# Implicit Iterative Method for Hierarchical Variational Inequalities

Authors: L.-C. Ceng; Q. H. Ansari; N.-C. Wong; J.-C. Yao
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We introduce a new implicit iterative scheme with
perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the
common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence
generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence
of the sequence are obtained.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥ and C a nonempty closed convex subset of H. For a given nonlinear operator A:C→H, the classical variational inequality problem (VIP) [1] is to find x*∈C such that
(1.1)〈Ax*,x-x*〉≥0,∀x∈C.
The set of solutions of VIP is denoted by VI(C,A). If the set C is replaced by the set Fix(T) of fixed points of a mapping T; then the VIP is called a hierarchical variational inequality problem (HVIP). The signal recovery [2], the power control problem [3], and the beamforming problem [4] can be written in the form of a hierarchical variational inequality problem. In the recent past, several authors paid their attention toward this kind of problem and developed different kinds of solution methods with applications; see [2, 5–11] and the references therein.LetF:H→H be η-strongly monotone (i.e., if there exists a constant η>0 such that 〈Fx-Fy,x-y〉≥η∥x-y∥2, for all x,y∈H) and κ-Lipschitz continuous (i.e., if there exists a constant κ>0 such that ∥Fx-Fy∥≤κ∥x-y∥, for all x,y∈H). Assume that C is the intersection of the sets of fixed points of N nonexpansive mappings Ti:H→H. For an arbitrary initial guess x0∈H, Yamada [10] proposed the following hybrid steepest-descent method:
(1.2)xn+1∶=Tn+1xn-λn+1μF(Tn+1xn),∀n≥0.
Here, Tk:=TkmodN, for every integer k>N, with the mod function taking values in the set {1,2,…,N}; that is, if k=jN+q for some integers j≥0 and 0≤q<N, then Tk=TN if q=0 and Tk=Tq if 1<q<N. Moreover, μ∈(0,2η/κ2) and the sequence {λn}⊂(0,1) of parameters satisfies the following conditions:(i)
limn→∞λn=0;(ii)
∑n=0∞λn=∞;(iii)
∑n=0∞|λn-λn+N| is convergent.Under these conditions, Yamada [10] proved the strong convergence of the sequence {xn} to the unique element of VI(C,F).Xu and Kim [12] replaced the condition (iii) by the following condition:(iii)′
limn→∞(λn/λn+N)=1, or equivalently, limn→∞((λn-λn+N)/λn+N)=0and proved the strong convergence of the sequence{xn} to the unique element of VI(C,F).On the other hand, letK be a nonempty convex subset of H, and let {Ti}i=1N be a finite family of nonexpansive self-maps on K. Xu and Ori [13] introduced the following implicit iteration process: for x0∈K and {αn}n=1∞⊂(0,1), the sequence {xn}n=1∞ is generated by the following process:
(1.3)xn=αnxn-1+(1-αn)Tnxn,∀n≥1,
where we use the convention Tn:=TnmodN. They also studied the weak convergence of the sequence generated by the above scheme to a common fixed point of the mappings {Ti}i=1N under certain conditions. Subsequently, Zeng and Yao [14] introduced another implicit iterative scheme with perturbation for finding the approximate common fixed points of a finite family of nonexpansive self-maps on H.Motivated and inspired by the above works, in this paper, we propose a new implicit iterative scheme with perturbation for finding the approximate solutions of the hierarchical variational inequalities, that is, variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps onH. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme with perturbation. In particular, necessary and sufficient conditions for strong convergence of the sequence generated by the proposed implicit iterative scheme with perturbation are obtained.


## 2. Preliminaries

Throughout the paper, we writexn⇀x to indicate that the sequence {xn} converges weakly to x in a Banach space E. Meanwhile, xn→x implies that {xn} converges strongly to x. For a given sequence {xn}⊂E, ωw(xn) denotes the weak ω-limit set of {xn}, that is,
(2.1)ωw(xn)∶={x∈E:xnj⇀xforsomesubsequence{nj}of{n}}.
