# Algorithms for a System of General Variational Inequalities in Banach Spaces

Authors: Jin-Hua Zhu; Shih-Sen Chang; Min Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.


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## Body

## 1. Introduction

Throughout this paper, we denote byℕ and ℝ the sets of positive integers and real numbers, respectively. We also assume that E is a real Banach space, E* is the dual space of E,C is a nonempty closed convex subset of E, and 〈·,·〉 is the pairing between E and E*.In this paper, we are concerned a finite family of a general system of nonlinear variational inequalities in Banach spaces, which involves finding(x1*,x2*,…,xn*)∈C×C×⋯×C such that (1.1)〈λ1A1x2*+x1*-x2*,j(x-x1*)〉≥0,∀x∈C,〈λ2A2x3*+x2*-x3*,j(x-x2*)〉≥0,∀x∈C,〈λ3A3x4*+x3*-x4*,j(x-x3*)〉≥0,∀x∈C,⋮〈λN-1AN-1xN*+xN-1*-xN*,j(x-xN-1*)〉≥0,∀x∈C,〈λNANx1*+xN*-x1*,j(x-xN*)〉≥0,∀x∈C,
where {Ai:C→E,i=1,2,…,N} is a finite family of nonlinear mappings and λi(i=1,2,…,N) are positive real numbers.As special cases of the problem (1.1), we have the following.(I) IfE is a real Hilbert space and N=2, then (1.1) reduces to(1.2)〈λ1A1x2*+x1*-x2*,x-x1*〉≥0,∀x∈C,〈λ2A2x1*+x2*-x1*,x-x2*〉≥0,∀x∈C,
which was considered by Ceng et al. [1]. In particular, if A1=A2=A, then the problem (1.2) reduces to finding (x1*,x2*)∈C×C such that(1.3)〈λ1Ax2*+x1*-x2*,x-x1*〉≥0,∀x∈C,〈λ2Ax1*+x2*-x1*,x-x2*〉≥0,∀x∈C,
which is defined by Verma [2]. Furthermore, if x1*=x2*, then (1.3) reduces to the following variational inequality (VI) of finding x*∈C such that(1.4)〈Ax*,x-x*〉≥0,∀x∈C.This problem is a fundamental problem in variational analysis and, in particular, in optimization theory. Many algorithms for solving this problem are projection algorithms that employ projections onto the feasible setC of the VI or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich’s extragradient method which was introduced by Korpelevich [3] in 1976 generates a sequence {xn} via the recursion(1.5)yn=PC[xn-λAxn],xn+1=PC[xn-λAyn],n≥0,
where PC is the metric projection from ℝn onto C, A:C→H is a monotone operator, and λ is a constant. Korpelevich [3] proved that the sequence {xn} converges strongly to a solution of VI(C,A). Note that the setting of the space is Euclid space ℝn.The literature on the VI is vast, and Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways. See, for example, [4–16] and references therein.(II) IfE is still a real Banach space and N=1, then the problem (1.1) reduces to finding x*∈C such that(1.6)〈Ax*,j(x-x*)〉≥0,∀x∈C,
which was considered by Aoyama et al. [17]. Note that this problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of a nonlinear operator, and so on. It is clear that problem (1.6) extends problem (1.4) from Hilbert spaces to Banach spaces.In order to find a solution for problem (1.6), Aoyama et al. [17] introduced the following iterative scheme for an accretive operator A in a Banach space E:(1.7)xn+1=αnxn+(1-αn)ΠC(xn-λnAxn),n≥1,
where ΠC is a sunny nonexpansive retraction from E to C. Then they proved a weak convergence theorem in a Banach space. For related works, please see [18] and the references therein.It is an interesting problem of constructing some algorithms with strong convergence for solving problem (1.1) which contains problem (1.6) as a special case.Our aim in this paper is to construct two algorithms for solving problem (1.1). For this purpose, we first prove that the system of variational inequalities (1.1) is equivalent to a fixed point problem of some nonexpansive mapping. Finally, we prove the strong convergence of the proposed methods which solve problem (1.1).


