# Modified Noor's Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces

Authors: Shunhou Fan; Shin Min Kang; Muhammad Aslam Noor; Yonghong Yao
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Motivated and inspired by Korpelevich's and Noor's extragradient methods, we suggest an extragradient method by using the sunny nonexpansive
retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.


---

## Body

## 1. Introduction

In the present paper, we focus on the following generalized variational inequality:(1.1)Findx*∈Csuchthat〈Ax*,J(x-x*)〉≥0,∀x∈C,
where C is a nonempty closed convex subset of a real Banach space E, A:C→E is a nonlinear mapping, and J:E→2E* is the normalized duality mapping defined by(1.2)J(x)={f∈E*:〈x,f〉=‖x‖2,‖f‖=‖x‖},∀x∈E.
We use S(C,A) to denote the solution set of (1.1). It is clear that (1.1) is reduced to the following variational inequality in Hilbert spaces:(1.3)Findx*∈Csuchthat〈Ax*,x-x*〉≥0,∀x∈C,
which was introduced and studied by Stampacchia [1]. Variational inequalities are being used as mathematical programming tools and models to study a wide class of unrelated problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences. See, for instance, [2–23]. In order to solve (1.3), especially, Korpelevich [24] introduced the following well-known extragradient method:(1.4)yn=PC(I-λA)xn,xn+1=PC(xn-λAyn),n≥0,
where PC is the metric projection from Rn onto its subset C, λ∈(0,1/k) and A:C→Rn is a monotone operator. He showed that the sequence {xn} converges to some solution of the above variational inequality (1.3). Noor [10] further suggested and analyzed the following new iterative methods:(1.5)xn+1=PC(yn-λAyn),yn=PC[xn-λAxn],n≥0,
which is known as the modified Noor's extragradient method. We would like to point out that this algorithm (1.5) is quite different from the method of Koperlevich. However, these two algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.The generalized variational inequality (1.1) was introduced by Aoyama et al. [25] which is connected with the fixed point problem for nonlinear mapping. For solving the aforementioned generalized variational inequality (1.1), Aoyama et al. [25] introduced an iterative algorithm:(1.6)xn+1=αnxn+(1-αn)QC[xn-λnAxn],n≥0,
where QC is a sunny nonexpansive retraction from E onto C, and {αn}⊂(0,1) and {λn}⊂(0,∞) are two real number sequences. Aoyama et al. [25] obtained on the aforementioned method (1.6) for solving variational inequality (1.1). Motivated by (1.6), Yao and Maruster [26] presented a modification of (1.5):(1.7)xn+1=βnxn+(1-βn)QC[(1-αn)(xn-λAxn)],n≥0.
Yao and Maruster [26] proved that (1.7) converges strongly to the solution of the generalized variational inequality (1.1). Yao et al. [27] further considered the following extended extragradient method for solving (1.1):(1.8)yn=QC[xn-λnAxn],xn+1=αnu+βnxn+γnQC[yn-λAyn],n≥0.In this paper, motivated and inspired by Korpelevich's and Noor's extragradient methods, (1.7) and (1.8), we suggest a modified Noor's extragradient method via the sunny nonexpansive retraction for solving the variational inequalities (1.1) in Banach spaces.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Banach space E. Recall that a mapping A of C into E is said to be accretive if there exists j(x-y)∈J(x-y) such that(2.1)〈Ax-Ay,j(x-y)〉≥0,
