# Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

Authors: Zhao-Rong Kong; Lu-Chuan Ceng; Qamrul Hasan Ansari; Chin-Tzong Pang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.


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## Body

## 1. Introduction and Formulations

Throughout the paper, we will adopt the following terminology and notations.ℋ is a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. The strong (resp., weak) convergence of the sequence {xn} to x will be denoted by xn→x (resp., xn⇀x). We shall use ωw(xn) to denote the weak ω-limit set of the sequence {xn}; namely,
(1)ωw(xn):={x:xnk⇀xfor  some  subsequence{xnk}of{xn}}.Throughout the paper, unless otherwise specified, we assume thatC is a nonempty, closed, and convex subset of a Hilbert space ℋ and A:C→ℋ is a nonlinear mapping. The variational inequality problem (VIP) on C is defined as follows:
(2)findx*∈Csuch  that〈Ax*,x-x*〉≥0,∀x∈C.We denote byΓ the set of solutions of VIP. In particular, if C is the set of fixed points of a nonexpansive mapping T, denoted by Fix(T), then (VIP) is called a hierarchical variational inequality problem (HVIP), also known as a hierarchical fixed point problem (HFPP). If we replace the mapping A by I-S, where I is the identity mapping and S is a nonexpansive mapping (not necessarily with fixed points), then the VIP becomes as follows:
(3)findx*∈Fix(T)such  that〈(I-S)x*,x-x*〉≥0,∀x∈Fix(T).This problem, first introduced and studied in [1, 2], is called a hierarchical variational inequality problem, also known as a hierarchical fixed point problem. Observe that if S has fixed points, then they are solutions of VIP (3). It is worth mentioning that many practical problems can be written in the form of a hierarchical variational inequality problem; see, for example, [1–18] and the references therein.IfS is a ρ-contraction with coefficient ρ∈[0,1) (i.e., ∥Sx-Sy∥≤ρ∥x-y∥ for some ρ∈[0,1)), then the set of solutions of VIP (3) is a singleton, and it is well known as a viscosity problem, which was first introduced by Moudafi [19] and then developed by Xu [20]. It is not hard to verify that solving VIP (3) is equivalent to finding a fixed point of the nonexpansive mapping PFix(T)S, where PFix(T) is the metric projection on the closed and convex set Fix(T).LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone, where κ>0, η>0 are constants, that is, for all x,y∈C(4)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2.A mappingT:C→C is called ζ-strictly pseudocontractive if there exists a constant ζ∈[0,1) such that
(5)∥Tx-Ty∥2≤∥x-y∥2+ζ∥(I-T)x-(I-T)y∥2,∀x,y∈C;see [21] for more details. We denote by Fix(T) the fixed point set of T; that is, Fix(T)={x∈C:Tx=x}.We introduce and consider the following triple hierarchical variational inequality problem (THVIP).Problem I.
LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of ℋ, where κ and η are positive constants. Let V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and, for i=1,2, Ti:C→C be ζi-strictly pseudocontractive mapping with Fix(T1)∩Fix(T2)≠∅. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Then the objective is to find x*∈Ξ such that
(6)〈(μF-γV)x*,x-x*〉≥0,∀x∈Ξ,
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding z*∈Fix(T) such that
(7)〈(μF-γS)z*,z-z*〉≥0,∀z∈Fix(T1)∩Fix(T2).
In particular, wheneverT1=T a nonexpansive mapping, and T2=I an identity mapping, Problem I reduces to the THVIP considered by Ceng et al. [22]. By combining the regularization method, the hybrid steepest-descent method, and the projection method, they proposed an iterative algorithm that generates a sequence via the explicit scheme and studied the convergence analysis of the sequences generated by the proposed method.
We consider and study the following triple hierarchical variational inequality problem.Problem II.
LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of ℋ, where κ and η are positive constants. Let A:C→ℋ be a monotone and L-Lipschitzian mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping with Fix(T)∩Γ≠∅. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Then the objective is to find x*∈Ξ such that
(8)〈(μF-γV)x*,x-x*〉≥0,∀x∈Ξ,
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding z*∈Fix(T)∩Γ such that
(9)〈(μF-γS)z*,z-z*〉≥0,∀z∈Fix(T)∩Γ.We remark that Problem II is a generalization of Problem I. Indeed, in Problem II we putT=T1 and A=I-T2, where Ti:C→C is a ζi-strictly pseudocontractive mapping for i=1,2. Then from the definition of strictly pseudocontractive mapping, it follows that
(10)〈T2x-T2y,x-y〉≤∥x-y∥2-1-ζ22∥(I-T2)x-(I-T2)y∥2,∀x,y∈C.It is clear that the mappingA=I-T2 is (1-ζ2)/2-inverse strongly monotone. Taking L=2/(1-ζ2), we know that A:C→ℋ is a monotone and L-Lipschitzian mapping. In this case, Γ=Fix(T2). Therefore, Problem II reduces to Problem I.Motivated and inspired by Korpelevich’s extragradient method [23] and the iterative method proposed in [22], we propose the following multistep hybrid extragradient method for solving Problem II.Algorithm I.
LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of ℋ, A:C→ℋ be a monotone and L-Lipschitzian mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping. Let {αn}⊂[0,∞), {νn}⊂(0,1/L), {γn}⊂[0,1) and {βn}, {δn}, {σn}, {λn}⊂(0,1), 0<μ<2η/κ2, and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). The sequence {xn} is generated by the following iterative scheme:
(11)x0=x∈Cchosen  arbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn],∀n≥0,
where An=αnI+A for all n≥0. In particular, if V≡0, then (11) reduces to the following iterative scheme:
(12)x0=x∈Cchosen  arbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[λn(1-δn)γSxn+(I-λnμF)zn],∀n≥0.
Further, ifS=V, then (11) reduces to the following iterative scheme:
(13)x0=x∈Cchosen  arbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[λnγVxn+(I-λnμF)zn],∀n≥0;
moreover, if S=V≡0, then (12) reduces to the following iterative scheme:
(14)x0=x∈Cchosen  arbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[(I-λnμF)zn],∀n≥0.
We prove that under appropriate conditions the sequence{xn} generated by Algorithm I converges strongly to a unique solution of Problem II. Our result improves and extends Theorem  4.1 in [22] in the following aspects.(a)
Problem II generalizes Problem I from the fixed point setFix(T) of a nonexpansive mapping T to the intersection Fix(T)∩Γ of the fixed point set of a strictly pseudocontractive mapping T and the solution set Γ of VIP (2).(b)
The Korpelevich extragradient algorithm is extended to develop the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving Problem II by virtue of the iterative schemes in Theorem  4.1 in [22].(c)
The strong convergence of the sequence{xn} generated by Algorithm I holds under the lack of the same restrictions as those in Theorem  4.1 in [22].(d)
The boundedness requirement of the sequence{xn} in Theorem  4.1 in [22] is replaced by the boundedness requirement of the sequence {Sxn}.We also consider and study the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving the following system of hierarchical variational inequalities (SHVI).Problem III.
LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of ℋ, where κ and η are positive constants. Let A:C→ℋ be a monotone and L-Lipschitzian mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping with Fix(T)∩Γ≠∅. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Then the objective is to find x*∈Fix(T)∩Γ such that
(15)〈(μF-γV)x*,x-x*〉≥0,∀x∈Fix(T)∩Γ,〈(μF-γS)x*,y-x*〉≥0,∀y∈Fix(T)∩Γ.In particular, ifT=T1 and A=I-T2 where Ti:C→C is ζi-strictly pseudocontractive for i=1,2, Problem III reduces to the following Problem IV.Problem IV.
LetF:C→ℋ be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of ℋ, where κ and η are positive constants. Let V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and, for i=1,2, Ti:C→C be ζi-strictly pseudocontractive mapping with Fix(T1)∩Fix(T2)≠∅. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Then the objective is to find x*∈Fix(T1)∩Fix(T2) such that
(16)〈(μF-γV)x*,x-x*〉≥0,∀x∈Fix(T1)∩Fix(T2),〈(μF-γS)x*,y-x*〉≥0,∀y∈Fix(T1)∩Fix(T2).
We prove that under very mild conditions the sequence{xn} generated by Algorithm I converges strongly to a unique solution of Problem III.


## 2. Preliminaries

LetC be a nonempty, closed, and convex subset of ℋ and V:C→ℋ be a (possibly nonself) ρ-contraction mapping with coefficient ρ∈[0,1); that is, there exists a constant ρ∈[0,1) such that ∥Vx-Vy∥≤ρ∥x-y∥, for allx,y∈C. Now we present some known results and definitions which will be used in the sequel.The metric (or nearest point) projection fromℋ onto C is the mapping PC:ℋ→C which assigns to each point x∈ℋ the unique point PCx∈C satisfying the property
(17)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).The following properties of projections are useful and pertinent to our purpose.Proposition 1 (see [21]).
Given anyx∈ℋ and z∈C. One has(i)
z=PCx⇔〈x-z,y-z〉≤0,for ally∈C,(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2, for ally∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2, for allx,y∈ℋ, which hence implies thatPCis nonexpansive and monotone.Definition 2.
A mappingT:ℋ→ℋ is said to be(a)
nonexpansive if(18)∥Tx-Ty∥≤∥x-y∥,∀x,y∈ℋ;(b)
firmly nonexpansive if2T-I is nonexpansive, or, equivalently,
(19)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈ℋ;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(20)T=12(I+S),
where S:ℋ→ℋ is nonexpansive; projections are firmly nonexpansive.Definition 3.
LetT be a nonlinear operator whose domain is D(T)⊆ℋ and whose range is R(T)⊆ℋ.(a)
T is said to be monotone if
(21)〈x-y,Tx-Ty〉≥0,∀x,y∈D(T).(b)
Given a numberβ>0, T is said to be β-strongly monotone if
(22)〈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
(23)〈x-y,Tx-Ty〉≥ν∥Tx-Ty∥2,∀x,y∈D(T).
It can be easily seen that ifT is nonexpansive, then I-T is monotone. It is also easy to see that a projection PC 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 [24, 25].Definition 4.
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,
(24)T≡(1-α)I+αS,
where α∈(0,1) and S:ℋ→ℋ are 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 5 (see [26]).
LetT:ℋ→ℋ be a given mapping.(i)
Tis nonexpansive if and only if the complementI-Tis1/2-ism.(ii)
IfTisν-ism, then,forγ>0,γTisν/γ-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>1/2. Indeed,forα∈(0,1),Tisα-averaged if and only ifI-Tis1/2α-ism.Proposition 6 (see [26, 27]).
LetS,T,V:ℋ→ℋ be given operators.(i)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis averaged andVis nonexpansive, thenTis averaged.(ii)
Tis firmly nonexpansive if and only if the complementI-Tis firmly nonexpansive.(iii)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis firmly nonexpansive andVis nonexpansive, thenTis averaged.(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}i=1Nis averaged, then so is the compositeT1∘⋯∘TN. In particular, ifT1isα1-averaged andT2isα2-averaged, whereα1,α2∈(0,1), then the compositeT1∘T2isα-averaged, whereα=α1+α2-α1α2.On the other hand, it is clear that, in a real Hilbert spaceℋ, T:C→C is ζ-strictly pseudocontractive if and only if there holds the following inequality:
(25)〈Tx-Ty,x-y〉≤∥x-y∥2-1-ζ2∥(I-T)x-(I-T)y∥2,∀x,y∈C.This immediately implies that ifT is a ζ-strictly pseudocontractive mapping, then I-T is (1-ζ)/2-inverse strongly monotone; for further detail, we refer to [21] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. The so-called demiclosedness principle for strict pseudocontractive mappings in the following lemma will often be used.Lemma 7 (see [21, Proposition 2.1]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ and T:C→C be a mapping.(i)
IfT is a ζ-strictly pseudocontractive mapping, then T satisfies the Lipschitz condition:
(26)∥Tx-Ty∥≤1+ζ1-ζ∥x-y∥,∀x,y∈C.(ii)
IfT is a ζ-strictly pseudocontractive mapping, then the mapping I-T is semiclosed at 0; that is, if {xn} is a sequence in C such that xn⇀x~ and (I-T)xn→0, then (I-T)x~=0.(iii)
IfT is a ζ-(quasi-)strict pseudocontraction, then the fixed point set Fix(T) of T is closed and convex so that the projection PFix(T) is well defined.The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.Lemma 8 (see [28]).
Let{an} be a sequence of nonnegative real numbers satisfying the property:
(27)an+1≤(1-sn)an+snbn+rn,n≥0,
where {sn}⊂(0,1] and {bn} are such that(i)
∑n=0∞‍sn=∞;(ii)
eitherlimsupn→∞bn≤0 or ∑n=0∞‍|snbn|<∞;(iii)
∑n=0∞‍rn<∞ where rn≥0, for alln≥0.
Then,limn→∞an=0.The following lemma is not hard to prove.Lemma 9 (see [20]).
LetV:C→ℋ be a ρ-contraction with ρ∈[0,1) and T:C→C be a nonexpansive mapping. Then(i)
I-V is (1-ρ)-strongly monotone:
(28)〈(I-V)x-(I-V)y,x-y〉≥(1-ρ)∥x-y∥2,∀x,y∈C;(ii)
I-T is monotone:
(29)〈(I-T)x-(I-T)y,x-y〉≥0,∀x,y∈C.The following lemma plays an important role in proving strong convergence of the sequences generated by our algorithm.Lemma 10 (see [29]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ. Let T:C→C be a ζ-strictly pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ+δ)ζ≤γ. Then
(30)∥γ(x-y)+δ(Tx-Ty)∥≤(γ+δ)∥x-y∥,∀x,y∈C.Lemma 11 (see [30, Lemma 3.1]).
