# Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities

Authors: Peixia Yang; Yonghong Yao; Yeong-Cheng Liou; Rudong Chen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space and C  a nonempty closed convex subset of H. Recall that a mapping T:C→C is called nonexpansive if
(1.1)‖Tx-Ty‖≤‖x-y‖
for every x,y∈C. A family S={T(τ):0<τ<∞} of mappings from C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:(i)
T(0)x=x for all x∈C,(ii)
T(s+t)=T(s)T(t) for all s,t≥0,(iii)
∥T(s)x-T(s)y∥≤∥x-y∥ for all x,y∈C,s≥0,(iv)
for allx∈C,s↦T(s)x is continuous.We denote by Fix(S) the set of all common fixed points of S, that is, Fix(S)=⋂0≤τ<∞‍Fix(T(τ)). It is known that Fix(S) is closed and convex.Approximation of fixed points of nonexpansive mappings has been considered extensively by many authors, see, for instance, [1–18]. Nonlinear ergodic theorem for nonexpansive semigroups have been researched by some authors, see, for example, [19–23]. Our main purpose in the present paper is devoted to finding the common fixed points of nonexpansive semigroups.LetF:C→C be a nonlinear operator. The variational inequality problem is formulated as finding a point x*∈C such that
(1.2)VI⁡(F,C):〈Fx*,υ-x*〉≥0,∀υ∈C.
Now it is well known that VI problem is an interesting problem and it covers as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance. Several numerical methods including the projection and its variant forms have been developed for solving the variational inequalities and related problems, see [24–41].It is clear that theVI(F,C) is equivalent to the fixed point equation
(1.3)x*=PC[x*-μF(x*)],
where PC is the projection of H onto the closed convex set C and μ>0 is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the VI(F,C) provided F satisfies some conditions and μ>0 is chosen appropriately. The fixed point formulation (1.3) involves the projection PC, which may not be easy to compute, due to the complexity of the convex set C. In order to reduce the complexity probably caused by the projection PC, Yamada [24] (see also [42]) recently introduced a hybrid steepest-descent method for solving the VI(F,C).Assume thatF is an η-strongly monotone and κ-Lipschitzian mapping with κ>0,η>0 on C. An equally important problem is how to find an approximate solution of the VI(F,C) if any. A great deal of effort has been done in this problem; see [43, 44].Take a fixed numberμ such that 0<μ<2η/κ2. Assume that a sequence {λn} of real numbers in (0,1) satisfies the following conditions:(C1)
limn→∞λn=0,(C2)
∑n=0∞‍λn=∞,(C3)
limn→∞(λn-λn+1)/λn2=0.Starting with an arbitrary initial guessx0∈H, one can generate a sequence {xn} by the following algorithm:
(1.4)xn+1=Txn-λn+1μF(Txn),n≥0.
Yamada [24] proved that the sequence {xn} generated by (1.4) converges strongly to the unique solution of the VI(F,C). Xu and Kim [30] proved the strong convergence of {xn} to the unique solution of the VI(F,C) if {λn} satisfies conditions (C1), (C2), and (C4): limn→∞λn/λn+1=1, or equivalently, limn→∞(λn-λn+1)/λn+1=1. Recently, Yao et al. [25] presented the following hybrid algorithm:
(1.5)yn=xn-λnF(xn),xn+1=(1-αn)yn+αnWnyn,n≥0,
where F is a κ-Lipschitzian and η-strongly monotone operator on H  and Wn is a W-mapping. It is shown that the sequences {xn} and {yn} defined by (1.5) converge strongly to x*∈⋂n=1∞F(Tn), which solves the following variational inequality:
(1.6)〈Fx*,x-x*〉≥0,∀x∈⋂n=1∞‍F(Tn).
Very recently, Wang [26] proved that the sequence {yn} generated by the iterative algorithm (1.5) converges to a common fixed point of an infinite family of nonexpansive mappings under some weaker assumptions.Motivated and inspired by the above works, in this paper, we introduce two hybrid algorithms for finding a common fixed point of a nonexpansive semigroup{T(τ)}τ≥0 in Hilbert spaces. We prove that the presented algorithms converge strongly to a common fixed point x* of {T(τ)}τ≥0. Such common fixed point x* is the unique solution of some variational inequality in Hilbert spaces.