A Banach space E is said to satisfy Opial’s property if
(2.2)lim⁡inf⁡n→∞‖xn-x‖<lim⁡inf⁡n→∞‖xn-y‖∀y∈E,y≠x,
whenever a sequence xn⇀x in E. It is well known that every Hilbert space H satisfies Opial’s property; see for example [15].A mappingA:H→H is said to be hemicontinuous if for any x,y∈H, the mapping g:[0,1]→H, defined by g(t)∶=A(tx+(1-t)y)(forallt∈[0,1]), is continuous in the weak topology of the Hilbert space H. The metric projection onto a nonempty, closed and convex set C⊆H, denoted by PC, is defined by, for all x∈H,PCx∈C and ∥x-PCx∥=infy∈C∥x-y∥.Proposition 2.1.
LetC⊆H be a nonempty closed and convex set and A:H→H monotone and hemicontinuous. Then,(a)
[1] VI(C,A)={x*∈C:〈Ay,y-x*〉≥0,forally∈C},(b)
[1] VI(C,A)≠∅ when C is bounded,(c)
[16, Lemma  2.24] VI(C,A)=Fix(PC(I-λA)) for all λ>0, where I stands for the identity mapping on H,(d)
[16, Theorem  2.31] VI(C,A) consists of one point if A is strongly monotone and Lipschitz continuous.On the other hand, it is well known that the metric projectionPC onto a given nonempty closed and convex set C⊆H is nonexpansive with Fix(PC)=C [17, Theorem  3.1.4 (i)]. The fixed point set of a nonexpansive mapping has the following properties.Proposition 2.2.
LetC⊆H be a nonempty closed and convex subset and T:C→C a nonexpansive map.(a)
[18, Proposition  5.3] Fix(T) is closed and convex.(b)
[18, Theorem  5.1] Fix(T)≠∅ when C is bounded.The following proposition provides an example of a nonexpansive mapping in which the set of fixed points is equal to the solution set of a monotone variational inequality.Proposition 2.3 (see [6, Proposition  2.3]).
LetC⊆H be a nonempty closed and convex set and A:H→H an α-inverse-strongly monotone operator. Then, for any given λ∈(0,2α], the mapping Sλ:H→H, defined by
(2.3)Sλx∶=PC(I-λA)x,∀x∈H,
is nonexpansive and Fix(Sλ)=VI(C,A).The following lemmas will be used in the proof of the main results of this paper.Lemma 2.4 (see [18, Demiclosedness Principle]).
Assume thatT is a nonexpansive self-mapping on a closed convex subset K of a Hilbert space H. If T has a fixed point, then I-T is demiclosed, that is, whenever {xn} is a sequence in K weakly converging to some x∈K and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y, where I is the identity operator of H.Lemma 2.5 (see [19, page 80]).
Let{an}n=1∞, {bn}n=1∞, and {δn}n=1∞ be sequences of nonnegative real numbers satisfying the inequality
(2.4)an+1≤(1+δn)an+bn,∀n≥1.
If ∑n=1∞δn<∞ and ∑n=1∞bn<∞, then limn→∞an exists. If in addition {an}n=1∞ has a subsequence, which converges to zero, then limn→∞an=0.LetT:H→H be a nonexpansive mapping and F:H→H  κ-Lipschitz continuous and η-strongly monotone for some constants κ>0, η>0. For any given numbers λ∈[0,1) and μ∈(0,2η/κ2), we define the mapping Tλ:H→H by(2.5)Tλx∶=Tx-λμF(Tx),∀x∈H.Lemma 2.6 (see [12]).
If0≤λ<1 and 0<μ<2η/κ2, then
(2.6)‖Tλx-Tλy‖≤(1-λτ)‖x-y‖,∀x,y∈H,
where τ=1-1-μ(2η-μκ2)∈(0,1).


## 3. An Iterative Scheme and Convergence Results

Let{Ti}i=1N be a finite family of nonexpansive self-maps on H. Let A:H→H be an α-inverse-strongly monotone mapping (i.e., if there exists a constant α>0 such that 〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2, for all x,y∈H). Let F:H→H be κ-Lipschitz continuous and η-strongly monotone for some constants κ>0, η>0. Let {αn}n=1∞⊂(0,1), {βn}n=1∞⊂(0,2α], {λn}n=1∞⊂[0,1), and take a fixed number μ∈(0,2η/κ2). We introduce the following implicit iterative scheme with perturbation F. For an arbitrary initial point x0∈H, the sequence {xn}n=1∞ is generated by the following process:
(3.1)xn=αnxn-1+(1-αn)[Tn(xn-βnAxn)-λnμF∘Tn(xn-βnAxn)],∀n≥1.