## 2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence{xn} by xn→x and xn⇀x, respectively.Forq>1, the generalized duality mapping Jq:E→2E* is defined by(2.1)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1}
for all x∈E. In particular, J=J2 is called the normalized duality mapping. It is known that Jq(x)=||x||q-2 for all x∈E. If E is a Hilbert space, then J=I, the identity mapping. Let U={x∈E:||x||=1}. A Banach space E is said to be uniformly convex if, for any ε∈(0,2], there exists δ>0 such that, for any x,y∈U,(2.2)‖x-y‖≥εimplies‖x+y2‖≤1-δ.It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach spaceE is said to be smooth if the limit(2.3)lim⁡n→0‖x+ty‖-‖x‖t
exists for all x,y∈U. It is also said to be uniformly smooth if the previous limit is attained uniformly for x,y∈U. The norm of E is said to be Fréchet differentiable if, for each x∈U, the previous limit is attained uniformly for all y∈U. The modulus of smoothness of E is defined by(2.4)ρ(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x,y∈E,‖x‖=1,‖y‖=τ},
where ρ:[0,∞)→[0,∞) is function. It is known that E is uniformly smooth if and only if lim⁡τ→0(ρ(τ)/τ)=0. Let q be a fixed real number with 1<q≤2. Then a Banach space E is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0. Note the following.(1)
E is a uniformly smooth Banach space if and only if J is single valued and uniformly continuous on any bounded subset of E.(2)
All Hilbert spaces,Lp (or lp) spaces (p≥2) and the Sobolev spaces Wmp(p≥2) are 2-uniformly smooth, while Lp (or lp) and WmP spaces (1<p≤2) are p-uniformly smooth.(3)
Typical examples of both uniformly convex and uniformly smooth Banach spaces areLp, where p>1. More precisely, Lp is min{p,2}-uniformly smooth for every p>1.In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constantK.LetE be a real Banach space, C a nonempty closed convex subset of E, T:C→C a mapping, and F(T) the set of fixed points of T.Recall that a mappingT:C→C is called nonexpansive if(2.5)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
A bounded linear operator F:C∈E is called strongly positive if there exists a constant γ>0 with the property(2.6)〈F(x),j(x)〉≥γ‖x‖2,∀x∈C.
A mapping A:C→E is said to be accretive if there exists j(x-y)∈J(x-y) such that(2.7)〈Ax-Ay,j(x-y)〉≥0,
for all x,y∈C, where J is the duality mapping.A mappingA of C into E is said to be α-strongly accretive if, for α>0,(2.8)〈Ax-Ay,j(x-y)〉≥α‖x-y‖2,
for all x,y∈C.A mappingA of C into E is said to be α-inverse-strongly accretive if, for α>0,(2.9)〈Ax-Ay,j(x-y)〉≥α‖Ax-Ay‖2,
for all x,y∈C.Remark 2.1.
Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for instance, [6, 19, 20].
LetD be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if
(2.10)Π[Π(x)+t(x-Π(x))]=Π(x)
whenever Π(x)+t(x-Π(x))∈C for x∈C and t≥0. A mapping Π of C into itself is called a retraction if Π2=Π. If a mapping Π of C into itself is a retraction, then Π(z)=z for every z∈R(Π), where R(Π) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. Then following lemma concerns the sunny nonexpansive retraction.Lemma 2.2 (see [21]).
LetC be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C, and let Π be a retraction from C onto D. Then Π is sunny and nonexpansive if and only if
(2.11)〈u-Π(u),j(y-Π(u))〉≤0,
for all u∈C and y∈D.Remark 2.3.
(1) It is well known that ifE is a Hilbert space, then a sunny nonexpansive retraction ΠC is coincident with the metric projection from E onto C.
(2) LetC be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with the set F(T)≠∅. Then the set F(T) is a sunny nonexpansive retract of C.In what follows, we need the following lemmas for proof of our main results.Lemma 2.4 (see [22]).
Assume that{αn} is a sequence of nonnegative real numbers such that
(2.12)αn+1≤(1-γn)αn+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that(a)
Σn=1∞γn=∞,(b)
limsup⁡n→∞(δn/γn)≤0 or Σn=1∞|δn|<∞.Then lim⁡n→∞αn=0.Lemma 2.5 (see [23]).
LetX be a Banach space, {xn},{yn} be two bounded sequences in X and {βn} be a sequence in [0,1] satisfying
(2.13)0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Suppose that xn+1=βnxn+(1-βn)yn, for all n≥1 and
(2.14)limsup⁡n→∞{‖yn+1-yn‖-‖xn+1-xn‖}≤0,
then lim⁡n→∞∥yn-xn∥=0.Lemma 2.6 (see [24]).