## 2. Preliminaries

In this section, we will collect some basic concepts and several lemmas that will be used in the next section.Suppose thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. For the sequence {xn} in H, we write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x means that {xn} converges strongly to x. We denote by ωw(xn) the weak ω-limit set of {xn}, that is
(2.1)ωw(xn)={x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.
Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping F:C→C is called κ-Lipschitzian if there exists a positive constant κ such that
(2.2)‖Fx-Fy‖≤κ‖x-y‖,∀x,y∈C.F is said to be η-strongly monotone if there exists a positive constant η such that
(2.3)〈Fx-Fy,x-y〉≥η‖x-y‖2,∀x,y∈C.
The following equalities are well known:
(2.4)∥x-y∥2=∥x∥2-2〈x,y〉+∥y∥2,‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2
for all x,y∈H and λ∈[0,1] (see [45]).In the sequel, we will make use of the following well-known lemmas.Lemma 2.1 (see [46]).
LetC be a nonempty bounded closed convex subset of H and let S={T(s)∣0≤s<∞} be a nonexpansive semigroup on C. Then, for any h≥0,
(2.5)lim⁡t→∞sup⁡x∈C‖1t∫0t‍T(s)xds-T(h)(1t∫0t‍T(s)xds)‖=0.Lemma 2.2 (see [47]).
Assume thatT:H→H is a nonexpansive mapping. If T has a fixed point, then I-T is demiclosed. That is, whenever {xn} is a sequence in H weakly converging to some x∈H and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y. Here, I is the identity operator of H.Lemma 2.3 (see [27]).
Let{γn} be a real sequence satisfying 0<liminfn→∞γn≤limsupn→∞γn<1. Assume that {xn} and {zn} are bounded sequences in Banach space E, which satisfy the following condition: xn+1=(1-γn)xn+γnzn. If  limsupn→∞(∥zn+1-zn∥)-∥xn+1-xn∥)≤0, then limn→∞∥xn-zn∥=0.Lemma 2.4 (see [48]).
LetF be a κ-Lipschitzian and η-strongly monotone operator on a Hilbert space H with 0<η≤κ and 0<t<η/κ2. Then, S=(I-tF):H→H is a contraction with contraction coefficient τt=1-t(2η-tκ2).Lemma 2.5 (see [49]).
Let{an} be a sequence of nonnegative real numbers satisfying
(2.6)an+1≤(1-λn)an+λnδn+γn,n≥0,
where {λn} and {γn} satisfy the following conditions:(i)
λn⊂[0,1] and ∑n=0∞‍λn=∞,(ii)
limsupn→∞δn≤0or∑n=0∞‍λnδn<∞,(iii)
γn≥0(n≥0),∑n=0∞‍γn<∞.
Then,limn→∞an=0.


## 3. Main Results

In this section we will show our main results.Theorem 3.1.
LetH be a real Hilbert space. Let S={T(τ)∣0≤τ<∞}:H→H be a nonexpansive semigroup such that Fix(S)≠∅. Let F be a κ-Lipschitzian and η-strongly monotone operator on H with 0<η≤κ. Let {γt}0<t<1 be a continuous net of positive real numbers such that limt→0+γt=+∞. Putting τt=1-t(2η-tκ2), for each t∈(0,η/κ2), let the net {xt} be defined by the following implicit scheme:
(3.1)xt=1γt∫0γt‍T(τ)[(I-tF)xt]dτ.
Then, as t→0+, the net {xt} converges strongly to a fixed point x* of S, which is the unique solution of the following variational inequality:
(3.2)〈Fx*,x*-u〉≤0,∀u∈Fix⁡(S).Proof.
First, we note that the net{xt} defined by (3.1) is well defined. We define a mapping
(3.3)Ptx:=1γt∫0γt‍T(τ)[(I-tF)x]dτ,t∈(0,ηκ2),x∈H.
It follows that
(3.4)‖Ptx-Pty‖≤1γt∫0γt‍‖T(τ)[(I-tF)x]-T(τ)[(I-tF)y]‖dτ≤‖(I-tF)x-(I-tF)y‖.