Here, we use the convention Tn∶=TnmodN. If A≡0, then the implicit iterative scheme (3.1) reduces to the implicit iterative scheme studied in [14].LetA:H→H be an α-inverse-strongly monotone mapping and s∈(0,2α]. By Lemma 2.6, for every u∈H and t∈(0,1), the mapping St:H→H defined by
(3.2)Stx∶=tu+(1-t)Tλx̃withx̃=x-sAx,
satisfies
(3.3)‖Stx-Sty‖=(1-t)‖Tλx̃-Tλỹ‖≤(1-t)(1-λτ)‖x̃-ỹ‖≤(1-t)‖x̃-ỹ‖=(1-t)‖(x-sAx)-(y-sAy)‖=(1-t)‖(x-y)-s(Ax-Ay)‖≤(1-t)‖x-y‖2-s(2α-s)‖Ax-Ay‖2≤(1-t)‖x-y‖,∀x,y∈H,
where 0≤λ<1, 0<μ<2η/κ2, and τ=1-1-μ(2η-μκ2)∈(0,1). By Banach’s contraction principle, there exists a unique xt∈H such that
(3.4)xt=tu+(1-t)Tλx̃t=tu+(1-t)[T(xt-sAxt)-λμF∘T(xt-sAxt)].
This shows that the implicit iterative scheme (3.1) with perturbation F is well defined and can be employed for finding the approximate solutions of the variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps on H.We now state and prove the main results of this paper.Theorem 3.1.
LetH be a real Hilbert space, A an α-inverse-strongly monotone mapping, and F:H→H a κ-Lipschitz continuous and η-strongly monotone mapping for some constants κ, η>0. Let {Ti}i=1N be N nonexpansive self-maps on H with a nonempty common fixed point set ⋂i=1NFix(Ti). Suppose VI(⋂i=1NFix(Ti),A)≠∅. Denote by Tn:=TnmodN for n>N. Let μ∈(0,2η/κ2), x0∈H, {λn}n=1∞⊂[0,1), {αn}n=1∞⊂(0,1), and {βn}n=1∞⊂(0,2α] be such that ∑n=1∞λn<∞, βn≤λn and a≤αn≤b,foralln≥1, for some a,b∈(0,1). Then, the sequence {xn}n=1∞, defined by
(3.5)xn∶=αnxn-1+(1-αn)Tnλnx̃n=αnxn-1+(1-αn)[Tn(xn-βnAxn)-λnμF∘Tn(xn-βnAxn)],∀n≥1,
converges weakly to an element of ⋂i=1NFix(Ti).
If, in addition,∥xn-Tnx̃n∥=o(βn), then {xn} converges weakly to an element of VI(⋂i=1NFix(Ti),A).Proof.
Notice first that the following identity:(3.6)‖tx+(1-t)y‖2=t‖x‖2+(1-t)‖y‖2-t(1-t)‖x-y‖2
holds for all x,y∈H and all t∈[0,1]. Let x̂ be an arbitrary element of ⋂i=1NFix(Ti). Observe that
(3.7)‖xn-x̂‖2=‖αnxn-1+(1-αn)Tnλnx̃n-x̂‖2=αn‖xn-1-x̂‖2+(1-αn)‖Tnλnx̃n-x̂‖2-αn(1-αn)‖xn-1-Tnλnx̃n‖2.
Since A is α-inverse strongly monotone and {βn}n=1∞⊂(0,2α], we have
(3.8)‖xn-x̂-βn(Axn-Ax̂)‖2=‖xn-x̂‖2-2βn〈Axn-Ax̂,xn-x̂〉+βn2‖Axn-Ax̂‖2≤‖xn-x̂‖2-βn(2α-βn)‖Axn-Ax̂‖2≤‖xn-x̂‖2.
