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

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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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, and let PC 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
(1.1)〈Ax-Ay,x-y〉≥0,∀x,y∈C.
A mapping A:C→H is called L-Lipschitz continuous if there exists a constant L>0, such that
(1.2)∥Ax-Ay∥≤L∥x-y∥,∀x,y∈C.
For a given mapping A:C→H, we consider the following variational inequality (VI) of finding x*∈C, such that
(1.3)〈Ax*,x-x*〉≥0,∀x∈C.
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 mappingA:C→H is called α-inverse strongly monotone if there exists a constant α>0, such that
(1.4)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C;
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
(1.5)∥Sx-Sy∥2≤∥x-y∥2+k∥(I-S)x-(I-S)y∥2,∀x,y∈C;
see [10]. In particular, if k=0, then S is called a nonexpansive mapping; see [11].A set-valued mappingM 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
(1.6)(x,f),(y,g)∈G(M)⇒〈x-y,f-g〉≥0.
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.LetΦ 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
(1.7)0∈Φ(x*)+Mx*.
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 ofFix(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:
(1.8)xn+1=αnxn+(1-αn)SPC(xn-λnAxn),∀n≥0,
where x0∈C chosen arbitrarily, {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). 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]).
LetC 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 {xn}, {yn} and {zn} be the sequences generated by
(1.9)yn=PC(xn-λnAxn),zn=αnxn+(1-αn)SPC(xn-λnAyn),Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where x0∈C is  chosen arbitrarily, {λn}⊂[a,b] for some a,b∈(0,1/k), and {αn}⊂[0,c] for some c∈[0,1). Then the sequences {xn}, {yn}, and {zn} converge strongly to PFix(S)∩VI(C,A)x0.It is easy to see that the class ofα-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.LetB1,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
(1.10)〈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 (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, ifB1=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
(1.11)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈y*-x*,x-y*〉≥0,∀x∈C.
Thus we must have x*=y*. As a matter of fact, if x*≠y*, then by setting x=x* we have
(1.12)0>-∥x*-y*∥2=〈y*-x*,x*-y*〉≥0,
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 givenx-,y-∈C,(x-,y-) is a solution of problem (1.10) if and only if x- is a fixed point of the mapping G:C→C defined by
(1.13)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 Lemma1.1, 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]).
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
(1.14)zn=PC(xn-λnAxn),yn=αnQxn+(1-αn)PC[PC(zn-μ2B2zn)-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 (1.14) converges strongly to x-=PFix(S)∩Ξ∩VI(C,A)Qx-, and (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).On the other hand, letA: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
(1.15)yn=PC(xn-λnAxn),tn=PC(xn-λnAyn),t^n=JM,μn(tn-μnΦ(tn)),zn=(1-αn-α^n)xn+αnt^n+α^nSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where JM,μn=(I+μnM)-1,x0∈C chosen arbitrarily, {λn}⊂(0,1/L),{μn}⊂(0,2α], and {αn},{α^n}⊂(0,1] such that αn+α^n≤1. It was proven in [36] that under very mild conditions three sequences {xn}, {yn}, and {zn} generated by (1.15) converge strongly to the same point PFix(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 setΞ 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 thatFix(S)∩Ω∩Ξ≠∅. Let μi∈(0,2βi) for i=1,2,{μn}⊂(0,2α], and {σn},{βn},{γn},{δn}⊂[0,1] such that βn+γn+δn=1, for all n≥0. For given x0∈C arbitrarily, let {xn},{yn}, and {zn} be the sequences generated by the hybrid extragradient iterative scheme
(1.16)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn)),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where JM,μn=(I+μnM)-1, for all n≥0.Under very appropriate assumptions, it is proven that all the sequences{xn},{yn}, and {zn} converge strongly to the same point x-=PFix(S)∩Ω∩Ξx0. Furthermore, (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).LetT: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 σn=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 thatFix(S)∩Fix(Γ)∩Fix(T)≠∅. Let μ1∈(0,1-k), {μn}⊂(0,1-κ], and {βn},{γn},{δn}⊂[0,1] such that βn+γn+δn=1, for all n≥0. For given x0∈C arbitrarily, let {xn},{yn}, and {zn} be the sequences generated by the hybrid extragradient iterative scheme
(1.17)yn=xn-μ1(xn-Txn),tn=yn-μ1(yn-Tyn),t^n=tn-μn(tn-Γtn),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0.
Under quite mild conditions, it is shown that all the sequences{xn}, {yn}, and {zn} converge strongly to the same point PFix(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<liminfn→∞βn≤limsupn→∞βn<1 and limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=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

LetH 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 {xn} converges strongly to x and ⇀ to indicate that the sequence {xn} converges weakly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,
(2.1)ωw(xn):={x:xni⇀xforsomesubsequence{xni}of{xn}}.For every pointx∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.2)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. We know that PC is a firmly nonexpansive mapping of H onto C; that is, there holds the following relation
(2.3)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.