LetE be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
(2.15)‖x+y‖2≤‖x‖2+2〈y,Jx〉+2‖Ky‖2,∀x,y∈E.Lemma 2.7 (see [25]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let G be a nonexpansive mapping of C into itself. If {xn} is a sequence of C such that xn⇀x and xn-Gxn→0, then x is a fixed point of G.Lemma 2.8 (see [26]).
LetC be a nonempty closed convex subset of a real Banach space E. Assume that the mapping F:C→E is accretive and weakly continuous along segments (i.e., F(x+ty)⇀F(x) as t→0). Then the variational inequality
(2.16)x*∈C,〈Fx*,j(x-x*)〉≥0,x∈C,
is equivalent to the dual variational inequality
(2.17)x*∈C,〈Fx,j(x-x*)〉≥0,x∈C.Lemma 2.9.
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E onto C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive. For given (x1*,x2*,…,xn*)∈C×C×⋯×C, where x*=x1*,xi*=ΠC(I-λiAi)xi+1*,i∈{i,2,…,N-1},xN*=ΠC(I-λNAN)x1*, then (x1*,x2*,…,xn*) is a solution of the problem (1.1) if and only if x* is a fixed point of the mapping Q defined by
(2.18)Q(x)=ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)(x),
where λi(i=1,2,…,N) are real numbers.Proof.
We can rewrite (1.1) as
(2.19)〈x1*-(x2*-λ1A1x2*),j(x-x1*)〉≥0,∀x∈C,〈x2*-(x3*-λ2A2x3*),j(x-x2*)〉≥0,∀x∈C,〈x3*-(x4*-λ3A3x4*),j(x-x3*)〉≥0,∀x∈C,⋮〈xN-1*-(xN*-λN-1AN-1xN*),j(x-xN-1*)〉≥0,∀x∈C,〈xN*-(x1*-λNANx1*),j(x-xN*)〉≥0,∀x∈C.
By Lemma 2.2, we can check (2.19) is equivalent to
(2.20)x1*=ΠC(I-λ1A1)x2*,x2*=ΠC(I-λ2A2)x3*,⋮xN-1*=ΠC(I-λN-1AN-1)xN*,xN*=ΠC(I-λNAN)x1*.⟺Q(x*)=ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)(x*)=x*.
This completes the proof.Throughout this paper, the set of fixed points of the mappingQ is denoted by Ω.Lemma 2.10.
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E onto C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive. Let Q be defined as Lemma 2.9. If 0≤λi≤γi/K2, then Q:C→C is nonexpansive.Proof.
First, we show that for alli∈{i,2,…,N}, the mapping ΠC(I-λiAi) is nonexpansive. Indeed, for all x,y∈C, from the condition λi∈[0,γi/K2] and Lemma 2.6, we have
(2.21)‖ΠC(I-λiAi)x-ΠC(I-λiAi)y‖2≤‖(I-λiAi)x-(I-λiAi)y‖2=‖(x-y)-λi(Aix-Aiy)‖2≤‖x-y‖2-2λi〈Aix-Aiy,j(x-y)〉+2K2λi2‖Aix-Aiy‖2≤‖x-y‖2-2λiγi‖Aix-Aiy‖2+2K2λi2‖Aix-Aiy‖2=‖x-y‖2+2λi(K2λi-γi)‖Aix-Aiy‖2≤‖x-y‖2,
which implies for all i∈{1,2,…,N}, the mapping ΠC(I-λiAi) is nonexpansive, so is the mapping Q.


## 3. Main Results

In this section, we introduce our algorithms and show the strong convergence theorems.Algorithm 3.1.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive. Let B:C→E be a strongly positive bounded linear operator with coefficient α>0 and F:C→E be a strongly positive bounded linear operator with coefficient ρ∈(0,α). For any t∈(0,1), define a net {xt} as follows:
(3.1)xt=ΠC(tF+(I-tB))yt,yt=ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)xt,
where, for any i,λi∈(0,γi/K2) is a real number.Remark 3.2.
We notice that the net{xt} defined by (3.1) is well defined. In fact, we can define a self-mapping Wt:C→C as follows:
(3.2)Wtx:=ΠC(tF+(I-tB))ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)x,∀x∈C.