By Lemma 2.6, we have
(3.9)‖Tnλnx̃n-x̂‖=‖Tnλnx̃n-Tnλnx̂+Tnλnx̂-x̂‖≤‖Tnλnx̃n-Tnλnx̂‖+‖Tnλnx̂-x̂‖≤(1-λnτ)‖x̃n-x̂‖+λnμ‖F(x̂)‖≤(1-λnτ)[‖xn-x̂-βn(Axn-Ax̂)‖+βn‖Ax̂‖]+λnμ‖F(x̂)‖≤(1-λnτ)[‖xn-x̂‖+βn‖Ax̂‖]+λnμ‖F(x̂)‖≤(1-λnτ)‖xn-x̂‖+βn‖Ax̂‖+λnμ‖F(x̂)‖≤(1-λnτ)‖xn-x̂‖+λn(‖Ax̂‖+μ‖F(x̂)‖)=(1-λnτ)‖xn-x̂‖+λnτ⋅[‖Ax̂‖+μ‖F(x̂)‖τ].
It follows
(3.10)‖Tnλnx̃n-x̂‖2≤(1-λnτ)‖xn-x̂‖2+λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τ.
This together with (3.7) yields
(3.11)‖xn-x̂‖2≤αn‖xn-1-x̂‖2+(1-αn)[(1-λnτ)‖xn-x̂‖2+λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τ]-αn(1-αn)‖xn-1-Tnλnx̃n‖2≤αn‖xn-1-x̂‖2+(1-αn)‖xn-x̂‖2+(1-αn)λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τ-αn(1-αn)‖xn-1-Tnλnx̃n‖2,
and so,
(3.12)‖xn-x̂‖2≤‖xn-1-x̂‖2+(1-αn)λnαn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τ-(1-αn)‖xn-1-Tnλnx̃n‖2≤‖xn-1-x̂‖2+λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τa-‖xn-xn-1‖2.
Since ∑n=1∞λn·[(∥Ax̂∥+μ∥F(x̂)∥)2/τa] converges, by Lemma 2.5, limn→∞∥xn-x̂∥ exists. As a consequence, the sequence {xn} is bounded. Moreover, we have
(3.13)‖xn-xn-1‖2≤‖xn-1-x̂‖2-‖xn-x̂‖2+λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τa⟶0asn⟶∞.
Therefore,
(3.14)lim⁡n→∞‖xn-xn-1‖=0.
Obviously, it is easy to see that limn→∞∥xn-xn+i∥=0 for each i=1,2,…,N. Now observe that
(3.15)(1-b)‖xn-1-Tnλnx̃n‖≤(1-αn)‖xn-1-Tnλnx̃n‖=‖xn-xn-1‖⟶0asn⟶∞.
Also note that the boundedness of {xn} implies that {Tnx̃n} and {F(Tnx̃n)} are both bounded. Thus, we have
(3.16)‖xn-1-Tnx̃n‖≤‖xn-1-Tnλnx̃n‖+‖Tnλnx̃n-Tnx̃n‖≤‖xn-1-Tnλnx̃n‖+λnμ‖F(Tnx̃n)‖⟶0asn⟶∞.
This implies
(3.17)‖xn-Tnx̃n‖≤‖xn-xn-1‖+‖xn-1-Tnx̃n‖⟶0asn⟶∞.
Consequently,
(3.18)‖xn-Tnxn‖≤‖xn-Tnx̃n‖+‖Tnx̃n-Tnxn‖≤‖xn-Tnx̃n‖+‖x̃n-xn‖=‖xn-Tnx̃n‖+βn‖Axn‖≤‖xn-Tnx̃n‖+λn‖Axn‖⟶0asn⟶∞,
and hence, for each i=1,2,…,N,
(3.19)‖xn-Tn+ixn‖≤‖xn-xn+i‖+‖xn+i-Tn+ixn+i‖+‖Tn+ixn+i-Tn+ixn‖≤2‖xn-xn+i‖+‖xn+i-Tn+ixn+i‖⟶0asn⟶∞.