Consequently, PC is nonexpansive and monotone. It is also known that PC is characterized by the following properties: PCx∈C and
(2.4)〈x-PCx,PCx-y〉≥0,(2.5)∥x-y∥2≥∥x-PCx∥2+∥y-PCx∥2,
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
(2.6)x∈VI(C,A)⟺x=PC(x-λAx),∀λ>0.It is also known that the norm of every Hilbert spaceH satisfies the weak lower semicontinuity [4]. That is, for any sequence {xn} with xn⇀x, the inequality
(2.7)liminfn→∞∥xn∥≥∥x∥
holds.Recall that a set-valued mappingM:D(M)⊂H→2H 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.LetA:C→H be a monotone, L-Lipschitz-continuous mapping, and let NCv be the normal cone to C at v∈C, that is,
(2.8)NCv={Av+NCv,ifv∈C,∅,ifv∉C.
Then, T is maximal monotone and 0∈Tv if and only if v∈VI(C,A); see [38].Assume thatM:D(M)⊂H→2H is a maximal monotone mapping. Then, for λ>0, associated with M, the resolvent operator JM,λ can be defined as
(2.9)JM,λx=(I+λM)-1x,∀x∈H.
In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator JM,λ:H→H.Lemma 2.1.
J
M
,
λ is single valued and firmly nonexpansive, that is,
(2.10)〈JM,λx-JM,λy,x-y〉≥∥JM,λx-JM,λy∥2,∀x,y∈H.
Consequently, JM,λ is nonexpansive and monotone.Lemma 2.2 (see [39]).
There holds the relation:(2.11)∥λx+μy+νz∥2=λ∥x∥2+μ∥y∥2+ν∥z∥2-λμ∥x-y∥2-μν∥y-z∥2-λν∥x-z∥2,
for all x,y,z∈H and λ,μ,ν∈[0,1] with λ+μ+ν=1.Lemma 2.3 (see [36]).
LetM 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
(2.12)x*=JM,λ(x*-λΦ(x*)).Lemma 2.4 (see [13]).
LetM 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 xλ for λ>0.Lemma 2.5 (see [36]).
LetM 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 spaceH, S:C→C is k-strictly pseudo-contractive if and only if there holds the following inequality:
(2.13)〈Sx-Sy,x-y〉≤∥x-y∥2-1-k2∥(I-S)x-(I-S)y∥2,∀x,y∈C.
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]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let S:C→C be a mapping.(i)
IfS is a k-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
(2.14)∥Sx-Sy∥≤1+k1-k∥x-y∥,∀x,y∈C.(ii)
IfS is a k-strict pseudo-contractive 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-quasistrict pseudo-contraction, then the fixed point set Fix(S) of S is closed and convex, so that the projection PFix(S) is well defined.Lemma 2.7 (see [34]).
LetC 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
(2.15)∥γ(x-y)+δ(Sx-Sy)∥≤(γ+δ)∥x-y∥,∀x,y∈C.The following lemma is well known to us.Lemma 2.8 (see [11]).
Every Hilbert spaceH has the Kadec-Klee property; that is, for given x∈H and {xn}⊂H, we have
(2.16)xn⇀x∥xn∥→∥x∥}⇒xn→x.


## 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.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Bi:C→H be βi-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 {xn},{yn}, and {zn} be the sequences generated by
(3.1)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn)),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μi∈(0,2βi) for i=1,2, {μn}⊂[ϵ,2α] for some ϵ∈(0,2α], and {σn},{βn},{γn},{δn}⊂[0,1] such that {σn}⊂[0,c] for some c∈[0,1), {δn}⊂[d,1] for some d∈(0,1], βn+γn+δn=1 and (γn+δn)k≤γn, for all n≥0. Then the sequences {xn}, {yn}, and {zn} converge strongly to the same point x-=PFix(S)∩Ω∩Ξx0 if and only if ∥St^n-t^n∥→0. Furthermore, (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).Proof.
It is obvious thatCn is closed and Qn is closed and convex for every n=0,1,2,…. As
(3.2)Cn={z∈C:∥zn-xn∥2+2〈zn-xn,xn-z〉≤0},
we also know that Cn is convex for every n=0,1,2,…. As
(3.3)Qn={z∈C:〈xn-z,x-xn〉≥0},
we have 〈xn-z,x-xn〉≥0, for all z∈Qn, and hence xn=PQnx0 by (2.4).
First of all, assume that the sequences{xn},{yn}, and {zn} converge strongly to the same point x-=PFix(S)∩Ω∩Ξx0. Then it is clear that ∥xn-yn∥→0 and ∥xn-zn∥→0. Observe that from the nonexpansiveness of the mappings PC(I-μ1B1) and PC(I-μ2B2) (due to μi∈(0,2βi) for i=1,2), we have
(3.4)∥yn-tn∥=∥PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]21-PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]∥=∥PC(I-μ1B1)PC(I-μ2B2)xn-PC(I-μ1B1)PC(I-μ2B2)yn∥≤∥PC(I-μ2B2)xn-PC(I-μ2B2)yn∥≤∥xn-yn∥.