Letλ be a number in (0,1] and let μ>0. Let F:C→ℋ be an operator on C such that, for some constants κ,η>0,F is κ-Lipschitzian and η-strongly monotone, associating with a nonexpansive mapping T:C→C, define the mapping Tλ:C→ℋ by
(31)Tλx:=Tx-λμF(Tx),∀x∈C.
ThenTλ is a contraction provided that μ<2η/κ2, that is,
(32)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1]. In particular, if  T is the identity mapping I, then
(33)∥(I-λμF)x-(I-λμF)y∥≤(1-λτ)∥x-y∥,∀x,y∈C.The following lemma appears implicitly in Reineermann [31].Lemma 12.
Letℋ be a Hilbert space. Then
(34)∥λx+(1-λ)y-z∥2=λ∥x-z∥2+(1-λ)∥y-z∥2-λ(1-λ)∥x-y∥2,∀x,y,z∈ℋ,∀λ∈[0,1].The following lemma is not difficult to prove.Lemma 13 (see [32]).
Let{αn} and {βn} be a sequence of nonnegative real numbers and a sequence of real numbers, respectively, such that limsupn→∞αn<∞ and limsupn→∞βn≤0. Then limsupn→∞αnβn≤0.A set-valued mappingT~:H→2H is called monotone if, for all x,y∈H,f∈T~x, and g∈T~y imply 〈x-y,f-g〉≥0. A monotone mapping T~:H→2H is maximal if its graph G(T~) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T~ is maximal if and only if, for (x,f)∈H×H,〈x-y,f-g〉≥0 for all (y,g)∈G(T~) implies f∈T~x. Let A:C→H be a monotone and L-Lipschitzian mapping, and let NCv be the normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0, for allu∈C}. Define
(35)T~v={Av+NCv,ifv∈C,∅,ifv∉C.It is known that in this caseT~ is maximal monotone, and 0∈T~v if and only if v∈Γ; see [33].


## 3. Main Results

We are now in a position to present the convergence analysis of Algorithm I for solving Problem II.Theorem 14.
LetF:C→ℋ be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, respectively, A:C→ℋ be a 1/L-inverse strongly monotone mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Assume that the solution set Ξ of the HVIP (9) is nonempty and that the following conditions hold for the sequences {αn}⊂[0,∞),{νn}⊂(0,1/L),{γn}⊂[0,1) and {βn},{δn},{σn},{λn}⊂(0,1):(C1)
∑n=0∞‍αn<∞ and limn→∞(αn/λn2)=0;(C2)
0<liminfn→∞νn≤limsupn→∞νn<1/L;(C3)
βn+γn+σn=1 and (γn+σn)ζ≤γn  for all n≥0;(C4)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C5)
limn→∞λn=0,limn→∞δn=0 and ∑n=0∞‍δnλn=∞;(C6)
there are constantsk-,θ>0 satisfying ∥x-Tx∥≥k-[d(x,Fix(T)∩Γ)]θ for each x∈C;(C7)
limn→∞(λn1/θ/δn)=0.
One has the following.(a)
If{xn} is the sequence generated by scheme (11) and {Sxn} is bounded, then {xn} converges strongly to the point x*∈Fix(T)∩Γ which is a unique solution of Problem II provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2).(b)
If{xn} is the sequence generated by the scheme (12) and {Sxn} is bounded, then {xn} converges strongly to a unique solution x* of the following VIP provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2):
(14)findx*∈Ξsuchthat〈Fx*,x-x*〉≥0,∀x∈Ξ.Proof.
We treat only case (a); that is, the sequence{xn} is generated by the scheme (11). Obviously, from the condition Ξ≠∅ it follows that Fix(T)∩Γ≠∅. In addition, in terms of conditions (C2) and (C4), without loss of generality, we may assume that {νn}⊂[a,b] for some a,b∈(0,1/L), {βn}⊂[c,d] for some c,d∈(0,1).
First of all, we observe (see, e.g., [34]) that PC(I-ν(αI+A)) and PC(I-νnAn) are nonexpansive for all n≥0.
Next we divide the remainder of the proof into several steps.
Step  1 ({xn} is bounded). Indeed, take a fixed p∈Fix(T)∩Γ arbitrarily. Then, we get Tp=p and PC(I-νA)p=p for ν∈(0,2/L). From (11), it follows that
(37)∥yn-p∥=∥PC(I-νnAn)xn-PC(I-νnA)p∥≤∥PC(I-νnAn)xn-PC(I-νnAn)p∥+∥PC(I-νnAn)p-PC(I-νnA)p∥≤∥xn-p∥+∥(I-νnAn)p-(I-νnA)p∥=∥xn-p∥+νnαn∥p∥.
Puttn=PC(xn-νnAnyn) for each n≥0. Then, by Proposition 1 (ii), we have
(38)∥tn-p∥2≤∥xn-νnAnyn-p∥2-∥xn-νnAnyn-tn∥2=∥xn-p∥2-∥xn-tn∥2+2νn〈Anyn,p-tn〉=∥xn-p∥2-∥xn-tn∥2+2νn(〈Anyn-Anp,p-yn〉+〈Anp,p-yn〉+〈Anyn,yn-tn〉)≤∥xn-p∥2-∥xn-tn∥2+2νn(〈Anp,p-yn〉+〈Anyn,yn-tn〉)=∥xn-p∥2-∥xn-tn∥2+2νn[〈(αnI+A)p,p-yn〉+〈Anyn,yn-tn〉]≤∥xn-p∥2-∥xn-tn∥2+2νn[αn〈p,p-yn〉+〈Anyn,yn-tn〉]=∥xn-p∥2-∥xn-yn∥2-2〈xn-yn,yn-tn〉-∥yn-tn∥2+2νn[αn〈p,p-yn〉+〈Anyn,yn-tn〉]=∥xn-p∥2-∥xn-yn∥2-∥yn-tn∥2+2〈xn-νnAnyn-yn,tn-yn〉+2νnαn〈p,p-yn〉.
Further, by Proposition1 (i), we have
(39)〈xn-νnAnyn-yn,tn-yn〉=〈xn-νnAnxn-yn,tn-yn〉+〈νnAnxn-νnAnyn,tn-yn〉≤〈νnAnxn-νnAnyn,tn-yn〉≤νn∥Anxn-Anyn∥∥tn-yn∥≤νn(αn+L)∥xn-yn∥∥tn-yn∥.
So, we obtain(40)∥tn-p∥2≤∥xn-p∥2-∥xn-yn∥2-∥yn-tn∥2+2〈xn-νnAnyn-yn,tn-yn〉+2νnαn〈p,p-yn〉≤∥xn-p∥2-∥xn-yn∥2-∥yn-tn∥2+2νn(αn+L)∥xn-yn∥∥tn-yn∥+2νnαn∥p∥∥p-yn∥≤∥xn-p∥2-∥xn-yn∥2-∥yn-tn∥2+νn2(αn+L)2∥xn-yn∥2+∥yn-tn∥2+2νnαn∥p∥∥p-yn∥=∥xn-p∥2+2νnαn∥p∥∥p-yn∥+(νn2(αn+L)2-1)∥xn-yn∥2≤∥xn-p∥2+2νnαn∥p∥∥p-yn∥.