This shows that limn→∞∥xn-Tn+ixn∥=0 for each i=1,2,…,N. Therefore,
(3.20)lim⁡n→∞‖xn-Tlxn‖=0,foreachl=1,2,…,N.
On the other hand, since{xn} is bounded, it has a subsequence {xnj}, which converges weakly to some x¯∈H, and so, we have limj→∞∥xnj-Tlxnj∥=0. From Lemma 2.4, it follows that I-Tl is demiclosed at zero. Thus, x¯∈Fix(Tl). Since l is an arbitrary element in the finite set {1,2,…,N}, we get x¯∈⋂i=1NFix(Ti).
Now, letx* be an arbitrary element of ωw(xn). Then, there exists another subsequence {xnk} of {xn}, which converges weakly to x*∈H. Clearly, by repeating the same argument, we get x*∈⋂i=1NFix(Ti). We claim that x*=x¯. Indeed, if x*≠x¯, then by the Opial’s property of H, we conclude that
(3.21)lim⁡n→∞‖xn-x*‖=lim⁡inf⁡k→∞‖xnk-x*‖<lim⁡inf⁡k→∞‖xnk-x¯‖=lim⁡n→∞‖xn-x¯‖=lim⁡inf⁡j→∞‖xnj-x¯‖<lim⁡inf⁡j→∞‖xnj-x*‖=lim⁡n→∞‖xn-x*‖.
This leads to a contradiction, and so, we get x*=x¯. Therefore, ωw(xn) is a singleton set. Hence, {xn} converges weakly to a common fixed point of the mappings {Ti}i=1N, denoted still by x*.
Assume that∥xn-Tnx̃n∥=o(βn). Let y∈⋂i=1NFix(Ti) be arbitrary but fixed. Then, it follows from the nonexpansiveness of each Ti and the monotonicity of A that
(3.22)‖Tnx̃n-y‖2=‖Tn(xn-βnAxn)-Tn(y)‖2≤‖(xn-y)-βnAxn‖2=‖xn-y‖2+2βn〈Axn,y-xn〉+βn2‖Axn‖2=‖xn-y‖2+2βn(〈Ay,y-xn〉+〈Axn-Ay,y-xn〉)+βn2‖Axn‖2≤‖xn-y‖2+2βn〈Ay,y-xn〉+βn2‖Axn‖2,
which implies that
(3.23)0≤1βn(‖xn-y‖2-‖Tnx̃n-y‖2)+2〈Ay,y-xn〉+βn‖Axn‖2=(‖xn-y‖+‖Tnx̃n-y‖)‖xn-y‖-‖Tnx̃n-y‖βn+2〈Ay,y-xn〉+βn‖Axn‖2≤M‖xn-Tnx̃n‖βn+2〈Ay,y-xn〉+M2βn,
where M∶=sup{∥xn-y∥+∥Tnx̃n-y∥+∥Axn∥:n≥1}<∞. Note that λn→0,βn≤λn,foralln≥1, and ∥xn-Tnx̃n∥=o(βn). Thus, for any ɛ>0, there exists an integer m0≥1 such that M∥xn-Tnx̃n∥/βn+M2βn≤ɛ for all n≥m0. Consequently, 0≤ɛ+2〈Ay,y-xn〉 for all n≥m0. Since xn⇀x*, we have ɛ+2〈Ay,y-x*〉≥0 as n→∞. Therefore, from the arbitrariness of ɛ>0, we deduce that 〈Ay,y-x*〉≥0 for all y∈⋂i=1NFix(Ti). Proposition 2.1 (a) ensures that
(3.24)〈Ax*,y-x*〉≥0,∀y∈⋂i=1NFix⁡(Ti);
that is, x*∈VI(⋂i=1NFix(Ti),A).Corollary 3.2 (see [14], Theorem  2.1).