Hence, we conclude that ∥yn-tn∥→0 and tn→x-. Since x-∈Fix(S)∩Ω∩Ξ, we obtain that Sx-=x- and x-=JM,μn(x--μnΦ(x-)). Thus, from the nonexpansiveness of the mapping JM,μn(I-μnΦ), we have
(3.5)∥t^n-tn∥=∥σntn+(1-σn)JM,μn(tn-μnΦ(tn))-tn∥=(1-σn)∥JM,μn(tn-μnΦ(tn))-tn∥≤∥JM,μn(tn-μnΦ(tn))-x-∥+∥x--tn∥=∥JM,μn(I-μnΦ)tn-JM,μn(I-μnΦ)x-∥+∥tn-x-∥≤∥tn-x-∥+∥tn-x-∥=2∥tn-x-∥.
So, we deduce that ∥t^n-tn∥→0 and t^n→x-. Note that
(3.6)∥St^n-t^n∥≤∥St^n-x-∥+∥x--t^n∥=∥St^n-Sx-∥+∥x--t^n∥≤(1+k1-k+1)∥t^n-x-∥=21-k∥t^n-x-∥.
This implies that ∥St^n-t^n∥→0 as n→∞.
For the remainder of the proof, we divide it into several steps.
Step1. We claim that Fix(S)∩Ω∩Ξ⊂Cn∩Qn for every n=0,1,2,….
Indeed, take a fixedp∈Fix(S)∩Ω∩Ξ arbitrarily. Then Sp=p,JM,μn(p-μnΦ(p))=p, for all n≥0, and
(3.7)p=PC[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)].
For simplicity, we write q=PC(p-μ2B2p),x~n=PC(xn-μ2B2xn), and y~n=PC(yn-μ2B2yn),
(3.8)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]=PC(x~n-μ1B1x~n),tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]=PC(y~n-μ1B1y~n),
for each n≥0. Since Bi:C→H is βi-inverse strongly monotone, and 0<μi<2βi for i=1,2, we know that for all n≥0,
(3.9)∥yn-p∥2=∥PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]-p∥2=∥PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]2-PC[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)]∥2≤∥[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]2-[PC(p-μ2B2p)-μ1B1PC(p-μ2B2p)]∥2=∥[PC(xn-μ2B2xn)-PC(p-μ2B2p)]-μ1[B1PC(xn-μ2B2xn)-B1PC(p-μ2B2p)]∥2≤∥PC(xn-μ2B2xn)-PC(p-μ2B2p)∥2-μ1(2β1-μ1)∥B1PC(xn-μ2B2xn)-B1PC(p-μ2B2p)∥2≤∥(xn-μ2B2xn)-(p-μ2B2p)∥2-μ1(2β1-μ1)∥B1x~n-B1q∥2=∥(xn-p)-μ2(B2xn-B2p)∥2-μ1(2β1-μ1)∥B1x~n-B1q∥2≤∥xn-p∥2-μ2(2β2-μ2)∥B2xn-B2p∥2-μ1(2β1-μ1)∥B1x~n-B1q∥2≤∥xn-p∥2.
Repeating the same argument, we can obtain that for all n≥0,
(3.10)∥tn-p∥2≤∥yn-p∥2-μ2(2β2-μ2)∥B2yn-B2p∥2-μ1(2β1-μ1)∥B1y~n-B1q∥2≤∥yn-p∥2.
Furthermore, by Lemma 2.1 we derive from (3.9) and (3.10)
(3.11)∥t^n-p∥2=∥σn(tn-p)+(1-σn)(JM,μn(tn-μnΦ(tn))-p)∥2≤σn∥tn-p∥2+(1-σn)∥JM,μn(tn-μnΦ(tn))-p∥2=σn∥tn-p∥2+(1-σn)∥JM,μn(tn-μnΦ(tn))-JM,μn(p-μnΦ(p))∥2≤σn∥tn-p∥2+(1-σn)∥(tn-μnΦ(tn))-(p-μnΦ(p))∥2≤σn∥tn-p∥2+(1-σn)[∥tn-p∥2+μn(μn-2α)∥Φ(tn)-Φ(p)∥2]≤∥tn-p∥2≤∥yn-p∥2-μ2(2β2-μ2)∥B2yn-B2p∥2-μ1(2β1-μ1)∥B1y~n-B1q∥2≤∥xn-p∥2-μ2(2β2-μ2)∥B2xn-B2p∥2-μ1(2β1-μ1)∥B1x~n-B1q∥2-μ2(2β2-μ2)∥B2yn-B2p∥2-μ1(2β1-μ1)∥B1y~n-B1q∥2=∥xn-p∥2-μ2(2β2-μ2)(∥B2xn-B2p∥2+∥B2yn-B2p∥2)-μ1(2β1-μ1)(∥B1x~n-B1q∥2+∥B1y~n-B1q∥2).