From Lemma2.10, we know that if, for any i,λi∈(0,γi/K2), the mapping ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)=Q is nonexpansive and ||I-tB||≤1-tα. Then, for any x,y∈C, we have
(3.3)‖Wtx-Wty‖=‖ΠC(tF+(I-tB))Q(x)-ΠC(tF+(I-tB))Q(y)‖≤‖((tF+(I-tB))Q)x-((tF+(I-tB))Q)y‖=‖t(Fx-Fy)+(I-tB)(Qx-Qy)‖≤tρ‖x-y‖+‖I-tB‖‖Qx-Qy‖≤tρ‖x-y‖+(1-tα)‖x-y‖=(1-(α-ρ)t)‖x-y‖.
This shows that the mapping Wt is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.Theorem 3.3.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive. Let B:C→E be a strongly positive bounded linear operator with coefficient α>0, and let F:C→E be a strongly positive bounded linear operator with coefficient ρ∈(0,α). Assume that Ω≠∅ and λi∈(0,γi/K2). Then the net {xt} generated by the implicit method (3.1) converges in norm, as t→0+ to the unique solution x̃ of VI
(3.4)x̃∈Ω,〈(B-F)x̃,j(z-x̃)〉≥0,z∈Ω.Proof.
We divide the proof of Theorem3.3 into four steps.(I)
Next we prove that the net{xt} is bounded.
Take thatx*∈Ω, we have
(3.5)‖xt-x*‖=‖ΠC(tF+(I-tB))yt-x*‖=‖ΠC(tF+(I-tB))yt-ΠCx*‖≤‖(tF+(I-tB))yt-x*‖=‖t(F(yt)-F(x*))+(I-tB)(yt-x*)+tF(x*)-tB(x*)‖≤t‖F(yt)-F(x*)‖+‖I-tB‖‖yt-x*‖+t‖F(x*)-B(x*)‖≤tρ‖yt-x*‖+(1-tα)‖yt-x*‖+t‖F(x*)-B(x*)‖=(1-(α-ρ)t)‖yt-x*‖+t‖F(x*)-B(x*)‖=(1-(α-ρ)t)‖Q(xt)-Q(x*)‖+t‖F(x*)-B(x*)‖≤(1-(α-ρ)t)‖xt-x*‖+t‖F(x*)-B(x*)‖.
It follows that
(3.6)‖xt-x*‖≤‖F(x*)-B(x*)‖α-ρ.
Therefore, {xt} is bounded. Hence, {yt},{Byt},{Aixt}, and {F(yt)} are also bounded. We observe that
(3.7)‖xt-yt‖=‖ΠC(tF+(I-tB))yt-ΠCyt‖≤‖(tF+(I-tB))yt-yt‖=t‖F(yt)-B(yt)‖⟶0.
From Lemma2.10, we know that Q:C→C is nonexpansive. Thus, we have
(3.8)‖yt-Q(yt)‖=‖Q(xt)-Q(yt)‖≤‖xt-yt‖⟶0.
Therefore,
(3.9)lim⁡t→0‖xt-Q(xt)‖=0.
(II){xt} is relatively norm-compact as t→0+.
Let{tn}⊂(0,1) be any subsequence such that tn→0+ as n→∞. Then, there exists a positive integer n0 such that 0<tn<1/2, for all n≥n0. Let xn:=xtn. It follows from (3.9) that
(3.10)‖xn-Q(xn)‖⟶0.
We can rewrite (3.1) as
(3.11)xt=ΠC(tF+(I-tB))yt-(tF+(I-tB))yt+(tF+(I-tB))yt.
For anyx*∈Ω⊂C, by Lemma 2.2, we have
(3.12)〈(tF+(I-tB))yt-xt,j(x*-xt)〉=〈(tF+(I-tB))yt-ΠC(tF+(I-tB))yt,j(x*-ΠC(tF+(I-tB))yt)〉≤0.
With this fact, we derive that
(3.13)‖xt-x*‖2=〈x*-xt,j(x*-xt)〉=〈x*-(tF+(I-tB))yt,j(x*-xt)〉+〈(tF+(I-tB))yt-ΠC(tF+(I-tB))yt,j(x*-xt)〉≤〈(tF+(I-tB))(x*-yt),j(x*-xt)〉+t〈B(x*)-F(x*),j(x*-xt)〉≤(1-t(α-ρ))‖x*-yt‖‖x*-xt‖+t〈B(x*)-F(x*),j(x*-xt)〉=(1-t(α-ρ))‖Q(x*)-Q(xt)‖‖x*-xt‖+t〈B(x*)-F(x*),j(x*-xt)〉≤(1-t(α-ρ))‖x*-xt‖‖x*-xt‖+t〈B(x*)-F(x*),j(x*-xt)〉≤(1-t(α-ρ))‖x*-xt‖2+t〈B(x*)-F(x*),j(x*-xt)〉.