Since(γn+σn)ζ≤γn, utilizing Lemmas 10 and 12, from (37) and the last inequality, we conclude that
(41)∥zn-p∥2=∥βnxn+γntn+σnTtn-p∥2=∥βn(xn-p)+(γn+σn)1γn+σn×[γn(tn-p)+σn(Ttn-p)]1γn+σn∥2=βn∥xn-p∥2+(γn+σn)×∥1γn+σn[γn(tn-p)+σn(Ttn-p)]∥2-βn(γn+σn)×∥1γn+σn[γn(tn-xn)+σn(Ttn-xn)]∥2≤βn∥xn-p∥2+(1-βn)∥tn-p∥2-βn1-βn∥zn-xn∥2≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2+2νnαn∥p∥∥p-yn∥+(νn2(αn+L)2-1)∥xn-yn∥2]-βn1-βn∥zn-xn∥2≤∥xn-p∥2+2νnαn∥p∥∥p-yn∥+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2≤∥xn-p∥2+2νnαn∥p∥(∥xn-p∥+νnαn∥p∥)+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2≤∥xn-p∥2+2∥xn-p∥(2νnαn∥p∥)+(2νnαn∥p∥)2+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2=(∥xn-p∥+2νnαn∥p∥)2+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2≤(∥xn-p∥+2νnαn∥p∥)2.
Noticing the boundedness of{Sxn}, we get supn≥0∥γSxn-μFp∥≤M for some M≥0. Moreover, utilizing Lemma 11 we have from (11)
(42)∥xn+1-p∥=∥PC[λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn]-PCp∥≤∥λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn-p∥=∥λnγ(δnVxn+(1-δn)Sxn)-λnμFp+(I-λnμF)zn-(I-λnμF)p∥≤∥λnγ(δnVxn+(1-δn)Sxn)-λnμFp∥+∥(I-λnμF)zn-(I-λnμF)p∥=λn∥δn(γVxn-μFp)+(1-δn)(γSxn-μFp)∥+∥(I-λnμF)zn-(I-λnμF)p∥≤λnδn∥γVxn-μFp∥+(1-δn)∥γSxn-μFp∥+(1-λnτ)∥zn-p∥≤λn[δn(∥γVxn-γVp∥+∥γVp-μFp∥)+(1-δn)M]+(1-λnτ)∥zn-p∥≤λn[δnγρ∥xn-p∥+δn∥γVp-μFp∥+(1-δn)M]+(1-λnτ)[∥xn-p∥+2νnαn∥p∥]≤λn[δnγρ∥xn-p∥+max{M,∥γVp-μFp∥}]+(1-λnτ)[∥xn-p∥+2νnαn∥p∥]≤λnγρ∥xn-p∥+λnmax{M,∥γVp-μFp∥}+(1-λnτ)∥xn-p∥+2νnαn∥p∥=[1-(τ-γρ)λn]∥xn-p∥+λnmax{M,∥γVp-μFp∥}+2νnαn∥p∥.
So, calling(43)M~=max{∥x0-p∥,Mτ-γρ,∥γVp-μFp∥τ-γρ},
we claim that(44)∥xn+1-p∥≤M~+∑j=0n‍2νjαj∥p∥,∀n≥0.
Indeed, whenn=0, it is clear from (42) that (44) is valid, that is,
(45)∥x1-p∥≤M~+∑j=00‍2νjαj∥p∥.
Assume that (44) is valid for n(≥1), that is,
(46)∥xn-p∥≤M~+∑j=0n-1‍2νjαj∥p∥.
Then from (42) and (46) it follows that
(47)∥xn+1-p∥≤[1-(τ-γρ)λn]∥xn-p∥+λnmax{M,∥γVp-μFp∥}+2νnαn∥p∥≤[1-(τ-γρ)λn][M~+∑j=0n-1‍2νjαj∥p∥]+λnmax{M,∥γVp-μFp∥}+2νnαn∥p∥≤[1-(τ-γρ)λn]M~+∑j=0n-1‍2νjαj∥p∥+(τ-γρ)λnmax{Mτ-γρ,∥γVp-μFp∥τ-γρ}+2νnαn∥p∥≤M~+∑j=0n‍2νjαj∥p∥.
This shows that (44) is also valid for n+1. Hence, by induction we derive the claim. Consequently, {xn} is bounded (due to ∑n=0∞‍αn<∞) and so are {yn},{zn},{Axn}, and {Ayn}.
Step  2
(
lim
n
→
∞
∥
x
n
-
y
n
∥
=
lim
n
→
∞
∥
x
n
-
t
n
∥
=
lim
n
→
∞
∥
t
n
-
T
t
n
∥
=
0
). Indeed, from (11) and (41), it follows that
(48)∥xn+1-p∥2≤∥λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn-p∥2=∥λnγ(δnVxn+(1-δn)Sxn)-λnμFp+(I-λnμF)zn-(I-λnμF)p∥2≤{∥λnγ(δnVxn+(1-δn)Sxn)-λnμFp∥+∥(I-λnμF)zn-(I-λnμF)p∥}2≤{λn∥δn(γVxn-μFp)+(1-δn)(γSxn-μFp)∥+(1-λnτ)∥zn-p∥}2≤λn1τ[δn∥γVxn-μFp∥+(1-δn)∥γSxn-μFp∥]2+(1-λnτ)∥zn-p∥2≤λn1τ[∥γVxn-μFp∥+∥γSxn-μFp∥]2+∥zn-p∥2≤λn1τ[∥γVxn-μFp∥+∥γSxn-μFp∥]2+(∥xn-p∥+2νnαn∥p∥)2+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2≤(∥xn-p∥+2νnαn∥p∥)2+λnM1+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2,
where M1=supn≥0{(1/τ)[∥γVxn-μFp∥+∥γSxn-μFp∥]2}. This together with {νn}⊂[a,b]⊂(0,1/L) and {βn}⊂[c,d]⊂(0,1) implies that
(49)(1-d)(1-b2(αn+L)2)∥xn-yn∥2+c1-c∥zn-xn∥2≤(1-βn)(1-νn2(αn+L)2)∥xn-yn∥2+βn1-βn∥zn-xn∥2≤(∥xn-p∥+2νnαn∥p∥)2-∥xn+1-p∥2+λnM1=[(∥xn-p∥+2νnαn∥p∥)-∥xn+1-p∥]×[(∥xn-p∥+2νnαn∥p∥)+∥xn+1-p∥]+λnM1≤[∥xn+1-xn∥+2νnαn∥p∥]×[∥xn-p∥+∥xn+1-p∥+2νnαn∥p∥]+λnM1≤[∥xn+1-xn∥+2bαn∥p∥]×[∥xn-p∥+∥xn+1-p∥+2bαn∥p∥]+λnM1.