Since (γn+δn)k≤γn, for all n≥0, utilizing Lemmas 2.2 and 2.7, we get from (3.11)
(3.12)∥zn-p∥2=∥βn(xn-p)+γn(t^n-p)+δn(St^n-p)∥2=∥βn(xn-p)+(γn+δn)1γn+δn[γn(t^n-p)+δn(St^n-p)]∥2≤βn∥xn-p∥2+(γn+δn)∥1γn+δn[γn(t^n-p)+δn(St^n-p)]∥2≤βn∥xn-p∥2+(γn+δn)∥t^n-p∥2≤βn∥xn-p∥2+(γn+δn){∥xn-p∥2-μ2(2β2-μ2)(∥B2xn-B2p∥2+∥B2yn-B2p∥2)≤βn∥xn-p∥2+(γn+δn)-μ1(2β1-μ1)(∥B1x~n-B1q∥2+∥B1y~n-B1q∥2)}=∥xn-p∥2-(γn+δn){μ2(2β2-μ2)(∥B2xn-B2p∥2+∥B2yn-B2p∥2)=∥xn-p∥2-(γn+δn)+μ1(2β1-μ1)(∥B1x~n-B1q∥2+∥B1y~n-B1q∥2)}≤∥xn-p∥2,
for every n=0,1,2,…, and hence p∈Cn. So, Fix(S)∩Ω∩Ξ⊂Cn for every n=0,1,2,…. Next, let us show by mathematical induction that {xn} is well defined and Fix(S)∩Ω∩Ξ⊂Cn∩Qn for every n=0,1,2,…. For n=0, we have Q0=C. Hence we obtain Fix(S)∩Ω∩Ξ⊂C0∩Q0. Suppose that xn is given and Fix(S)∩Ω∩Ξ⊂Cn∩Qn for some integer n≥0. Since Fix(S)∩Ω∩Ξ is nonempty, Cn∩Qn is a nonempty closed convex subset of C. So, there exists a unique element xn+1∈Cn∩Qn such that xn+1=PCn∩Qnx0. It is also obvious that there holds 〈xn+1-z,x0-xn+1〉≥0 for z∈Fix(S)∩Ω∩Ξ, and hence Fix(S)∩Ω∩Ξ⊂Qn+1. Therefore, we derive Fix(S)∩Ω∩Ξ∩Cn+1∩Qn+1.
Step2. We claim that
(3.13)limn→∞∥xn+1-xn∥=limn→∞∥xn-zn∥=0.
Indeed, letl0=PFix(S)∩Ω∩Ξx0. From xn+1=PCn∩Qnx0, and l0∈Fix(S)∩Ω∩Ξ⊂Cn∩Qn, we have
(3.14)∥xn+1-x0∥≤∥xn+1-x0∥
for every n=0,1,2,…. Therefore, {xn} is bounded. From (3.9)–(3.12), we also obtain that {x~n}, {yn}, {y~n}, {tn}, {t^n}, and {zn} all are bounded. Since xn+1∈Cn∩Qn⊂Qn and xn=PQnx0, we have
(3.15)∥xn-x0∥≤∥xn+1-x0∥
for every n=0,1,2,…. Therefore, there exists limn→∞∥xn-x0∥. Since xn=PQnx0 and xn+1∈Qn, utilizing (2.5), we have
(3.16)∥xn+1-xn∥2≤∥xn+1-x0∥2-∥xn-x0∥2
for every n=0,1,2,…. This implies that
(3.17)limn→∞∥xn+1-xn∥=0.
Since xn+1∈Cn, we have ∥zn-xn+1∥≤∥xn-xn+1∥, and hence
(3.18)∥xn-zn∥≤∥xn-xn+1∥+∥xn+1-zn∥≤2∥xn+1-xn∥
for every n=0,1,2,…. From ∥xn+1-xn∥→0 it follows that
(3.19)limn→∞∥xn-zn∥=0.
Step3. We claim that
(3.20)limn→∞∥xn-yn∥=limn→∞∥xn-tn∥=limn→∞∥t^n-tn∥=0.
Indeed, for p∈Fix(S)∩Ω∩Ξ, we obtain from (3.12)
(3.21)∥zn-p∥2≤∥xn-p∥2-(γn+δn){μ2(2β2-μ2)(∥B2xn-B2p∥2+∥B2yn-B2p∥2)-(γn+δn)21+μ1(2β1-μ1)(∥B1x~n-B1q∥2+∥B1y~n-B1q∥2)}.
Therefore, we have
(3.22)(γn+δn){μ2(2β2-μ2)(∥B2xn-B2p∥2+∥B2yn-B2p∥2)(γn+δn)2+μ1(2β1-μ1)(∥B1x~n-B1q∥2+∥B1y~n-B1q∥2)}≤∥xn-p∥2-∥zn-p∥2=(∥xn-p∥-∥zn-p∥)(∥xn-p∥+∥zn-p∥)≤∥xn-zn∥(∥xn-p∥+∥zn-p∥).
Since {δn}⊂[d,1] for some d∈(0,1],∥xn-zn∥→0, and the sequences {xn} and {zn} are bounded, we deduce that
(3.23)limn→∞∥B2xn-B2p∥=limn→∞∥B2yn-B2p∥=limn→∞∥B1x~n-B1q∥=limn→∞∥B1y~n-B1q∥=0.
On the other hand, by firm nonexpansiveness of PC, we have
(3.24)∥x~n-q∥2=∥PC(xn-μ2B2xn)-PC(p-μ2B2p)∥2≤〈(xn-μ2B2xn)-(p-μ2B2p),x~n-q〉=12[∥xn-p-μ2(B2xn-B2p)∥2+∥x~n-q∥2-∥(xn-p)-μ2(B2xn-B2p)-(x~n-q)∥2]≤12[∥xn-p∥2+∥x~n-q∥2-∥(xn-x~n)-μ2(B2xn-B2p)-(p-q)∥2]=12[∥xn-p∥2+∥x~n-q∥2-∥xn-x~n-(p-q)∥2+2μ2〈xn-x~n-(p-q),B2xn-B2p〉-μ22∥B2xn-B2p∥2]≤12[∥xn-p∥2+∥x~n-q∥2-∥xn-x~n-(p-q)∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥∥xn-p∥2];
that is,
(3.25)∥x~n-q∥2≤∥xn-p∥2-∥xn-x~n-(p-q)∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥.