It turns out that(3.14)‖xt-x*‖2≤1α-ρ〈B(x*)-F(x*),j(x*-xt)〉,x*∈Ω.
In particular,**‖xn-x*‖2≤1α-ρ〈B(x*)-F(x*),j(x*-xn)〉,x*∈Ω.
Since{xn} is bounded, without loss of generality, xn⇀x̃∈C can be assumed. Noticing (3.10), we can use Lemma 2.7 to get x̃∈Ω=F(Q). Therefore, we can substitute x̃ for x* in (**) to get
(3.15)‖xn-x̃‖2≤1α-ρ〈B(x̃)-F(x̃),j(x̃-xn)〉.
Consequently, the weak convergence of {xn} to x̃ actually implies that xn→x̃ strongly. This has proved the relative norm compactness of the net {xt} as t→0+.
(III) Now, we prove thatx̃ solves the variational inequality (3.4). From (3.1), we have
(3.16)xt=ΠC(tF+(I-tB))yt-(tF+(I-tB))yt+(tF+(I-tB))yt⟹xt=ΠC(tF+(I-tB))yt-(tF+(I-tB))yt-(tF+(I-tB))(xt-yt)+tF(xt)+(I-tB)(xt)⟹F(xt)-B(xt)=1t[(tF+(I-tB))yt-ΠC(tF+(I-tB))yt-(tF+(I-tB))(yt-xt)].
For any z∈Ω, we obtain
(3.17)〈F(xt)-B(xt),j(z-xt)〉=1t〈(tF+(I-tB))yt-ΠC(tF+(I-tB))yt,j(z-xt)〉-1t〈(tF+(I-tB))(yt-xt),j(z-xt)〉≤-1t〈(tF+(I-tB))(yt-xt),j(z-xt)〉=-1t〈yt-xt,j(z-xt)〉+〈(B-F)(yt-xt),j(z-xt)〉.
Now we prove that〈yt-xt,j(z-xt)〉≥0. In fact, we can write yt=Q(xt). At the same time, we note that z=Q(z), so
(3.18)〈yt-xt,j(z-xt)〉=〈z-Q(z)-(xt-Q(xt)),j(z-xt)〉.
Since I-Q is accretive (this is due to the nonexpansivity of Q), we can deduce immediately that
(3.19)〈yt-xt,j(z-xt)〉=〈z-Q(z)-(xt-Q(xt),j(z-xt)〉≥0.
Therefore,
(3.20)〈F(xt)-B(xt),j(z-xt)〉≤〈(B-F)(yt-xt),j(z-xt)〉.
Since B,F is strongly positive, we have
(3.21)0≤(α-ρ)‖z-xt‖2≤〈(B-F)(z-xt),j(z-xt)〉=〈(F(xt)-B(xt))-(F(z)-B(z)),j(z-xt)〉.
It follows that
(3.22)〈F(z)-B(z),j(z-xt)〉≤〈F(xt)-B(xt),j(z-xt)〉.
Combining (3.20) and (3.22), we get
(3.23)〈F(z)-B(z),j(z-xt)〉≤〈(B-F)(yt-xt),j(z-xt)〉.
Now replacing t in (3.23) with tn and letting n→∞, noticing that xtn-ytn→0, we obtain
(3.24)〈F(z)-B(z),j(z-x̃)〉≤0,z∈Ω,
which is equivalent to its dual variational inequality (see Lemma 2.8)
(3.25)〈(B-F)x̃,j(z-x̃)〉≥0,z∈Ω,
that is, x̃∈Ω is a solution of (3.4).
(IV) Now we show that the solution set of (3.4) is singleton.
As a matter of fact, we assume thatx*∈Ω is also a solution of (3.4) Then, we have
(3.26)〈(B-F)x*,j(x̃-x*)〉≥0.
From (3.25), we have
(3.27)〈(B-F)x̃,j(x*-x̃)〉≥0.