Note thatlimn→∞αn=limn→∞λn=0. Hence, taking into account the boundedness of {xn} and limn→∞∥xn+1-xn∥=0, we deduce from (49) that
(50)limn→∞∥xn-yn∥=limn→∞∥zn-xn∥=0.
Furthermore, we obtain(51)∥yn-tn∥=∥PC(xn-νnAnxn)-PC(xn-νnAnyn)∥≤∥(xn-νnAnxn)-(xn-νnAnyn)∥=νn∥Anxn-Anyn∥≤νn(αn+L)∥xn-yn∥,
which together with (50) implies that
(52)limn→∞∥yn-tn∥=0,limn→∞∥xn-tn∥=0.
So, from (11) we get
(53)∥σn(Ttn-xn)∥=∥zn-xn-γn(tn-xn)∥≤∥zn-xn∥+γn∥tn-xn∥⟶0,
which together with liminfn→∞σn>0 implies that
(54)limn→∞∥Ttn-xn∥=0.
Note that(55)∥tn-Ttn∥≤∥tn-yn∥+∥yn-xn∥+∥xn-Ttn∥.
This together with (50)–(54) implies that
(56)limn→∞∥tn-Ttn∥=0.
Step  3
(
ω
w
(
x
n
)
⊂
Fix
(
T
)
∩
Γ
). Indeed, since A is L-Lipschitz continuous, we have
(57)limn→∞∥Ayn-Atn∥=0.
As{xn} is bounded, there is a subsequence {xni} of {xn} that converges weakly to some x^. By the same argument as that in [34], we can obtain that x^∈Fix(T)∩Γ from which it follows that
(58)ωw(xn)⊂Fix(T)∩Γ.
Step  4
(
ω
w
(
x
n
)
⊂
Ξ
). Indeed, we first note that 0<γ≤τ and
(59)μη≥τ⟺μη≥1-1-μ(2η-μκ2)⟺1-μ(2η-μκ2)≥1-μη⟺1-2μη+μ2κ2≥1-2μη+μ2η2⟺κ2≥η2⟺κ≥η.
It is clear that(60)〈(μF-γS)x-(μF-γS)y,x-y〉≥(μη-γ)∥x-y∥2,∀x,y∈C.
Hence, it follows from0<γ≤τ≤μη that μF-γS is monotone. Putting
(61)wn=λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)Tzn,∀n≥0,
and noticing from (11)
(62)xn+1=PCwn-wn+λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn,
we obtain(63)xn-xn+1=wn-PCwn+δnλn(μF-γV)xn+λn(1-δn)(μF-γS)xn+(I-λnμF)xn-(I-λnμF)zn.
Set(64)en=xn-xn+1λn(1-δn),∀n≥0.
It can be easily seen from (63) that
(65)en=wn-PCwnλn(1-δn)+(μF-γS)xn+δn1-δn(μF-γV)xn+(I-λnμF)xn-(I-λnμF)znλn(1-δn).
This yields that, for allw∈Fix(T)∩Γ (noticing xn=PCwn-1),
(66)〈en,xn-w〉=1λn(1-δn)〈wn-PCwn,PCwn-1-w〉+〈(μF-γS)xn,xn-w〉+δn1-δn〈(μF-γV)xn,xn-w〉+1λn(1-δn)〈(I-λnμF)xn-(I-λnμF)zn,xn-w〉=1λn(1-δn)〈wn-PCwn,PCwn-w〉+1λn(1-δn)〈wn-PCwn,PCwn-1-PCwn〉+〈(μF-γS)w,xn-w〉+〈(μF-γS)xn-(μF-γS)w,xn-w〉+δn1-δn〈(μF-γV)xn,xn-w〉+1λn(1-δn)〈(I-λnμF)xn-(I-λnμF)zn,xn-w〉.
In (66), the first term is nonnegative due to Proposition 1, and the fourth term is also nonnegative due to the monotonicity of μF-γS. We, therefore, deduce from (66) that (noticing again xn+1=PCwn)
(67)〈en,xn-w〉≥1λn(1-δn)〈wn-PCwn,PCwn-1-PCwn〉+〈(μF-γS)w,xn-w〉+δn1-δn〈(μF-γV)xn,xn-w〉+1λn(1-δn)〈(I-λnμF)xn-(I-λnμF)zn,xn-w〉=〈wn-PCwn,en〉+〈(μF-γS)w,xn-w〉+δn1-δn〈(μF-γV)xn,xn-w〉+1λn(1-δn)〈(I-λnμF)xn-(I-λnμF)zn,xn-w〉.
Note that(68)∥(I-λnμF)xn-(I-λnμF)zn∥≤(1-λnτ)∥xn-zn∥.
Hence it follows from∥xn-zn∥=o(λn) that
(69)∥(I-λnμF)xn-(I-λnμF)zn∥λn⟶0.
Also, sinceen→0 (due to ∥xn+1-xn∥=o(λn)), δn→0 and {xn} is bounded by Step 1 which implies that {wn} is bounded, we obtain from (67) that
(70)limsupn→∞〈(μF-γS)w,xn-w〉≤0,∀w∈Fix(T)∩Γ.
This suffices to guarantee thatωw(xn)⊂Ξ; namely, every weak limit point of {xn} solves the HVIP (9). As a matter of fact, if xni⇀x~∈ωw(xn) for some subsequence {xni} of {xn}, then we deduce from (70) that
(71)〈(μF-γS)w,x~-w〉≤limsupn→∞〈(μF-γS)w,xn-w〉≤0,∀w∈Fix(T)∩Γ,
that is,(72)〈(μF-γS)w,w-x~〉≥0,∀w∈Fix(T)∩Γ.
In addition, note thatωw(xn)⊂Fix(T)∩Γ by Step 3. Since μF-γS is monotone and Lipschitz continuous and Fix(T)∩Γ is nonempty, closed, and convex, by the Minty lemma [1] the last inequality is equivalent to (9). Thus, we get x~∈Ξ.
Step  5 ({xn} converges strongly to a unique solution x* of Problem II.) Indeed, we now take a subsequence {xni} of {xn} satisfying
(73)limsupn→∞〈(μF-γV)x*,xn-x*〉=limi→∞〈(μF-γV)x*,xni-x*〉.
Without loss of generality, we may further assume thatxni⇀x~; then x~∈Ξ as we just proved. Since x* is a solution of the THVIP (8), we get
(74)limsupn→∞〈(μF-γV)x*,xn-x*〉=〈(μF-γV)x*,x~-x*〉≥0.