Obviously, Pt is a contraction. Indeed, from Lemma 2.4, we have
(3.5)‖Ptx-Pty‖≤‖(I-tF)x-(I-tF)y‖≤‖x-y‖,
for all x,y∈C. So it has a unique fixed point. Therefore, the net {xt} defined by (3.1) is well defined.
We prove that{xt} is bounded. Taking u∈Fix(S) and using Lemma 2.4, we have
(3.6)‖xt-u‖=‖1γt∫0γt‍T(τ)[(I-tF)xt]dτ-u‖=‖1γt∫0γt‍T(τ)[(I-tF)xt]dτ-1γt∫0γt‍T(τ)udτ‖≤1γt∫0γt‍‖T(τ)[(I-tF)xt]dτ-T(τ)u‖dτ≤‖(I-tF)xt-u‖≤‖(I-tF)xt-(I-tF)u-tFu‖≤τt‖xt-u‖+t‖Fu‖.
It follows that
(3.7)‖xt-u‖≤t1-τt‖Fu‖.
Observe that
(3.8)lim⁡t→0+t1-τt=1η.
Thus, (3.7) and (3.8) imply that the net {xt} is bounded for small enough t. Without loss of generality, we may assume that the net {xt} is bounded for all t∈(0,η/κ2). Consequently, we deduce that {Fxt} is also bounded.
On the other hand, from (3.1), we have
(3.9)‖xt-T(τ)xt‖≤‖T(τ)xt-T(τ)(1γt∫0γt‍T(τ)xtdτ)‖+‖1γt∫0γt‍T(τ)xtdτ-xt‖+‖T(τ)(1γt∫0γt‍T(τ)xtdτ)-1γt∫0γt‍T(τ)xtdτ‖≤2‖1γt∫0γt‍T(τ)xtdτ-xt‖+‖T(τ)(1γt∫0γt‍T(τ)xtdτ)-1γt∫0γt‍T(τ)xtdτ‖=2‖1γt∫0γt‍T(τ)xtdτ-1γt∫0γt‍T(τ)[(I-tF)]xtdτ‖+‖T(τ)(1γt∫0γt‍T(τ)xtdτ)-1γt∫0γt‍T(τ)xtdτ‖≤21γt∫0γt‍‖T(τ)xt-T(τ)[(I-tF)xt]‖dτ+‖T(τ)(1γt∫0γt‍T(τ)xtdτ)-1γt∫0γt‍T(τ)xtdτ‖≤2t‖Fxt‖+‖T(τ)(1γt∫0γt‍T(τ)xtdτ)-1γt∫0γt‍T(τ)xtdτ‖.
This together with Lemma 2.1 implies that
(3.10)lim⁡t→0+‖xt-T(τ)xt‖=0.
Let {tn}⊂(0,1) be a sequence such that tn→0 as n→∞. Put xn:=xtn. Since {xn} is bounded, without loss of generality, we may assume that {xn} converges weakly to a point x~∈C. Noticing (3.10), we can use Lemma 2.2 to get x~∈Fix(S).
Again, from (3.1), we have
(3.11)‖xt-u‖2=‖1γt∫0γt‍T(τ)[(I-tF)xt]dτ-u‖2≤‖1γt∫0γt‍[T(τ)[(I-tF)xt]-T(τ)u]dτ‖2≤‖(I-tF)xt-(I-tF)u-tFu‖2≤τt2‖xt-u‖2+t2‖Fu‖2+2t〈(I-tF)u-(I-tF)xt,Fu〉≤τt‖xt-u‖2+t2‖Fu‖2+2t〈u-xt,Fu〉+2t2〈Fxt-Fu,Fu〉≤τt‖xt-u‖2+t2‖Fu‖2+2t〈u-xt,Fu〉+2κt2‖xt-u‖‖Fu‖.
Therefore,
(3.12)‖xt-u‖2≤t21-τt‖Fu‖2+2t1-τt〈u-xt,Fu〉+2t2κ1-τt‖xt-u‖‖Fu‖.