Repeating the same argument, we can also obtain
(3.26)∥y~n-q∥2≤∥yn-p∥2-∥yn-y~n-(p-q)∥2+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥.
Moreover, using the argument technique similar to the above one, we derive
(3.27)∥yn-p∥2=∥PC(x~n-μ1B1x~n)-PC(q-μ1B1q)∥2≤〈(x~n-μ1B1x~n)-(q-μ1B1q),yn-p〉=12[∥x~n-q-μ1(B1x~n-B1q)∥2+∥yn-p∥2-∥(x~n-q)-μ1(B1x~n-B1q)-(yn-p)∥2∥x~n-q-μ1(B1x~n-B1q)∥2]≤12[∥x~n-q∥2+∥yn-p∥2-∥(x~n-yn)-μ1(B1x~n-B1q)+(p-q)∥2]=12[∥x~n-q∥2+∥yn-p∥2-∥x~n-yn+(p-q)∥2+2μ1〈x~n-yn+(p-q),B1x~n-B1q〉-μ12∥B1x~n-B1q∥2]≤12[∥x~n-q∥2+∥yn-p∥2-∥x~n-yn+(p-q)∥2+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥∥x~n-q∥2];
that is,
(3.28)∥yn-p∥2≤∥x~n-q∥2-∥x~n-yn+(p-q)∥2+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥.
Repeating the same argument, we can also obtain
(3.29)∥tn-p∥2≤∥y~n-q∥2-∥y~n-tn+(p-q)∥2+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥.
Utilizing (3.11), (3.25)–(3.29), we have
(3.30)∥zn-p∥2=∥βn(xn-p)+γn(t^n-p)+δn(St^n-p)∥2≤βn∥xn-p∥2+(γn+δn)∥t^n-p∥2=βn∥xn-p∥2+(γn+δn)∥tn-p∥2≤βn∥xn-p∥2+(γn+δn)[∥y~n-q∥2-∥y~n-tn+(p-q)∥2+(γn+δn)2+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥∥y~n-q∥2]≤βn∥xn-p∥2+(γn+δn)[∥yn-p∥2-∥yn-y~n-(p-q)∥2+(γn+δn)2+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+(γn+δn)2-∥y~n-tn+(p-q)∥2+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥]≤βn∥xn-p∥2+(γn+δn){∥x~n-q∥2-∥x~n-yn+(p-q)∥2+(γn+δn)2+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥+(γn+δn)2-∥yn-y~n-(p-q)∥2+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+(γn+δn)2-∥y~n-tn+(p-q)∥2+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥}≤βn∥xn-p∥2+(γn+δn){∥xn-p∥2-∥xn-x~n-(p-q)∥2+(γn+δn)211+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+(γn+δn)211-∥x~n-yn+(p-q)∥2+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥+(γn+δn)211-∥yn-y~n-(p-q)∥2+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+(γn+δn)2-∥y~n-tn+(p-q)∥2+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥}≤∥xn-p∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥-(γn+δn)[∥xn-x~n-(p-q)∥2+∥x~n-yn+(p-q)∥2+(γn+δn)2+∥yn-y~n-(p-q)∥2+∥y~n-tn+(p-q)∥2],
which hence implies that
(3.31)(γn+δn)[∥xn-x~n-(p-q)∥2+∥x~n-yn+(p-q)∥221+∥yn-y~n-(p-q)∥2+∥y~n-tn+(p-q)∥2]≤∥xn-p∥2-∥zn-p∥2+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2μ2∥xn-x~n-(p-q)∥∥B2xn-B2p∥+2μ1∥x~n-yn+(p-q)∥∥B1x~n-B1q∥+2μ2∥yn-y~n-(p-q)∥∥B2yn-B2p∥+2μ1∥y~n-tn+(p-q)∥∥B1y~n-B1q∥.
Since {δn}⊂[d,1] for some d∈(0,1], ∥xn-zn∥→0, and {xn}, {yn}, {zn}, {x~n}, {y~n}, and {tn} all are bounded, it follows from (3.23) that
(3.32)limn→∞∥xn-x~n-(p-q)∥=limn→∞∥x~n-yn+(p-q)∥=0,limn→∞∥yn-y~n-(p-q)∥=limn→∞∥y~n-tn+(p-q)∥=0.
Consequently, it immediately follows that
(3.33)limn→∞∥xn-yn∥=0,limn→∞∥yn-tn∥=0.
This shows that
(3.34)limn→∞∥xn-tn∥=0.