LetH be a real Hilbert space and F:H→Hκ-Lipschitz continuous and η-strongly monotone for some constants κ>0, η>0. Let {Ti}i=1N be N nonexpansive self-maps on H such that C=⋂i=1NFix(Ti)≠∅. Let μ∈(0,2η/κ2), x0∈H, {λn}n=1∞⊂[0,1), and {αn}n=1∞⊂(0,1) be such that ∑n=1∞λn<∞ and a≤αn≤b,foralln≥1, for some a,b∈(0,1). Then, the sequence {xn}n=1∞, defined by
(3.25)xn∶=αnxn-1+(1-αn)Tnλnxn=αnxn-1+(1-αn)[Tnxn-λnμF(Tnxn)],∀n≥1,
converges weakly to a common fixed point of the mappings {Ti}i=1N.Proof.
In Theorem3.1, putting A≡0, we can see readily that for any given positive number α∈(0,∞), A:H→H is an α-inverse-strongly monotone mapping. In this case, we have
(3.26)VI⁡(⋂i=1NFix⁡(Ti),A)=⋂i=1NFix⁡(Ti).
Hence, for any given sequence {βn}n=1∞⊂(0,2α] with βn≤λn(∀n≥1), the implicit iterative scheme (3.5) reduces to (3.25). Therefore, by Theorem 3.1, we obtain the desired result.Lemma 3.3.
In the setting of Theorem3.1, we have(a)
limn→∞∥xn-x̂∥ exists for each x̂∈C,(b)
limn→∞d(xn,C) exists, where d(xn,C)=infp∈C∥xn-p∥,(c)
liminfn→∞∥xn-Tnxn∥=0,where C∶=VI(⋂i=1NFix(Ti),A).Proof.
Conclusion (a) follows from (3.12), and conclusion (c) follows from (3.18). We prove conclusion (b). Indeed, for each x̂∈C,
(3.27)(‖Ax̂‖+μ‖F(x̂)‖)2≤(‖Ax̂-Axn-1‖+‖Axn-1‖+μ‖F(x̂)-F(xn-1)‖+μ‖F(xn-1)‖)2≤(1α‖xn-1-x̂‖+‖Axn-1‖+μκ‖xn-1-x̂‖+μ‖F(xn-1)‖)2=[(1α+μκ)‖xn-1-x̂‖+(‖Axn-1‖+μ‖F(xn-1)‖)]2≤2(1α+μκ)2‖xn-1-x̂‖2+2(‖Axn-1‖+μ‖F(xn-1)‖)2.
This together with (3.12) implies that
(3.28)‖xn-x̂‖2≤‖xn-1-x̂‖2+λn⋅(‖Ax̂‖+μ‖F(x̂)‖)2τa≤‖xn-1-x̂‖2+λn⋅1τa[2(1α+μκ)2‖xn-1-x̂‖2+2(‖Axn-1‖+μ‖F(xn-1)‖)2]≤(1+λn⋅2((1/α)+μκ)2τa)‖xn-1-x̂‖2+λn⋅2τa(‖Axn-1‖+μ‖F(xn-1)‖)2≤(1+γn)‖xn-1-x̂‖2+γn,
and hence,
(3.29)[d(xn,C)]2≤(1+γn)[d(xn-1,C)]2+γn,
where
(3.30)γn=λn⋅max⁡{2((1/α)+μκ)2τa,2τa(‖Axn-1‖+μ‖F(xn-1)‖)2},∀n≥1.
Since ∑n=1∞λn<∞ and both {Axn-1} and {F(xn-1)} are bounded, it is known that ∑n=1∞γn<∞. On account of Lemma 2.5, we deduce that limn→∞d(xn,C) exists, that is, conclusion (b) holds.Finally, we give necessary and sufficient conditions for the strong convergence of the sequence generated by the implicit iterative scheme (3.5) with perturbation F.Theorem 3.4.
In the setting of Theorem3.1, the sequence {xn} converges strongly to an element of VI(⋂i=1NFix(Ti),A)if and only if liminfn→∞d(xn,C)=0 where C∶=VI(⋂i=1NFix(Ti),A).Proof.
From (3.28), we derive for each n≥1(3.31)‖xn-x̂‖2≤(1+γn)‖xn-1-x̂‖2+γn,∀x̂∈C,
where ∑n=1∞γn<∞. Put M̃=∏n=1∞(1+γn). Then 1≤M̃<∞.
Suppose that the sequence{xn} converges strongly to a common fixed point p of the family {Ti}i=1N. Then, limn→∞∥xn-p∥=0. Since
(3.32)0≤d(xn,C)≤‖xn-p‖,
we have liminfn→∞d(xn,C)=0.