From (11) and (41), it follows that (noticing that xn+1=PCwn and 0<γ≤τ)
(75)∥xn+1-x*∥2=〈wn-x*,xn+1-x*〉+〈PCwn-wn,PCwn-x*〉≤〈wn-x*,xn+1-x*〉=〈(I-λnμF)zn-(I-λnμF)x*,xn+1-x*〉+δnλnγ〈Vxn-Vx*,xn+1-x*〉+λn(1-δn)γ〈Sxn-Sx*,xn+1-x*〉+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉≤(1-λnτ)∥zn-x*∥∥xn+1-x*∥+[δnλnγρ+λn(1-δn)γ]×∥xn-x*∥∥xn+1-x*∥+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉≤(1-λnτ)12(∥zn-x*∥2+∥xn+1-x*∥2)+[δnλnγρ+λn(1-δn)γ]×12(∥xn-x*∥2+∥xn+1-x*∥2)+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉≤(1-λnτ)12[(∥xn-x*∥+2νnαn∥x*∥)2+∥xn+1-x*∥2(∥xn-x*∥+2νnαn∥x*∥)2]+[δnλnγρ+λn(1-δn)γ]×12(∥xn-x*∥2+∥xn+1-x*∥2)+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉≤(1-λnτ)12(∥xn-x*∥2+αnM2+∥xn+1-x*∥2)+[δnλnγρ+λn(1-δn)γ]×12(∥xn-x*∥2+∥xn+1-x*∥2)+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉≤[1-δnλnγ(1-ρ)]×12(∥xn-x*∥2+∥xn+1-x*∥2)+δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉+αnM2,
where M2=supn≥0{2νn∥x*∥(2∥xn-x*∥+νnαn∥x*∥)}<∞. It turns out that
(76)∥xn+1-x*∥2≤1-δnλnγ(1-ρ)1+δnλnγ(1-ρ)∥xn-x*∥2+21+δnλnγ(1-ρ)×[δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)×〈(γS-μF)x*,xn+1-x*〉+αnM2]≤[1-δnλnγ(1-ρ)]∥xn-x*∥2+21+δnλnγ(1-ρ)×[δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉]+2αnM2.
However, fromx*∈Ξ and condition (C6) we obtain that
(77)〈(γS-μF)x*,xn+1-x*〉=〈(γS-μF)x*,xn+1-PFix(T)∩Γxn+1〉+〈(γS-μF)x*,PFix(T)∩Γxn+1-x*〉≤〈(γS-μF)x*,xn+1-PFix(T)∩Γxn+1〉≤∥(γS-μF)x*∥d(xn+1,Fix(T)∩Γ)≤∥(γS-μF)x*∥(1k-∥xn+1-Txn+1∥)1/θ.
On the other hand, from (41) we have
(78)∥zn-p∥2≤(∥xn-p∥+2νnαn∥p∥)2+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2-βn1-βn∥zn-xn∥2≤(∥xn-p∥+2νnαn∥p∥)2+(1-βn)(νn2(αn+L)2-1)∥xn-yn∥2,
which, together with ∥xn-zn∥+αn=o(λn2), implies that
(79)(1-d)(1-b2(αn+L)2)∥xn-yn∥2λn2≤(1-βn)(1-νn2(αn+L)2)∥xn-yn∥2λn2≤(∥xn-p∥+2νnαn∥p∥)2-∥zn-p∥2λn2≤∥xn-zn∥+2bαn∥p∥λn2×[∥xn-p∥+∥zn-p∥+2bαn∥p∥]⟶0asn⟶∞.
That is,∥xn-yn∥=o(λn). Observe that
(80)zn-xn=γn(tn-xn)+σn(Ttn-xn)=γn(tn-xn)+σn(Ttn-Txn)+σn(Txn-xn).
Since(γn+σn)ζ≤γn, utilizing Lemma 10 we get
(81)∥σn(Txn-xn)∥λn=∥zn-xn-[γn(tn-xn)+σn(Ttn-Txn)]∥λn≤∥zn-xn∥+∥γn(tn-xn)+σn(Ttn-Txn)∥λn≤∥zn-xn∥+∥tn-xn∥λn≤∥zn-xn∥+∥tn-yn∥+∥yn-xn∥λn=∥zn-xn∥+∥PC(xn-νnAnyn)-PC(xn-νnAnxn)∥λn+∥yn-xn∥λn≤∥zn-xn∥+∥νnAnyn-νnAnxn∥+∥yn-xn∥λn≤∥zn-xn∥+νn(αn+L)∥yn-xn∥+∥yn-xn∥λn≤∥zn-xn∥+2∥yn-xn∥λn⟶0asn⟶∞.
That is,∥σn(Txn-xn)∥=o(λn). Taking into account liminfn→∞σn>0, we have ∥xn-Txn∥=o(λn). Furthermore, utilizing Lemma 7 (i), we have
(82)∥xn+1-Txn+1∥≤∥xn+1-Txn∥+∥Txn-Txn+1∥≤1+ζ1-ζ∥xn-xn+1∥+∥λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn-Txn∥≤1+ζ1-ζ∥xn-xn+1∥+∥zn-Txn∥+λn∥γ(δnVxn+(1-δn)Sxn)-μFzn∥≤1+ζ1-ζ∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥+λn∥γδn(Vxn-Sxn)+γSxn-μFzn∥≤1+ζ1-ζ∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥+λnM0,
where M0=supn≥0∥γδn(Vxn-Sxn)+γSxn-μFzn∥<∞. Hence, for a big enough constant k-1>0, from (77), we have
(83)〈(γS-μF)x*,xn+1-x*〉≤k-1(λn+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥)1/θ≤k-1λn1/θ(1+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥λn)1/θ.
Combining (76)–(83), we get
(84)∥xn+1-x*∥2≤[1-δnλnγ(1-ρ)]∥xn-x*∥2+21+δnλnγ(1-ρ)×[δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉]+2αnM2≤[1-δnλnγ(1-ρ)]∥xn-x*∥2+2δnλn1+δnλnγ(1-ρ)×[(1+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥λn)〈(γV-μF)x*,xn+1-x*〉+k-1λn1/θδn×(1+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥λn)1/θ]+2αnM2=(1-sn)∥xn-x*∥2+μn+2αnM2,
where sn=δnλnγ(1-ρ) and
(85)μn=2δnλn1+δnλnγ(1-ρ)×[(1+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥λn)〈(γV-μF)x*,xn+1-x*〉+k-1λn1/θδn×(1+∥xn-xn+1∥+∥zn-xn∥+∥xn-Txn∥λn)1/θ].
Now condition (C5) implies that∑n=0∞‍sn=∞. Moreover, since ∥xn+1-xn∥ + ∥zn-xn∥ + ∥xn-Txn∥ = o(λn), conditions (C7) and (74) imply that
(86)limsupn→∞μnsn≤0.
Therefore, we can apply Lemma8 to (84) to conclude that xn→x*. The proof of part (a) is complete. It is easy to see that part (b) now becomes a straightforward consequence of part (a) since, if V=0, THVIP (8) reduces to the VIP in part (b). This completes the proof.Next we consider a special case of Problem II. In Problem II, putμ=2,F=(1/2)I and γ=τ=1. In this case, the objective is to find x*∈Ξ such that
(87)〈(I-V)x*,x-x*〉≥0,∀x∈Ξ,
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding z*∈Fix(T)∩Γ such that
(88)〈(I-S)z*,z-z*〉≥0,∀z∈Fix(T)∩Γ.Corollary 15.
