βn<1, limsupn→∞σn<1, {λn}⊂[a,b], αn→0, ∥B2xn-B2p∥→0, ∥B1x~n-B1q∥→0, ∥u¯n-u~n∥→0 and limn→∞∥xn-p∥ exists, it follows from the boundedness of {xn}, {x~n}, {un}, {u~n} and {u¯n} that limn→∞∥xn-x~n-(p-q)∥=0:
(105)limn→∞∥x~n-un+(p-q)∥=0,limn→∞∥γn(xn-yn)+δn(xn-Syn)∥=0.
Consequently, it immediately follows that
(106)limn→∞∥xn-un∥=0,limn→∞∥xn-u¯n∥=0.
This together with ∥yn-u¯n∥≤σn∥xn-u¯n∥→0 implies that
(107)limn→∞∥xn-yn∥=0.
Since
(108)∥δn(Syn-xn)∥=∥γn(xn-yn)+δn(xn-Syn)+γn(yn-xn)∥≤∥γn(xn-yn)+δn(xn-Syn)∥+γn∥xn-yn∥,
we have
(109)limn→∞∥Syn-xn∥=0,limn→∞∥Syn-yn∥=0.
Step
4.  {xn}, {un} and {u~n} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ.
Indeed, since{xn} is bounded, there exists a subsequence {xni} of {xn} that converges weakly to some x¯∈C. We obtain that x¯∈Fix(S)∩Ξ∩Γ. Taking into account that ∥xn-un∥→0 and ∥u~n-un∥→0 and ∥xn-yn∥→0, we deduce that yni→x¯ weakly and u~ni→x¯ weakly.
First, it is clear from Lemma15 and ∥Syn-yn∥→0 that x¯∈Fix(S). Now let us show that x¯∈Ξ. Note that
(110)∥xn-G(xn)∥=∥xn-PC[PC(xn-μ2B2xn)00000000000000-μ1B1PC(xn-μ2B2xn)]∥=∥xn-un∥→0,
as n→∞, where G:C→C is defined as that in Lemma 1. According to Lemma 15, we get x¯∈Ξ. Further, let us show that x¯∈Γ. As a matter of fact, define
(111)Tv={∇f(v)+NCv,ifv∈C,∅,ifv∉C,
where NCv={w∈ℋ1:〈v-u,w〉≥0,∀u∈C}. Utilizing the argument similar to that of Step 4 in the proof of Theorem 20, from the relation
(112)u~n=PC(un-λn∇fαn(un)),v∈C,
we can easily conclude that
(113)〈v-x¯,w〉≥0.
It is easy to see that x¯∈Γ. Therefore, x¯∈Fix(S)∩Ξ∩Γ. Finally, utilizing the Opial condition [41], we infer that {xn} converges weakly to x¯∈Fix(S)∩Ξ∩Γ. Further, from ∥xn-un∥→0 and ∥xn-u~n∥→0, it follows that both {un} and {u~n} converge weakly to x¯. This completes the proof.Corollary 24.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and Bi:C→ℋ1 be βi-inverse strongly monotone for i=1,2. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ξ∩Γ≠∅. For given x0∈C arbitrarily, let the sequences {xn}, {un}, {u~n} be generated iteratively by
(114)un=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],u~n=PC(un-λn∇fαn(un)),yn=PC(un-λn∇fαn(u~n)),xn+1=βnxn+γnyn+δnSyn,∀n≥0,
where μi∈(0,2βi) for i=1,2, {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {βn},{γn},{δn}⊂[0,1] such that(i)
∑n=0∞‍αn<∞;(ii)
βn+γn+δn=1 and (γn+δn)k≤γn for all n≥0;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(iv)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then the sequences {xn}, {un}, {u~n} converge weakly to an element x¯∈Fix(S)∩Ξ∩Γ. Furthermore, (x¯,y¯) is a solution of GSVI (3), where y¯=PC(x¯-μ2B2x¯).Next, utilizing Corollary24, we derive the following result.Corollary 25.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ1. Let A∈B(ℋ1,ℋ2) and S:C→C be a nonexpansive mapping such that Fix(S)∩Γ≠∅. For given x0∈C arbitrarily, let the sequences {xn}, {u~n} be generated iteratively by
(115)u~n=PC(xn-λn∇fαn(xn)),xn+1=βnxn+(1-βn)SPC(xn-λn∇fαn(u~n)),000000000000000000000000000000000∀n≥0,
where {αn}⊂(0,∞), {λn}⊂(0,1/∥A∥2) and {βn}⊂[0,1] such that (i)
∑n=0∞‍αn<∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
0<liminfn→∞λn≤limsupn→∞λn<1/∥A∥2.Then, both the sequences {xn} and {u~n} converge weakly to an element x¯∈Fix(S)∩Γ.Proof.
In Corollary24, put B1=B2=0 and γn=0. Then, Ξ=C, βn+δn=1 for all n≥0, and the iterative scheme (114) is equivalent to
(116)un=xn,u~n=PC(un-λn∇fαn(un)),yn=PC(un-λn∇fαn(u~n)),xn+1=βnxn+δnSyn,∀n≥0.
This is equivalent to (115). Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k=0. In this case, it is easy to see that all the conditions (i)–(iv) in Corollary 24 are satisfied. Therefore, in terms of Corollary 24, we obtain the desired result.Remark 26.
Compared with the Ceng and Yao [31, Theorem 3.1], our Corollary 25 coincides essentially with [31, Theorem 3.1]. This shows that our Theorem 23 includes [31, Theorem 3.1] as a special case.Remark 27.
Our Theorems20 and 23 improve, extend, and develop [20, Theorem 5.7], [31, Theorem 3.1], [7, Theorem 3.2], and [14, Theorem 3.1] in the following aspects. (i)
Compared with the relaxed extragradient iterative algorithm in [7, Theorem 3.2], our Mann-type extragradient iterative algorithms with regularization remove the requirement of boundedness for the domain C in which various mappings are defined.(ii)
Because [31, Theorem 3.1] is the supplementation, improvement, and extension of [20, Theorem 5.7] and our Theorem 23 includes [31, Theorem 3.1] as a special case, beyond question our results are very interesting and quite valuable.(iii)
The problem of finding an element ofFix(S)∩Ξ∩Γ in our Theorems 20 and 23 is more general than the corresponding problems in [20, Theorem 5.7] and [31, Theorem 3.1], respectively.(iv)
The hybrid extragradient method for finding an element ofFix(S)∩Ξ∩VI(C,A) in [14, Theorem 3.1] is extended to develop our Mann-type extragradient iterative algorithms (16) and (17) with regularization for finding an element of Fix(S)∩Ξ∩Γ.(v)
The proof of our results are very different from that of [14, Theorem 3.1] because our argument technique depends on the Opial condition, the restriction on the regularization parameter sequence {αn}, and the properties of the averaged mappings PC(I-λn∇fαn) to a great extent.(vi)
Because our iterative algorithms (16) and (17) 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 [20, Theorem 5.7] and [31, Theorem 3.1], respectively.



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*Source: 378750-2013-03-19.xml*