Conversely, suppose thatliminfn→∞d(xn,C)=0. Then, by Lemma 3.3 (b), we deduce that limn→∞d(xn,C)=0. Thus, for arbitrary ɛ>0, there exists a positive integer N0 such that
(3.33)d(xn,C)<ɛ8M̃,∀n≥N0.
Furthermore, the condition ∑n=1∞γn<∞ implies that there exists a positive integer N1 such that
(3.34)∑j=n∞γj<ɛ2(8M̃),∀n≥N1.
Choose N*=max{N0,N1}. Observe that (3.31) yields
(3.35)‖xn-x̂‖2≤(1+γn)(1+γn-1)‖xn-2-x̂‖2+(1+γn)γn-1+γn≤∏j=N*+1n(1+γj)‖xN*-x̂‖2+∑j=N*+1n-1γj∏i=j+1n(1+γj)+γn≤M̃[‖xN*-x̂‖2+∑j=N*+1nγj].
Note that d(xN*,C)<ɛ/8M̃ and ∑j=N*∞γj<ɛ2/(8M̃). Thus, for all n,m≥N* and all x̂∈C, we have from (3.35) that
(3.36)‖xn-xm‖2≤(‖xn-x̂‖+‖xm-x̂‖)2≤2‖xn-x̂‖2+2‖xm-x̂‖2≤2M̃[‖xN*-x̂‖2+∑j=N*+1nγj]+2M̃[‖xN*-x̂‖2+∑j=N*+1mγj]≤4M̃[‖xN*-x̂‖2+∑j=N*∞γj]<4M̃(‖xN*-x̂‖2+ɛ28M̃).
Taking the infimum over all x̂∈C, we obtain
(3.37)‖xn-xm‖2≤4M̃([d(xN*,C)]2+ɛ28M̃)≤4M̃(ɛ28M̃+ɛ28M̃)=ɛ2,
and hence, ∥xn-xm∥≤ɛ. This shows that {xn}n=1∞ is a Cauchy sequence in H. Let xn→p∈H as n→∞. Then, we derive from (3.20) that for each l=1,2,…,N,
(3.38)‖p-Tlp‖≤‖p-xn‖+‖xn-Tlxn‖+‖Tlxn-Tlp‖≤2‖xn-p‖+‖xn-Tlxn‖⟶0asn⟶∞.
Therefore, p∈Fix(Tl) for each l=1,2,…,N, and hence, p∈⋂i=1NFix(Ti).
On the other hand, choose a positive sequence{ɛn}n=1∞⊂(0,∞) such that ɛn→0 as n→∞. For each n≥1, from the definition of d(xn,C), it follows that there exists a point pn∈C such that
(3.39)‖xn-pn‖≤d(xn,C)+ɛn.
Since d(xn,C)→0 and ɛn→0 as n→∞, it is clear that ∥xn-pn∥→0 as n→∞. Note that
(3.40)‖pn-p‖≤‖pn-xn‖+‖xn-p‖⟶0asn⟶∞.
Hence, we get
(3.41)lim⁡n→∞‖pn-p‖=0.
Furthermore, for each βn∈(0,2α], the mapping Sβn:H→H is defined as follows:
(3.42)Sβnx∶=P⋂i=1NFix⁡(Ti)(I-βnA)x,∀x∈H.
From Proposition 2.3, we deduce that Sβn is nonexpansive and
(3.43)Fix⁡(Sβn)=VI⁡(⋂i=1NFix⁡(Ti),A)(=C).
From Proposition 2.2 (a), we conclude that Fix(Sβn) is closed and convex. Thus, from the condition VI(⋂i=1NFix(Ti),A)≠∅, it is known that C is a nonempty closed and convex set. Since {pn} lies in C and converges strongly to p, we must have p∈C.Remark 3.5.
SettingA=0 in Lemma 3.3 and Theorem 3.4 above, we shall derive Lemma  2.1 and Theorem  2.2 in [14] as direct consequences, respectively.



---

*Source: 472935-2012-04-09.xml*