LetA be a 1/L-inverse strongly monotone mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping. Assume that the solution set Ξ of the HVIP (88) is nonempty and that the following conditions hold for the sequences {αn}⊂[0,∞),{νn}⊂(0,1/L),{γn}⊂[0,1), and {βn},{δn},{σn},{λn}⊂(0,1):(C1)
∑n=0∞‍αn<∞ and  limn→∞(αn/λn2)=0;(C2)
0<liminfn→∞νn≤limsupn→∞νn<1/L;(C3)
βn+γn+σn=1 and (γn+σn)ζ≤γn for all n≥0;(C4)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C5)
limn→∞λn=0,limn→∞δn=0 and ∑n=0∞‍δnλn=∞;(C6)
there are constantsk-,θ>0 satisfying ∥x-Tx∥≥k-[d(x,Fix(T)∩Γ)]θ for each x∈C;(C7)
limn→∞(λn1/θ/δn)=0.
One has(a)
If{xn} is the sequence generated by the iterative scheme
(89)x0=x∈Cchosenarbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[λn(δnVxn+(1-δn)Sxn)+(1-λn)zn],∀n≥0
and {Sxn} is bounded, then {xn} converges strongly to the point x*∈Fix(T)∩Γ which is a unique solution of THVIP (87) provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2).(b)
If{xn} is the sequence generated by the iterative scheme
(90)x0=x∈Cchosenarbitrarily,yn=PC(xn-νnAnxn),zn=βnxn+γnPC(xn-νnAnyn)+σnTPC(xn-νnAnyn),xn+1=PC[λn(1-δn)Sxn+(1-λn)zn],∀n≥0
and{Sxn} is bounded, then {xn} converges strongly to a unique solution x* of the following VIP provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2):
(91)findx*∈Ξsuchthat〈x*,x-x*〉≥0,∀x∈Ξ;
that is,x* is the minimum-norm solution of HVIP (88).Furthermore, applying Theorem14 to Problem I, we get the result as below.Corollary 16.
LetF:C→ℋ be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, respectively, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and Ti:C→C be a ζi-strictly pseudocontractive mapping for i=1,2. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Assume that the solution set Ξ of the HVIP in Problem I is nonempty and that the following conditions hold for the sequences {νn}⊂(0,(1-ζ2)/2),{γn}⊂[0,1) and {βn},{δn},{σn},{λn}⊂(0,1):(C1)
0<liminfn→∞νn≤limsupn→∞νn<(1-ζ2)/2;(C2)
βn+γn+σn=1 and (γn+σn)ζ≤γn for all n≥0;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C4)
limn→∞λn=0,limn→∞δn=0 and ∑n=0∞‍δnλn=∞;(C5)
there are constantsk-,θ>0 satisfying ∥x-T1x∥≥k-[d(x,Fix(T1)∩Fix(T2))]θ for each x∈C;(C6)
limn→∞(λn1/θ/δn)=0.
One has the following.(a)
If{xn} is the sequence generated by
(92)x0=x∈Cchosenarbitrarily,yn=xn-νn(I-T2)xn,zn=βnxn+γnPC(xn-νn(I-T2)yn)+σnTPC(xn-νn(I-T2)yn),xn+1=PC[λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn],∀n≥0,
such that {Sxn} is bounded, then {xn} converges strongly to the point x*∈Fix(T1)∩Fix(T2) which is a unique solution of Problem I provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2).(b)
If{xn} is the sequence generated by
(93)x0=x∈Cchosenarbitrarily,yn=xn-νn(I-T2)xn,zn=βnxn+γnPC(xn-νn(I-T2)yn)+σnTPC(xn-νn(I-T2)yn),xn+1=PC(λn(1-δn)γSxn+(I-λnμF)zn],∀n≥0,
such that{Sxn} is bounded, then {xn} converges strongly to a unique solution x* of the following VIP provided that ∥xn+1-xn∥+∥xn-zn∥=o(λn2):
(94)findx*∈Ξsuchthat〈Fx*,x-x*〉≥0,∀x∈Ξ.Proof.
In Theorem14, we put T=T1 and A=I-T2 where Ti:C→C is ζi-strictly pseudocontractive for i=1,2. Taking L=2/(1-ζ2) and αn=0 for all n≥0, we know that A:C→ℋ is a 1/L-inverse strongly monotone mapping such that Γ=Fix(T2). In the scheme (11), we have
(95)yn=PC(xn-νnAnxn)=PC((1-νn)xn+νnT2xn)=xn-νn(I-T2)xn.
Utilizing Theorem14, we obtain desired result.On the other hand, we also derive the following strong convergence result of Algorithm I for finding a unique solution of Problem III.Theorem 17.
LetF:C→ℋ be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, respectively, A be a 1/L-inverse strongly monotone mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and T:C→C be a ζ-strictly pseudocontractive mapping. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Assume that Problem III has a solution and that the following conditions hold for the sequences {αn}⊂(0,∞),{νn}⊂(0,1/L),{γn}⊂[0,1) and {βn},{δn},{σn},{λn}⊂(0,1):(C1)
∑n=0∞‍αn<∞;(C2)
0<liminfn→∞νn≤limsupn→∞νn<1/L;(C3)
βn+γn+σn=1 and (γn+σn)ζ≤γn  for all n≥0;(C4)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C5)
0<liminfn→∞δn≤limsupn→∞δn<1;(C6)
limn→∞λn=0 and ∑n=0∞‍λn=∞.
One has the following.(a)
If{xn} is the sequence generated by the scheme (11) and {Sxn} is bounded, then {xn} converges strongly to a unique solution of Problem III provided that limn→∞∥xn+1-xn∥=0.(b)
If{xn} is the sequence generated by the scheme (12) and {Sxn} is bounded, then {xn} converges strongly to a unique solution x*∈Fix(T)∩Γ of the following system of variational inequalities provided that limn→∞∥xn+1-xn∥=0:
(96)〈Fx*,x-x*〉≥0,∀x∈Fix(T)∩Γ,〈(μF-γS)x*,y-x*〉≥0,∀y∈Fix(T)∩Γ.Proof.
We treat only case (a); that is, the sequence{xn} is generated by scheme (11). First of all, it is seen easily that 0<γ≤τ and κ≥η⇔μη≥τ. Hence it follows from the ρ-contractiveness of V and γρ<γ≤τ≤μη that μF-γV is (μη-γρ)-strongly monotone and Lipschitz continuous. So, there exists a unique solution x* of the following VIP:
(97)findx*∈Fix(T)∩Γsuch  that〈(μF-γV)x*,x-x*〉≥0,∀x∈Fix(T)∩Γ.
Consequently, it is easy to see that Problem III has a unique solutionx*∈Fix(T)∩Γ. In addition, taking into account condition (C5), without loss of generality we may assume that {δn}⊂[a,b] for some a,b∈(0,1).
Next we divide the rest of the proof into several steps.
Step  1 ({xn} is bounded). Indeed, repeating the same argument as in Step 1 of the proof of Theorem 14 we can derive the claim.