for all x,y∈C. A mapping A of C into E is said to be α-strongly accretive if, for α>0,(2.2)〈Ax-Ay,j(x-y)〉≥α‖x-y‖2,
for all x,y∈C. A mapping A of C into E is said to be α-inverse-strongly accretive if, for α>0,(2.3)〈Ax-Ay,j(x-y)〉≥α‖Ax-Ay‖2,
for all x,y∈C.LetU={x∈E:∥x∥=1}. A Banach space E is said to uniformly convex if, for each ϵ∈(0,2], there exists δ>0 such that for any x,y∈U,(2.4)‖x-y‖≥ϵimplies‖x+y2‖≤1-δ.
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit(2.5)lim⁡t→0⁡‖x+ty‖-‖x‖t
exists for all x,y∈U. It is also said to be uniformly smooth if the limit (2.5) is attained uniformly for x,y∈U. The norm of E is said to be Fréchet differentiable if, for each x∈U, the limit (2.5) is attained uniformly for y∈U. And we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of E as follows:(2.6)ρ(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x,y∈X,‖x‖=1,‖y‖=τ}.
It is known that E is uniformly smooth if and only if limτ→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.We need the following lemmas for proof of our main results.Lemma 2.1 (see [28]).
Letq be a given real number with 1<q≤2 and let E be a q-uniformly smooth Banach space. Then
(2.7)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+2‖Ky‖q
for all x,y∈E, where K is the q-uniformly smoothness constant of E and Jq is the generalized duality mapping from E into 2E* defined by
(2.8)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1},∀x∈E.LetD be a subset of C and let Q be a mapping of C into D. Then Q is said to be sunny if(2.9)Q(Qx+t(x-Qx))=Qx,
whenever Qx+t(x-Qx)∈C for x∈C and t≥0. A mapping Q of C into itself is called a retraction if Q2=Q. If a mapping Q of C into itself is a retraction, then Qz=z for every z∈R(Q), where R(Q) is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. One knows the following lemma concerning sunny nonexpansive retraction.Lemma 2.2 (see [29]).
LetC be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C, and let Q be a retraction from C onto D. Then Q is sunny and nonexpansive if and only if
(2.10)〈u-Qu,j(y-Qu)〉≤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 QC 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 F(T)≠∅. Then the set F(T) is a sunny nonexpansive retract of C.The following lemma characterized the set of solution of (1.1) by using sunny nonexpansive retractions.Lemma 2.4 (see [25]).
LetC be a nonempty closed convex subset of a smooth Banach space X. Let QC be a sunny nonexpansive retraction from X onto C and let A be an accretive operator of C into X. Then for all λ>0,
(2.11)S(C,A)=F(QC(I-λA)),
where S(C,A)={x*∈C:〈Ax*,J(x-x*)〉≥0,∀x∈C}.Lemma 2.5 (see [30]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of C into itself. If {xn} is a sequence of C such that xn→x weakly and xn-Txn→0 strongly, then x is a fixed point of T.Lemma 2.6 (see [31]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.12)an+1≤(1-γn)an+δn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a sequence in R such that(a)
∑n=0∞γn=∞;(b)
limsupn→∞(δn/γn)≤0 or ∑n=0∞|δn|<∞. Then limn→∞an=0.