Also, note that
(3.35)∥zn-xn∥2=∥γn(t^n-xn)+δn(St^n-xn)∥2=∥(γn+δn)1γn+δn[γn(t^n-xn)+δn(St^n-xn)]∥2=(γn+δn)2[γnγn+δn∥t^n-xn∥2+δnγn+δn∥St^n-xn∥2-γnδn(γn+δn)2∥t^n-St^n∥2]=(γn+δn)[γn∥t^n-xn∥2+δn∥St^n-xn∥2]-γnδn∥t^n-St^n∥2.
Thus we have
(3.36)d2∥St^n-xn∥2]≤(γn+δn)[γn∥t^n-xn∥2+δn∥St^n-xn∥2]=∥zn-xn∥2+γnδn∥t^n-St^n∥2≤∥zn-xn∥2+∥t^n-St^n∥2.
This together with ∥zn-xn∥→0 and ∥t^n-St^n∥→0 implies that
(3.37)limn→∞∥St^n-xn∥=0,limn→∞∥t^n-xn∥=0.
Consequently, from (3.34) we immediately derive
(3.38)limn→∞∥t^n-tn∥=0.
Step4. We claim that ωw(xn)⊂Fix(S)∩Ω∩Ξ.
Indeed, as{xn} is bounded, there is a subsequence {xni} of {xn} such that {xni} converges weakly to some u∈ωw(xn). 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
(3.39)∥xn-G(xn)∥=∥xn-PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]∥=∥xn-yn∥→0(n→∞),
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
(3.40)t^ni=σnitni+(1-σni)JM,μni(tni-μniΦ(tni))
implies
(3.41)1μni(tni-sni-μniΦ(tni))∈Msni,
where sni=tni+(t^ni-tni)/(1-σni), we have
(3.42)〈y-sni,g-Φ(y)-1μni(tni-sni-μniΦ(tni))〉≥0,
which hence yields
(3.43)〈y-sni,g〉≥〈y-sni,Φ(y)+1μni(tni-sni-μniΦ(tni))〉=〈y-sni,Φ(y)-Φ(tni)〉+〈y-sni,1μni(tni-sni)〉≥α∥Φ(y)-Φ(sni)∥2+〈y-sni,Φ(sni)-Φ(tni)〉+〈y-sni,1μni(tni-sni)〉≤〈y-sni,Φ(sni)-Φ(tni)〉+〈y-sni,1μni(tni-sni)〉.
Observe that
(3.44)|〈y-sni,Φ(sni)-Φ(tni)〉+〈y-sni,1μni(tni-sni)〉|≤∥y-sni∥∥Φ(sni)-Φ(tni)∥+∥y-sni∥∥1μni(tni-sni)∥≤1α∥y-sni∥∥sni-tni∥+1ϵ∥y-sni∥∥tni-sni∥=(1α+1ϵ)∥y-sni∥∥tni-sni∥.
From ∥sni-tni∥=(1/(1-σni))∥t^ni-tni∥≤(1/(1-c))∥t^ni-tni∥→0, it follows that
(3.45)limi→∞|〈y-sni,Φ(sni)-Φ(tni)〉+〈y-sni,1μni(tni-sni)〉|=0.
Since ∥xn-tn∥→0,∥t^n-tn∥→0, and xni⇀u, we derive sni=tni+((t^ni-tni)/(1-σni))⇀u, and hence by letting i→∞ we get from (3.43)
(3.46)〈y-u,g〉≥0.
This shows that 0∈Φ(u)+Mu. Thus, u∈Ω. Therefore, u∈Fix(S)∩Ω∩Ξ.
Step5. We claim that
(3.47)limn→∞∥xn-l0∥=limn→∞∥yn-l0∥=limn→∞∥zn-l0∥=0,
where l0=PFix(S)∩Ω∩Ξx0.
Indeed, Sincel0=PFix(S)∩Ω∩Ξx0, and u∈Fix(S)∩Ω∩Ξ, from (3.14) we have
(3.48)∥l0-x0∥≤∥u-x0∥≤liminfi→∞∥xni-x0∥≤limsupi→∞∥xni-x0∥≤∥l0-x0∥.
So, we obtain
(3.49)limi→∞∥xni-x0∥=∥u-x0∥.
From xni-x0⇀u-x0, we have xni-x0→u-x0 (due to the Kadec-Klee property of Hilbert spaces [37]), and hence xni→u. Since xn=PQnx0, and l0∈Fix(S)∩Ω∩Ξ⊂Cn∩Qn⊂Qn, we have
(3.50)-∥l0-xni∥2=〈l0-xni,xni-x0〉+〈l0-xni,x0-l0〉≥〈l0-xni,x0-l0〉.
As i→∞, we obtain -∥l0-xni∥2≥〈l0-u,x0-l0〉≥0 by l0=PFix(S)∩Ω∩Ξx0, and u∈Fix(S)∩Ω∩Ξ. Hence we have u=l0. This implies that xn→l0. It is easy to see that yn→l0 and zn→l0. This completes the proof.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Bi:C→H be βi-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 {xn}, {yn}, and {zn} be the sequences generated by
(3.51)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn)),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μi∈(0,2βi) for i=1,2, {μn}⊂[ϵ,2α] for some ϵ∈(0,2α], and {σn}, {βn}, {γn}, {δn}⊂[0,1] such that {σn}⊂[0,c] for some c∈[0,1), {γn},{δn}⊂[d,1] for some d∈(0,1], and βn+γn+δn=1 for all n≥0. Then the sequences {xn},{yn}, and {zn} converge strongly to the same point x-=PFix(S)∩Ω∩Ξx0. Furthermore, (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).Proof.