Step  2
(
lim
n
→
∞
∥
x
n
-
y
n
∥
=
lim
n
→
∞
∥
x
n
-
t
n
∥
=
lim
n
→
∞
∥
t
n
-
T
t
n
∥
=
0
). Indeed, repeating the same argument as in Step 2 of the proof of Theorem 14 we can derive the claim.
Step  3
(
ω
w
(
x
n
)
⊂
Fix
(
T
)
∩
Γ
). Indeed, repeating the same argument as in Step 3 of the proof of Theorem 14 we can derive the claim.
Step  4 ({xn} converges strongly to a unique solution x* of Problem III). Indeed, according to ∥xn+1-xn∥→0, we can take a subsequence {xni} of {xn} satisfying
(98)limsupn→∞〈(γV-μF)x*,xn+1-x*〉=limsupn→∞〈(γV-μF)x*,xn-x*〉=limi→∞〈(γV-μF)x*,xni-x*〉.
Without loss of generality, we may further assume thatxni⇀x~; then x~∈Fix(T)∩Γ due to Step 3. Since x* is a solution of Problem III, we get
(99)limsupn→∞〈(γV-μF)x*,xn+1-x*〉=〈(γV-μF)x*,x~-x*〉≤0.
Repeating the same argument as that of (99), we have
(100)limsupn→∞〈(γS-μF)x*,xn+1-x*〉≤0.
Repeating the same argument as that of (76) in the proof of Theorem 14, we obtain
(101)∥xn+1-x*∥2≤[1-δnλnγ(1-ρ)]∥xn-x*∥2+21+δnλnγ(1-ρ)×[δnλn〈(γV-μF)x*,xn+1-x*〉+λn(1-δn)〈(γS-μF)x*,xn+1-x*〉]+2αnM2.
Putrn=2αnM2,sn=δnλnγ(1-ρ) and
(102)bn=2γ(1-ρ)[1+δnλnγ(1-ρ)]×〈(γV-μF)x*,xn+1-x*〉+2(1-δn)δnγ(1-ρ)[1+δnλnγ(1-ρ)]×〈(γS-μF)x*,xn+1-x*〉.
Then (101) is rewritten as
(103)∥xn+1-x*∥2≤(1-sn)∥xn-x*∥2+snbn+rn.
In terms of conditions (C5) and (C6), we conclude from0<1-ρ≤1 that
(104){sn}⊂(0,1],∑n=0∞‍sn=∞.
Note that(105)2γ(1-ρ)[1+δnλnγ(1-ρ)]≤2γ(1-ρ),2(1-δn)δnγ(1-ρ)[1+δnλnγ(1-ρ)]≤2aγ(1-ρ).
Consequently, utilizing Lemma13 we obtain that
(106)limsupn→∞bn≤limsupn→∞2γ(1-ρ)[1+δnλnγ(1-ρ)]×〈(γV-μF)x*,xn+1-x*〉+limsupn→∞2(1-δn)δnγ(1-ρ)[1+δnλnγ(1-ρ)]×〈(γS-μF)x*,xn+1-x*〉≤0.
So, this together with Lemma8 leads to limn→∞∥xn-x*∥=0. The proof is complete.Utilizing Theorem17 we immediately derive the following result.Corollary 18.
LetF:C→ℋ be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, respectively, A be a 1/L-inverse strongly monotone mapping, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), and T:C→C be a ζ-strictly pseudocontractive mapping such that Fix(T)∩Γ≠∅. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Assume that the following conditions hold for the sequences {αn}⊂(0,∞),{νn}⊂(0,1/L),{γn}⊂[0,1) and {βn},{σn},{λn}⊂(0,1):(C1)
∑n=0∞‍αn<∞;(C2)
0<liminfn→∞νn≤limsupn→∞νn<1/L;(C3)
βn+γn+σn=1 and (γn+σn)ζ≤γn  for all n≥0;(C4)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C5)
limn→∞λn=0 and ∑n=0∞‍λn=∞.
One has the following.(a)
If{xn} is the sequence generated by the scheme (13) and {Vxn} is bounded, then {xn} converges strongly to a unique solution of the following VIP provided that limn→∞∥xn+1-xn∥=0:
(107)findx*∈Fix(T)∩Γsuchthat〈(μF-γV)x*,x-x*〉≥0,∀x∈Fix(T)∩Γ.(b)
If{xn} is the sequence generated by the scheme (14), then {xn} converges strongly to a unique solution of the following VIP provided that limn→∞∥xn+1-xn∥=0:
(108)findx*∈Fix(T)∩Γsuchthat〈Fx*,x-x*〉≥0,∀x∈Fix(T)∩Γ.Proof.
In Theorem17, putting S=V, we know that the iterative scheme (11) reduces to (13) since there holds for any {δn}⊂(0,1)(109)xn+1=PC[λnγ(δnVxn+(1-δn)Sxn)+(I-λnμF)zn]=PC[λnγ(δnVxn+(1-δn)Vxn)+(I-λnμF)zn]=PC[λnγVxn+(I-λnμF)zn].
In this case, the SVI (15) with VIP constraint is equivalent to the VIP (107). Thus, utilizing Theorem 17 (a) we obtain the desired conclusion (a). As for the conclusion (b), we immediately derive it from S=V≡0 and Theorem 17 (b).In addition, applying Theorem17 to Problem IV, we derive the result as below.Corollary 19.
LetF:C→ℋ be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, respectively, V:C→ℋ be a ρ-contraction with coefficient ρ∈[0,1), S:C→C be a nonexpansive mapping, and Ti:C→C be a ζi-strictly pseudocontractive mapping for i=1,2. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Assume that Problem IV has a solution and that the following conditions hold for the sequences {νn}⊂(0,(1-ζ2)/2),{γn}⊂[0,1) and {βn},{δn},{σn},{λn}⊂(0,1):(C1)
0<liminfn→∞νn≤limsupn→∞νn<(1-ζ2)/2;(C2)
βn+γn+σn=1 and (γn+σn)ζ≤γn for all n≥0;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞σn>0;(C4)
0<liminfn→∞δn≤limsupn→∞δn<1;(C5)
limn→∞λn=0 and ∑n=0∞‍λn=∞.
One has the following.(a)
If{xn} is the sequence generated by the scheme in Corollary 16 (a) such that {Sxn} is bounded, then {xn} converges strongly to a unique solution of Problem IV provided that limn→∞∥xn+1-xn∥=0.(b)
If{xn} is the sequence generated by the scheme in Corollary 16 (b) such that {Sxn} is bounded, then {xn} converges strongly to a unique solution x*∈Fix(T1)∩Fix(T2) of the following system of variational inequalities provided that limn→∞∥xn+1-xn∥=0:
(110)〈Fx*,x-x*〉≥0,∀x∈Fix(T1)∩Fix(T2),〈(μF-γS)x*,y-x*〉≥0,∀y∈Fix(T1)∩Fix(T2).



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*Source: 718624-2013-04-17.xml*