## 3. Main Results

In this section, we will state and prove our main result.Theorem 3.1.
LetE be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and let A:C→E be an α-strongly accretive and L-Lipschitz continuous mapping with S(C,A)≠∅. For given x0∈C, let the sequence {xn} be generated iteratively by
(3.1)yn=QC[xn-λAxn],xn+1=QC[(1-αn)(yn-λAyn)],n≥0,
where {αn} and {βn} are two sequences in (0,1) and λ is a constant in [a,b] for some a,b with 0<a<b<α/K2L2. Assume that the following conditions hold:(a)
limn→∞αn=0 and ∑n=1∞αn=∞;(b)
limn→∞(αn+1/αn)=1.
Then{xn} defined by (3.1) converges strongly to Q′(0), where Q′ is a sunny nonexpansive retraction of E onto S(C,A).Proof.
First, we note thatA must be α/L2-inverse-strongly accretive mapping. Take p∈S(C,A). By using Lemmas 2.1 and 2.4, we easily obtain the following facts.(1)
p=QC[p-λAp] for all λ>0; in particular,
(3.2)p=QC[p-λ(1-αn)Ap]=QC[αnp+(1-αn)(p-λAp)],n≥0.(2)
Ifλ∈(0,α/K2L2], then I-λA is nonexpansive and for all x,y∈C(3.3)‖(I-λA)x-(I-λA)y‖2≤‖x-y‖2+2λ(K2λ-αL2)‖Ax-Ay‖2.
Indeed, from Lemma2.1, we have
(3.4)‖(I-λA)x-(I-λA)y‖2=‖(x-y)-λ(Ax-Ay)‖2≤‖x-y‖2-2λ〈Ax-Ay,j(x-y)〉+2K2λ2‖Ax-Ay‖2≤‖x-y‖2-2λαL2‖Ax-Ay‖2+2K2λ2‖Ax-Ay‖2=‖x-y‖2+2λ(K2λ-αL2)‖Ax-Ay‖2.
From (3.1), we have
(3.5)‖yn-p‖=‖QC(xn-λAxn)-QC(p-λAp)‖≤‖(xn-λAxn)-(p-λAp)‖≤‖xn-p‖.
By (3.1) and (3.5), we have
(3.6)‖xn+1-p‖=‖QC[(1-αn)(yn-λAyn)]-QC[αnp+(1-αn)(p-λAp)]‖≤‖[(1-αn)(yn-λAyn)]-[αnp+(1-αn)(p-λAp)]‖≤αn‖p‖+(1-αn)‖(yn-λAyn)-(p-λAp)‖≤αn‖p‖+(1-αn)‖yn-p‖≤αn‖p‖+(1-αn)‖xn-p‖≤max⁡⁡{‖p‖,‖x0-p‖}.
Therefore, {xn} is bounded. We observe that
(3.7)‖yn-yn-1‖=‖QC[xn-λAxn]-QC[xn-1-λAxn-1]‖≤‖(xn-λAxn)-(xn-1-λAxn-1)‖≤‖xn-xn-1‖,
and hence
(3.8)‖xn+1-xn‖=‖QC[(1-αn)(yn-λAyn)]-QC[(1-αn-1)(yn-1-λAyn-1)]‖≤‖[(1-αn)(yn-λAyn)]-[(1-αn-1)(yn-1-λAyn-1)]‖=‖(1-αn)[(yn-λAyn)-(yn-1-λAyn-1)]+(αn-1-αn)(yn-1-λAyn-1)‖≤(1-αn)‖(yn-λAyn)-(yn-1-λAyn-1)‖+|αn-αn-1|‖yn-1-λAyn-1‖≤(1-αn)‖yn-yn-1‖+|αn-αn-1|‖yn-1-λAyn-1‖≤(1-αn)‖xn-xn-1‖+|αn-αn-1|‖yn-1-λAyn-1‖.
By Lemma 2.6, we obtain
(3.9)lim⁡n→∞‖xn+1-xn‖=0.
From (3.1), we also have
(3.10)‖yn-p‖2=‖QC[xn-λAxn]-QC[p-λAp]‖2≤‖(xn-λAxn)-(p-λAp)‖2≤‖xn-p‖2+2λ(K2λ-αL2)‖Axn-Ap‖2.
By (3.1) and (3.10), we obtain
(3.11)‖xn+1-p‖2≤‖αn(-p)+(1-αn)[(yn-λAyn)-(p-λAp)]‖2≤αn‖p‖2+(1-αn)‖(yn-λAyn)-(p-λAp)‖2≤αn‖p‖2+(1-αn)[‖yn-p‖2+2λ(K2λ-αL2)‖Ayn-Ap‖2]≤αn‖p‖2+(1-αn)[‖xn-p‖2+2λ(K2λ-αL2)‖Axn-Ap‖2]+2(1-αn)λ(K2λ-αL2)‖Ayn-Ap‖2≤αn‖p‖2+‖xn-p‖2+2(1-αn)λ(K2λ-αL2)‖Axn-Ap‖2+2(1-αn)λ(K2λ-αL2)‖Ayn-Ap‖2.
Therefore, we have
(3.12)0≤-2(1-αn)λ(K2λ-αL2)‖Axn-Ap‖2-2(1-αn)λ(K2λ-αL2)‖Ayn-Ap‖2≤αn‖p‖2+‖xn-p‖2-‖xn+1-p‖2=αn‖p‖2+(‖xn-p‖+‖xn+1-p‖)(‖xn-p‖-‖xn+1-p‖)≤αn‖p‖2+(‖xn-p‖+‖xn+1-p‖)‖xn-xn+1‖.
Since liminfn→∞2(1-αn)λ(K2λ-α/L2)>0,  αn→0 and ∥xn-xn+1∥→0, we obtain
(3.13)lim⁡n→∞‖Axn-Ap‖=lim⁡n→∞‖Ayn-Ap‖=0.
It follows that