It follows that
(3.13)‖xn-x~‖2≤tn21-τtn‖Fx~‖2+2tn1-τtn〈x~-xn,Fx~〉+2tn2κ1-τtn‖xn-x~‖‖Fx~‖.
Thus, xn⇀x~ implies that xn→x~.
Again, from (3.12), we obtain
(3.14)‖xn-u‖2≤tn21-τtn‖Fu‖2+2tn1-τtn〈u-xn,Fu〉+2tn2κ1-τtn‖xn-u‖‖Fu‖.
It is clear that limn→∞(tn2/(1-τtn))=0, limn→∞(2tn/(1-τtn))=2/η, and limn→∞(2tn2κ/(1-τtn))=0. We deduce immediately from (3.14) that
(3.15)〈Fu,x~-u〉≤0,
which is equivalent to its dual variational inequality
(3.16)〈Fx~,x~-u〉≤0.
That is, x~∈Fix(S) is a solution of the variational inequality (3.2).
Suppose thatx*∈Fix(S) and x~∈Fix(S) both are solutions to the variational inequality (3.2); then
(3.17)〈Fx*,x*-x~〉≤0,〈Fx~,x~-x*〉≤0.
Adding up (3.17) and the last inequality yields
(3.18)〈Fx*-Fx~,x*-x~〉≤0.
The strong monotonicity ofF implies that x*=x~ and the uniqueness is proved. Later, we use x*∈Fix(S) to denote the unique solution of (3.2).
Therefore,x~=x* by uniqueness. In a nutshell, we have shown that each cluster point of {xt}(t→0) equals x*. Hence xt→x* as t→0. This completes the proof.Next we introduce an explicit algorithm for finding a solution of the variational inequality (3.2).Algorithm 3.2.
For givenx0∈C arbitrarily, define a sequence {xn} iteratively by
(3.19)yn=xn-λnF(xn),xn+1=(1-αn)yn+αn1tn∫0tn‍T(τ)yndτ,n≥0,
where {λn} and {tn} are sequences in (0,∞) and {αn} is a sequence in [0,1].Theorem 3.33.3.
LetH be a real Hilbert space. Let F be a κ-Lipschitzian and η-strongly monotone operator on H with 0<η≤κ. Let S={T(τ)∣0≤τ<∞}:H→H be a nonexpansive semigroup with Fix(S)≠∅. Assume that(i)
limsupn→∞λn<η/κ2 and ∑n=1∞‍λn=∞,(ii)
limn→∞tn=∞ and limn→∞(tn+1/tn)=1,(iii)
0<γ≤liminfn→∞αn≤limsupn→∞αn<1, for some γ∈(0,1).
Then, the sequences{xn} and {yn} generated by (3.19) converge strongly to x*∈Fix(S) if and only if λnF(xn)→0, where x* solves the variational inequality (3.2).Proof.
The necessity is obvious. We only need to prove the sufficiency. Suppose thatλnF(xn)→0. First, we show that xn is bounded. In fact, letting u∈Fix(S), we have
(3.20)‖xn+1-u‖=‖(1-αn)yn+αntn∫0tn‍T(τ)yndτ-u‖=‖(1-αn)(yn-u)+αn(1tn∫0tn‍T(τ)yndτ-u)‖≤(1-αn)‖yn-u‖+αn‖1tn∫0tn‍T(τ)yndτ-u‖≤(1-αn)‖yn-u‖+αn1tn∫0tn‍‖T(τ)yn-T(τ)u‖dτ≤(1-αn)‖yn-u‖+αn‖yn-u‖=‖yn-u‖.
From condition (i), without loss of generality, we can assume that λn≤a<η/κ2 for all n. By (3.19) and Lemma 2.4, we have
(3.21)‖yn-u‖=‖xn-λnF(xn)-u‖=‖(I-λnF)xn-(I-λnF)u-λnFu‖≤τλn‖xn-u‖+λn‖Fu‖,
where τλn=1-λn(2η-λnκ2)∈(0,1).
Then, from (3.20) and (3.21), we obtain
(3.22)‖xn+1-u‖≤τλn‖xn-u‖+λn‖Fu‖=[1-(1-τλn)]‖xn-u‖+(1-τλn)λn1-τλn‖Fu‖≤max⁡{‖xn-u‖,λn‖Fu‖1-τλn}.