SinceS 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 {xn} is bounded and the relation holds
(3.52)∥t^n-p∥≤∥xn-p∥,∀n≥0
(due to (3.11)). Moreover, in Step 2 for the proof of Theorem 3.1, we have proven that
(3.53)limn→∞∥zn-xn∥=0.
Now, utilizing Lemma 2.2, from the nonexpansiveness of S we deduce that
(3.54)∥zn-p∥2=∥βn(xn-p)+γn(t^n-p)+δn(St^n-p)∥2≤βn∥xn-p∥2+γn∥t^n-p∥2+δn∥St^n-p∥2-γnδn∥St^n-t^n∥2≤βn∥xn-p∥2+γn∥t^n-p∥2+δn∥t^n-p∥2-γnδn∥St^n-t^n∥2=βn∥xn-p∥2+(γn+δn)∥t^n-p∥2-γnδn∥St^n-t^n∥2≤βn∥xn-p∥2+(γn+δn)∥xn-p∥2-γnδn∥St^n-t^n∥2=∥xn-p∥2-γnδn∥St^n-t^n∥2.
This together with {γn},{δn}⊂[d,1] implies that
(3.55)d2∥St^n-t^n∥2≤γnδn∥St^n-t^n∥2≤∥xn-p∥2-∥zn-p∥2≤∥xn-zn∥(∥xn-p∥+∥zn-p∥).
So, we immediately derive
(3.56)limn→∞∥St^n-t^n∥=0.
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.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Bi:C→H be βi-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 {xn}, {yn}, and {zn} be the sequences generated by
(3.57)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],zn=βnxn+γnPC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]+δnSPC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μi∈(0,2βi) for i=1,2, {βn},{γn},{δn}⊂[0,1] such that {δn}⊂[d,1] for some d∈(0,1], βn+γn+δn=1, and (γn+δn)k≤γn for all n≥0. Then the sequences {xn}, {yn}, and {zn} converge strongly to the same point x-=PFix(S)∩Ξx0. Furthermore, (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).Proof.
PuttingΦ=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 {σn}⊂[0,c] for some c∈[0,1) and any sequence {μn}⊂[ϵ,2α] for some ϵ∈(0,2α]. Then Φ is α-inverse strongly monotone, and we have
(3.58)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn))=σntn+(1-σn)(I+μnM)-1tn=tn,zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
which is just equivalent to (3.57). In this case, we have
(3.59)zn=βnxn+γnt^n+δnSt^n=βnxn+γntn+δnStn.
Note that in Steps 2 and 3 for the proof of Theorem 3.1, we have proven that
(3.60)limn→∞∥zn-xn∥=0,limn→∞∥tn-xn∥=0,
respectively. Thus, we have
(3.61)∥δn(Stn-xn)∥≤∥zn-xn∥+γn∥tn-xn∥→0.
Consequently, it follows from {δn}⊂[d,1] that ∥Stn-xn∥→0, and hence ∥Stn-tn∥→0. This shows that ∥St^n-t^n∥→0. Utilizing Theorem 3.1, we obtain the desired result.Remark 3.4.
Our Theorems3.1 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<liminfn→∞βn≤limsupn→∞βn<1 and limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.(ii)
The problem of finding an element ofFix(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 ofFix(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 JM,λ to a great extent.(v)
Because our iterative algorithms involve two inverse strongly monotone mappingsB1 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 Theorem3.1, we prove some strong convergence theorems in a real Hilbert space.Theorem 4.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Bi:C→H be βi-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 {xn}, {yn}, and {zn} be the sequences generated by
(4.1)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],zn=βnxn+(1-βn)[σntn+(1-σn)JM,μn(tn-μnΦ(tn))],Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μi∈(0,2βi) for i=1,2, {μn}⊂[ϵ,2α] for some ϵ∈(0,2α], and {σn},{βn}⊂[0,1] such that {σn}⊂[0,c] for some c∈[0,1), and {βn}⊂[0,d] for some d∈[0,1). Then the sequences {xn},{yn}, and {zn} converge strongly to the same point x-=PΩ∩Ξx0. Furthermore, (x-,y-) is a solution of the GSVI (1.10), where y-=PC(x--μ2B2x-).Proof.
In Corollary3.2, putting S=I, we have
(4.2)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)],tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)],t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn)),zn=βnxn+γnt^n+δnSt^n=βnxn+(1-βn)t^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
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]).
LetC 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 {xn} and {zn} be the sequences generated by
(4.3)zn=(1-δn)xn+δnSxn,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where {δn}⊂[d,1] for some d∈(0,1]. Then the sequences {xn} and {zn} converge strongly to PFix(S)x0.Proof.
PuttingB1=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 μi∈(0,2βi) for i=1,2 and any sequence {μn}⊂[ϵ,2α] for some ϵ∈(0,2α]. Then Bi:C→H is βi-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
(4.4)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]=xn,tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]=yn,t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn))=tn,zn=βnxn+γnt^n+δnSt^n=(1-δn)xn+δnSxn,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
which is just equivalent to (4.3). Therefore, by Corollary 3.2 we obtain the desired result.Remark 4.3.