(3.14)lim⁡n→∞‖Ayn-Axn‖=0.
Since A is α-strongly accretive, we deduce
(3.15)‖Ayn-Axn‖≥α‖yn-xn‖,
which implies that
(3.16)lim⁡n→∞‖yn-xn‖=0,
that is,
(3.17)lim⁡n→∞‖QC(xn-λAxn)-xn‖=0.
Next, we show that
(3.18)lim⁡⁡sup⁡n→∞〈Q′(0),j(xn-Q′(0))〉≥0.
To show (3.18), since {xn} is bounded, we can choose that a sequence {xni} of {xn} converges weakly to z such that
(3.19)limsup⁡n→∞〈Q′(0),j(xn-Q′(0))〉=limsup⁡i→∞〈Q′(0),j(xni-Q′(0))〉.
We first prove z∈S(C,A). It follows that
(3.20)lim⁡i→∞‖QC(I-λA)xni-xni‖=0.
By Lemma 2.5 and (3.20), we have z∈F(QC(I-λA)); it follows from Lemma 2.4 that z∈S(C,A).
Now, from (3.19) and Lemma 2.2, we have
(3.21)limsup⁡n→∞〈Q′(0),j(xn-Q′(0))〉=limsup⁡i→∞〈Q′(0),j(xni-Q′(0))〉=〈Q′(0),j(z-Q′(0))〉≥0.
Since xn+1=QC[(1-αn)(yn-λAyn)] and x*=QC[αnx*+(1-αn)(x*-λAx*)] for all n≥0, we can deduce from Lemma 2.2 that
(3.22)〈QC[(1-αn)(yn-λnAyn)]-[(1-αn)(yn-λnAyn)],j(xn+1-x*)〉≤0,〈[αnx*+(1-αn)(x*-λnAx*)]-QC[αnx*+(1-αn)(x*-λnAx*)],j(xn+1-x*)〉≤0.
Therefore, we have
(3.23)‖xn+1-x*‖2=〈QC[(1-αn)(yn-λnAyn)]-QC[αnx*+(1-αn)(x*-λnAx*)],j(xn+1-x*)〉=〈QC[(1-αn)(yn-λnAyn)]-[(1-αn)(yn-λnAyn)],j(xn+1-x*)〉+〈[(1-αn)(yn-λnAyn)]-[αnx*+(1-αn)(x*-λnAx*)],j(xn+1-x*)〉+〈[αnx*+(1-αn)(x*-λnAx*)]-QC[αnx*+(1-αn)(x*-λnAx*)],j(xn+1-x*)〉≤〈(1-αn)(yn-λnAyn)-(1-αn)(x*-λnAx*)-αnx*,j(xn+1-x*)〉≤(1-αn)‖yn-x*‖‖xn+1-x*‖-αn〈x*,j(xn+1-x*)〉≤(1-αn)‖xn-x*‖‖xn+1-x*‖-αn〈x*,j(xn+1-x*)〉≤1-αn2(‖xn-x*‖2+‖xn+1-x*‖2)-αn〈x*,j(xn+1-x*)〉,
which implies that
(3.24)‖xn+1-z‖2≤(1-αn)‖xn-z‖2+2αn〈-z,j(xn+1-z)〉.
Finally, by Lemma2.6 and (3.24), we conclude that xn converges strongly to Q′(0). This completes the proof.


## 4. Conclusion

Variational inequality theory provides a simple, natural, and unified framework for a general treatment of unrelated problems. These activities have motivated to generalize and extend the variational inequalities and related optimization problems in several directions using new and novel techniques. A well-known method to solve the VI is the following gradient method:(4.1)xn+1=PC(xn-αnA(xn)),n≥0,
This method requires some monotonicity properties of A. However, we remark that there is no chance of relaxing the assumption on A to plain monotonicity. To overcome this weakness of the method, Korpelevich proposed a so-called Korpelevich's method which has been extensively extended and studied. Noor [10] especially, suggested another method referred as Noor's method which is different from Korpelevich's method. However, these two algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces. In the present paper, we suggested a modified Noor's method which has strong convergence in Banach spaces. We hope that the ideas and technique of this paper may stimulate further research in this field.



---

*Source: 493862-2012-04-05.xml*