Since limn→∞(λn/(1-τλn))=1/η, we have by induction
(3.23)‖xn+1-u‖≤max⁡{‖x0-u‖,M1‖Fu‖},
where M1=supn{λn/(1-τλn)}<∞. Hence, {xn} is bounded. We also obtain that {yn},{T(τ)yn}, and {Fxn} are all bounded.
Definexn+1=(1-αn)xn+αnzn for all n. Observe that
(3.24)‖zn+1-zn‖=‖xn+2-(1-αn+1)xn+1αn+1-xn+1-(1-αn)xnαn‖=‖(1-αn+1)yn+1+αn+1(1/tn+1)∫0tn+1‍T(τ)yn+1dτ-(1-αn+1)xn+1αn+1-(1-αn)yn+αn(1/tn)∫0tn‍T(τ)yndτ-(1-αn)xnαn‖=‖αn+1(1/tn+1)∫0tn+1‍T(τ)yn+1dτ-(1-αn+1)λn+1F(xn+1)αn+1-αn(1/tn)∫0tn‍T(τ)yndτ-(1-αn)λnF(xn)αn‖≤1-αn+1αn+1‖λn+1F(xn+1)‖+1-αnαn‖λnF(xn)‖+‖1tn+1∫0tn+1‍T(τ)yn+1dτ-1tn∫0tn‍T(τ)yndτ‖.
Next, we estimate ∥(1/tn+1)∫0tn+1‍T(τ)yn+1dτ-(1/tn)∫0tn‍T(τ)yndτ∥. As a matter of fact, we have
(3.25)‖1tn+1∫0tn+1‍T(τ)yn+1dτ-1tn∫0tn‍T(τ)yndτ‖≤‖1tn+1∫0tn+1‍T(τ)yn+1dτ-1tn+1∫0tn+1‍T(τ)yndτ‖+‖1tn+1∫0tn+1‍T(τ)yndτ-1tn∫0tn‍T(τ)yndτ‖≤‖yn+1-yn‖+‖1tn+1∫0tn+1‍T(τ)yndτ-1tn∫0tn‍T(τ)yndτ‖≤‖yn+1-yn‖+|1tn+1-1tn|‖∫0tn‍T(τ)yndτ‖+1tn+1‖∫tntn+1‍T(τ)yndτ‖≤‖xn+1-λn+1F(xn+1)-xn+λnF(xn)‖+|tntn+1-1|M2≤‖xn+1-xn‖+‖λn+1F(xn+1)‖+‖λnF(xn)‖+M2|tntn+1-1|,
where M2=supn{2∥T(τ)yn∥}<∞. From (3.24) and (3.25), we have
(3.26)‖zn+1-zn‖≤1-γγ‖λn+1F(xn+1)‖+1-γγ‖λnF(xn)‖+‖xn+1-xn‖+‖λn+1F(xn+1)‖+‖λnF(xn)‖+M2|tntn+1-1|≤1γ‖λn+1F(xn+1)‖+1γ‖λnF(xn)‖+‖xn+1-xn‖+M2|tntn+1-1|.
Namely,
(3.27)‖zn+1-zn‖-‖xn+1-xn‖≤1γ‖λn+1F(xn+1)‖+1γ‖λnF(xn)‖+M2|tntn+1-1|.
Since λnF(xn)→0 and (tn/tn+1)-1→0, we get
(3.28)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Consequently, by Lemma 2.3, we deduce limn→∞∥zn-xn∥=0. Therefore,
(3.29)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞αn‖zn-xn‖=0.
Next, we claim thatlimn→∞∥xn-T(τ)xn∥=0. Observe that
(3.30)‖T(τ)xn-xn‖≤‖T(τ)xn-T(τ)(1tn∫0tn‍T(τ)xndτ)‖+‖1tn∫0tn‍T(τ)xndτ-xn‖+‖T(τ)(1tn∫0tn‍T(τ)xndτ)-1tn∫0tn‍T(τ)xndτ‖≤2‖1tn∫0tn‍T(τ)xndτ-xn‖+‖T(τ)(1tn∫0tn‍T(τ)xndτ)-1tn∫0tn‍T(τ)xndτ‖.