Originally Theorem4.2 is the result of Nakajo and Takahashi [22].Theorem 4.4.
LetH 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→2H 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 {xn} and {zn} be the sequences generated by
(4.5)tn=xn-μ[Axn+A(xn-μAxn)],zn=βnxn+γnJM,μn(tn-μnΦ(tn))+δnSJM,μn(tn-μnΦ(tn)),Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μ∈(0,2λ), {μn}⊂[ϵ,2α] for some ϵ∈(0,2α], and {βn},{γn},{δn}⊂[0,1] such that {γn},{δn}⊂[d,1] for some d∈(0,1], and βn+γn+δn=1 for all n≥0. Then the sequences {xn} and {zn} converge strongly to PFix(S)∩Ω∩A-10x0.Proof.
PuttingC=H, B1=A, B2=0, μ1=μ, and σn=0, for all n≥0 in Corollary 3.2, we know that PC=PH=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
(4.6)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]=xn-μAxn,tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]=xn-μAxn-μA(xn-μAxn),t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn))=JM,μn(tn-μnΦ(tn)),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0.
Therefore, by Corollary 3.2 we obtain the desired result.LetB:H→2H be a maximal monotone mapping. Then, for any x∈H and r>0, consider JB,rx=(I+rB)-1x. It is known that such a JB,r is the resolvent of B.Theorem 4.5.
LetH 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→2H be two maximal monotone mappings such that A-10∩B-10∩Ω≠∅. Let JB,r be the resolvent of B for each r>0. For given x0∈H arbitrarily, let {xn} and {zn} be the sequences generated by
(4.7)tn=xn-μ[Axn+A(xn-μAxn)],zn=βnxn+γnJM,μn(tn-μnΦ(tn))+δnJB,rJM,μn(tn-μnΦ(tn)),Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μ∈(0,2λ), {μn}⊂[ϵ,2α] for some ϵ∈(0,2α], and {βn},{γn},{δn}⊂[0,1] such that {γn},{δn}⊂[d,1] for some d∈(0,1], and βn+γn+δn=1 for all n≥0. Then the sequences {xn} and {zn} converge strongly to PA-10∩B-10∩Ωx0.Proof.
PuttingS=JB,r in Theorem 4.4, we know that Fix(S)=Fix(JB,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 mappingT: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
(4.8)〈Tx-Ty,x-y〉≤∥x-y∥2-1-k2∥(I-T)x-(I-T)y∥2,
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.
LetC 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 {xn}, {yn}, and {zn} be the sequences generated by
(4.9)yn=xn-μ1(xn-Txn),tn=yn-μ1(yn-Tyn),t^n=tn-μn(tn-Γtn),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where μ1∈(0,1-k), {μn}⊂[ϵ,1-κ] for some ϵ∈(0,1-κ], and {βn},{γn},{δn}⊂[0,1] such that {γn},{δn}⊂[d,1] for some d∈(0,1], and βn+γn+δn=1 for all n≥0. Then the sequences {xn},{yn}, and {zn} converge strongly to PFix(T)∩Fix(S)∩Fix(Γ)x0.Proof.
PuttingB1=I-T, B2=0, Φ=I-Γ, M=0, and σn=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)xn+μ1Txn∈C. Also, noticing {μn}⊂[ϵ,1-κ]⊂(0,1-κ], we know that {μn}⊂(0,1], and hence (1-μn)tn+μnΓtn∈C. This implies that
(4.10)yn=PC[PC(xn-μ2B2xn)-μ1B1PC(xn-μ2B2xn)]=PC((1-μ1)xn+μ1Txn)=xn-μ1(xn-Txn),tn=PC[PC(yn-μ2B2yn)-μ1B1PC(yn-μ2B2yn)]=PC((1-μ1)yn+μ1Tyn)=yn-μ1(yn-Tyn),t^n=σntn+(1-σn)JM,μn(tn-μnΦ(tn))=tn-μn(tn-Γtn),zn=βnxn+γnt^n+δnSt^n,Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0.
Now let us show Fix(T)=VI(C,B1). In fact, we have, for λ>0,
(4.11)u∈VI(C,B1)⟺〈B1u,y-u〉≥0,∀y∈C⟺〈u-λB1u-u,u-y〉≥0,∀y∈C⟺u=PC(u-λB1u)⟺u=PC(u-λu+λTu)⟺〈u-λu+λTu-u,u-y〉≥0,∀y∈C⟺〈u-Tu,u-y〉≤0,∀y∈C⟺u=Tu⟺u∈Fix(T).
Next let us show Ω=Fix(Γ). In fact, noticing that M=0 and Φ=I-Γ, we have
(4.12)u∈Ω⟺0∈Φ(u)+Mu⟺0=Φ(u)=u-Γu⟺u∈Fix(Γ).
Consequently,
(4.13)Fix(S)∩Ω∩Ξ=Fix(S)∩Fix(Γ)∩VI(C,B1)=Fix(S)∩Fix(Γ)∩Fix(T).
Therefore, by Theorem 3.1 we obtain the desired result.



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*Source: 924309-2012-12-31.xml*