So,
(3.28)〈(B-F)x*,j(x̃-x*)〉+〈(B-F)x̃,j(x*-x̃)〉≥0⟹〈(B-F)(x̃-x*),j(x*-x̃)〉≥0⟹〈(B-F)(x*-x̃),j(x*-x̃)〉≤0⟹(α-ρ)‖x*-x̃‖2≤0.
Therefore, x*=x̃. In summary, we have shown that each cluster point of {xt} (as t→0) equals x̃. Therefore, xt→x̃ as t→0. This completes the proof.Next, we introduce our explicit method which is the discretization of the implicit method (3.1).Algorithm 3.4.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive. Let B:C→E be a strongly positive bounded linear operator with coefficient α>0, and let F:C→E be a strongly positive bounded linear operator with coefficient ρ∈(0,α). For arbitrarily given x0∈C, let the sequence {xn} be generated iteratively by
(3.29)xn+1=βnxn+(1-βn)ΠC(αnF+(I-αnB))ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)xn,n≥0,
where {αn} and {βn} are two sequences in [0,1] and, for any i,λi∈(0,γi/K2) is a real number.Theorem 3.5.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E, and let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai:C→E,i=1,2,…,N} be a finite family of γi-inverse-strongly accretive.Let B:C→E be a strongly positive bounded linear operator with coefficient α>0, and let F:C→E be a strongly positive bounded linear operator with coefficient ρ∈(0,α). Assume that Ω≠∅. For given x0∈C, let {xn} be generated iteratively by (3.29). Suppose the sequences {αn} and {βn} satisfy the following conditions:(1)
lim⁡n→∞αn=0 and Σn=1∞αn=∞,(2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn≤1.Then {xn} converges strongly to x̃∈Ω which solves the variational inequality (3.4).Proof.
Setyn=ΠC(I-λ1A1)ΠC(I-λ2A2)⋯ΠC(I-λNAN)xn for all n≥0. Then xn+1=βnxn+(1-βn)ΠC(αnF+(I-αnB))yn for all n≥0. Pick up x*∈Ω.
From Lemma2.10, we have
(3.30)‖yn-x*‖=‖Q(xn)-Q(x*)‖≤‖xn-x*‖.
Hence, it follows that
(3.31)‖xn+1-x*‖=‖βnxn+(1-βn)ΠC(αnF+(I-αnB))yn-x*‖=‖βn(xn-x*)+(1-βn)(ΠC(αnF+(I-αnB))yn-x*)‖≤βn‖xn-x*‖+(1-βn)‖ΠC(αnF+(I-αnB))yn-ΠCx*‖≤βn‖xn-x*‖+(1-βn)‖(αnF+(I-αnB))yn-x*‖=βn‖xn-x*‖+(1-βn)‖(αnF+(I-αnB))(yn-x*)+αn(F(x*)-B(x*))‖≤βn‖xn-x*‖+(1-βn)(αnρ+(1-αnα))‖yn-x*‖+(1-βn)αn‖F(x*)-B(x*)‖≤(1-αn(1-βn)(α-ρ))‖xn-x*‖+αn(1-βn)(α-ρ)‖F(x*)-B(x*)‖α-ρ.
By induction, we deduce that
(3.32)‖xn+1-x*‖≤max⁡{‖x0-x*‖,‖F(x*)-B(x*)‖α-ρ}.
Therefore, {xn} is bounded. Hence, {Aixi}(i=1,2,…,N),{yn},{Byn}, and {F(yn)} are also bounded. We observe that
(3.33)‖yn+1-yn‖=‖Q(xn+1)-Q(xn)‖≤‖xn+1-xn‖.
Set xn+1=βnxn+(1-βn)zn for all n≥0. Then zn=ΠC(αnF+(I-αnB))yn. It follows that
(3.34)‖zn+1-zn‖=‖ΠC(αn+1F+(I-αn+1B))yn+1-ΠC(αnF+(I-αnB))yn‖≤‖(αn+1F+(I-αn+1B))yn+1-(αnF+(I-αnB))yn‖=‖yn+1-yn+αn+1(F(yn+1)-B(yn+1))-αn(F(yn)-B(yn))‖≤‖yn+1-yn‖+αn+1‖F(yn+1)-B(yn+1)‖-αn‖F(yn)-B(yn)‖≤‖xn+1-xn‖+αn+1‖F(yn+1)-B(yn+1)‖-αn‖F(yn)-B(yn)‖.