Note that
(3.31)‖xn-1tn∫0tn‍T(τ)xndτ‖≤‖xn-xn+1‖+‖xn+1-1tn∫0tn‍T(τ)xndτ‖≤‖xn-xn+1‖+(1-αn)‖yn-1tn∫0tn‍T(τ)xndτ‖+αn‖1tn∫0tn‍T(τ)xndτ-1tn∫0tn‍T(τ)yndτ‖≤‖xn-xn+1‖+(1-αn)‖yn-xn‖+(1-αn)‖xn-1tn∫0tn‍T(τ)xndτ‖+αn‖yn-xn‖≤‖xn-xn+1‖+‖yn-xn‖+(1-αn)‖xn-1tn∫0tn‍T(τ)xndτ‖.
It follows that
(3.32)‖xn-1tn∫0tn‍T(τ)xndτ‖≤1αn(‖xn-xn+1‖+‖yn-xn‖)=1αn(‖xn-xn+1‖+‖λnF(xn)‖)⟶0.
By Lemma 2.1, (3.30), and (3.32), we derive
(3.33)lim⁡n→∞‖T(τ)xn-xn‖=0.
Next, we show thatlimsupn→∞〈Fx*,x*-xn〉≤0, where x*=limn→∞xtn and xtn is defined by xtn=(1/tn)∫0tn‍T(τ)[(I-tnF)xtn]dτ. Since {xn} is bounded, there exists a subsequence {xnk} of {xn} that converges weakly to ω. It is clear that T(τ)xnk⇀ω. From Lemma 2.2, we have ω∈Fix(S). Hence, by Theorem 3.1, we have
(3.34)limsup⁡n→∞〈Fx*,x*-xn〉=lim⁡k→∞〈Fx*,x*-xnk〉=〈Fx*,x*-ω〉≤0.
Finally, we prove that{xn} converges strongly to x*∈Fix(S). From (3.19), we have
(3.35)‖xn+1-x*‖2≤(1-αn)‖yn-x*‖2+αn‖1tn∫0tn‍T(τ)yndτ-x*‖2≤(1-αn)‖yn-x*‖2+αn1tn∫0tn‍‖T(τ)yn-T(τ)x*‖2dτ≤‖yn-x*‖2=‖xn-λnF(xn)-x*‖2=‖(I-λnF)xn-(I-λnF)x*-λnFx*‖2≤τλn2‖xn-x*‖2+λn2‖F(x*)‖2+2λn〈(I-λnF)x*-(I-λnF)xn,F(x*)〉≤τλn‖xn-x*‖2+λn2‖F(x*)‖2+2λn〈x*-xn,Fx*〉+2λn〈λnFxn,Fx*〉-2λn2‖Fx*‖2≤[1-(1-τλn)]‖xn-x*‖2+2λn〈x*-xn,Fx*〉+2λn‖λnF(xn)‖‖Fx*‖-λn2‖Fx*‖2≤[1-(1-τλn)]‖xn-x*‖2+(1-τλn)[2λn1-τλn〈x*-xn,Fx*〉+2λn‖Fx*‖1-τλn‖λnFxn‖]=(1-δn)‖xn-x*‖2+δnσn,
where δn=1-τλn and σn=(2λn/1-τλn)〈x*-xn,Fx*〉+(2λn∥Fx*∥/(1-τλn))∥λnFxn∥. Obviously, we can see that  ∑n=1∞‍δn=∞ and limsupn→∞σn≤0. Hence, all conditions of Lemma 2.5 are satisfied. Therefore, we immediately deduce that the sequence {xn} converges strongly to x*∈Fix(S).
Observe that(3.36)‖yn-x*‖≤‖yn-xn‖+‖xn-x*‖≤‖λnF(xn)‖+‖xn-x*‖⟶0(n⟶∞).
Consequently, it is clear that{yn} converges strongly to x*∈Fix(S). From x*=limt→0xt and Theorem 3.1, we get that  x* is the unique solution of the variational inequality
(3.37)〈Fx*,x*-u〉≤0,∀u∈Fix⁡(S).
This completes the proof.



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*Source: 634927-2012-06-28.xml*