This implies that
(3.35)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.5, we obtain lim⁡n→∞∥zn-xn∥=0. Consequently,
(3.36)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖zn-xn‖=0.
At the same time, we note that
(3.37)‖zn-yn‖=‖ΠC(αnF+(I-αnB))yn-yn‖=‖ΠC(αnF+(I-αnB))yn-ΠCyn‖≤‖(αnF+(I-αnB))yn-yn‖=αn‖F(yn)-B(yn)‖⟶0.
It follows that
(3.38)lim⁡n→∞‖xn-yn‖=0.
From Lemma 2.10, we know that Q:C→C is nonexpansive. Thus, we have
(3.39)‖yn-Q(yn)‖=‖Q(xn)-Q(yn)‖≤‖xn-yn‖⟶0.
Thus, lim⁡n→∞∥xn-Q(xn)∥=0. We note that
(3.40)‖zn-Q(zn)‖≤‖zn-xn‖+‖xn-Q(xn)‖+‖Q(xn)-Q(zn)‖≤2‖zn-xn‖+‖xn-Q(xn)‖=2‖ΠC(αnF+(I-αnB))yn-ΠCxn‖+‖xn-Q(xn)‖≤2(‖yn-xn‖+αn‖F(yn)-B(yn)‖)+‖xn-Q(xn)‖⟶0.
Next, we show that
(3.41)limsup⁡n→∞〈F(x̃)-B(x̃),j(zn-x̃)〉≤0,
where x̃∈Ω is the unique solution of VI(3.4).
To see this, we take a subsequence{znj} of {zn} such that
(3.42)lim⁡n→∞〈F(x̃)-B(x̃),j(zn-x̃)〉=lim⁡nj→∞〈F(x̃)-B(x̃),j(znj-x̃)〉.
We may also assume that znj⇀z. Note that z∈Ω in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that
(3.43)lim⁡n→∞〈F(x̃)-B(x̃),j(zn-x̃)〉=lim⁡nj→∞〈F(x̃)-B(x̃),j(znj-x̃)〉=〈F(x̃)-B(x̃),j(z-x̃)〉≤0.
Since zn=ΠC(αnF+(I-αnB))yn, according to Lemma 2.2, we have
(3.44)〈(αnF+(I-αnB))yn-ΠC(αnF+(I-αnB))yn,j(x̃-zn)〉≤0.
From (3.44), we have
(3.45)‖zn-x̃‖2=〈ΠC(αnF+(I-αnB))yn-x̃,j(zn-x̃)〉=〈ΠC(αnF+(I-αnB))yn-(αnF+(I-αnB))yn,j(zn-x̃)〉+〈(αnF+(I-αnB))yn-x̃,j(zn-x̃)〉≤〈(αnF+(I-αnB))yn-x̃,j(zn-x̃)〉=〈(αnF+(I-αnB))(yn-x̃),j(zn-x̃)〉+αn〈F(x̃)-B(x̃),j(zn-x̃)〉≤(1-αn(α-ρ))‖yn-x̃‖‖zn-x̃‖+αn〈F(x̃)-B(x̃),j(zn-x̃)〉≤(1-αn(α-ρ))22‖yn-x̃‖2+12‖zn-x̃‖2+αn〈F(x̃)-B(x̃),j(zn-x̃)〉.
It follows that
(3.46)‖zn-x̃‖2≤(1-αn(α-ρ))‖yn-x̃‖2+2αn〈F(x̃)-B(x̃),j(zn-x̃)〉,≤(1-αn(α-ρ))‖xn-x̃‖2+2αn〈F(x̃)-B(x̃),j(zn-x̃)〉.
Finally, we prove xn→x̃. From xn+1=βnxn+(1-βn)zn and (3.46), we have
(3.47)‖xn+1-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖zn-x̃‖2≤βn‖xn-x̃‖2+(1-βn)((1-αn(α-ρ))‖xn-x̃‖2+2αn〈F(x̃)-B(x̃),j(zn-x̃)〉)=(1-αn(1-βn)(α-ρ))‖xn-x̃‖2+αn(1-βn)(α-ρ)×{2α-ρ〈F(x̃)-B(x̃),j(zn-x̃)〉}.
We can apply Lemma 2.4 to the relation (3.47) and conclude that xn→x̃. This completes the proof.



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*Source: 580158-2012-04-22.xml*