Proof.
Note that the conditions αn→0 and λ∈(0,2/∥A∥2). We deduce αn<1-(λ∥A∥2/2) for enough large n. Without loss of generality, we may assume that, for all n∈ℕ, αn<1-(λ∥A∥2/2), that is, λ/(1-αn)∈(0,2/∥A∥2).

Pick up any x*∈Γ. From Proposition 3.1, we have x*=PC[x*-δA*(I-PQ)Ax*] for any δ>0. Thus,
(3.4)x*=PC[x*-λ1-αnA*(I-PQ)Ax*]=PC[αnx*+(1-αn)(x*-λ1-αnA*(I-PQ)Ax*)], ∀n≥0.

From (3.3) and (3.4), we have
(3.5)∥yn-x*∥=∥PC[(1-αn)xn-λA*(I-PQ)Axn]-x*∥=∥PC[(1-αn)(xn-λ1-αnA*(I-PQ)Axn)] -PC[αnx*+(1-αn)(x*-λ1-αnA*(I-PQ)Ax*)]∥≤∥αn(-x*)+(1-αn)[(xn-λ1-αnA*(I-PQ)Axn) -(x*-λ1-αnA*(I-PQ)Ax*)]∥.
From Propositions 3.1 and 3.2, we get that I-(λ/(1-αn))A*(I-PQ)A is nonexpansive. It follows that
(3.6)∥yn-x*∥≤αn∥x*∥+(1-αn)∥(I-λ1-αnA*(I-PQ)A)xn αn∥x*∥+(1-αn)-(I-λ1-αnA*(I-PQ)A)x*∥≤αn∥x*∥+(1-αn)∥xn-x*∥.

Thus,
(3.7)∥xn+1-x*∥=∥PC[xn-λA*(I-PQ)Ayn+μ(yn-xn)]-x*∥=∥PC[(1-μ)xn+μ(yn-λμA*(I-PQ)Ayn)] -PC[(1-μ)x*+μ(x*-λμA*(I-PQ)Ax*)]∥≤(1-μ)∥xn-x*∥+μ∥(yn-λμA*(I-PQ)Ayn) (1-μ)∥xn-x*∥+μ-(x*-λμA*(I-PQ)Ax*)∥≤(1-μ)∥xn-x*∥+μ∥yn-x*∥≤(1-μ)∥xn-x*∥+μαn∥x*∥+μ(1-αn)∥xn-x*∥=(1-μαn)∥xn-x*∥+μαn∥x*∥≤max {∥x*∥,∥x0-x*∥}.
Hence, {xn} is bounded.

Set S=2PC-I. Note that S is nonexpansive. Thus, we can rewrite xn+1 in (3.3) as
(3.8)xn+1=I+S2[(1-μ)xn+μ(yn-λμA*(I-PQ)Ayn)]=1-μ2xn+μ2(yn-λμA*(I-PQ)Ayn)+12S[(1-μ)xn+μ(yn-λμA*(I-PQ)Ayn)]=1-μ2xn+1+μ2zn,
where
(3.9)zn=μ(yn-(λ/μ)A*(I-PQ)Ayn)+S[(1-μ)xn+μ(yn-(λ/μ)A*(I-PQ)Ayn)]1+μ.
It follows that
(3.10)∥zn+1-zn∥=∥μ(yn+1-(λ/μ)A*(I-PQ)Ayn+1)1+μ+S[(1-μ)xn+1+μ(yn+1-(λ/μ)A*(I-PQ)Ayn+1)]1+μ-μ(yn-(λ/μ)A*(I-PQ)Ayn)1+μ +S[(1-μ)xn+μ(yn-(λ/μ)A*(I-PQ)Ayn)]1+μ∥≤μ1+μ∥(yn+1-λμA*(I-PQ)Ayn+1)-(yn-λμA*(I-PQ)Ayn)∥+11+μ∥S[(1-μ)xn+1+μ(yn+1-λμA*(I-PQ)Ayn+1)] -S[(1-μ)xn+μ(yn-λμA*(I-PQ)Ayn)]∥≤μ1+μ∥yn+1-yn∥+11+μ∥(1-μ)(xn+1-xn)+μ[(yn+1-λμA*(I-PQ)Ayn+1) (1-μ)(xn+1-xn)+μ -(yn-λμA*(I-PQ)Ayn)]∥≤μ1+μ∥yn+1-yn∥+1-μ1+μ∥xn+1-xn∥+μ1+μ∥yn+1-yn∥≤2μ1+μ∥yn+1-yn∥+1-μ1+μ∥xn+1-xn∥.
By (3.3), we have
(3.11)∥yn+1-yn∥=∥PC[(1-αn+1)xn+1-λA*(I-PQ)Axn+1]-PC[(1-αn)xn-λA*(I-PQ)Axn]∥≤∥[(1-αn+1)xn+1-λA*(I-PQ)Axn+1]-[(1-αn)xn-λA*(I-PQ)Axn]∥≤∥[xn+1-λA*(I-PQ)Axn+1]-[xn-λA*(I-PQ)Axn]∥+αn+1∥xn+1∥+αn∥xn∥≤∥xn+1-xn∥+αn+1∥xn+1∥+αn∥xn∥.
Hence, we deduce
(3.12)∥zn+1-zn∥≤2μ1+μ∥xn+1-xn∥+1-μ1+μ∥xn+1-xn∥+αn+1∥xn+1∥+αn∥xn∥=∥xn+1-xn∥+αn+1∥xn+1∥+αn∥xn∥.
Therefore,
(3.13)limsup n→∞ (∥zn+1-zn∥-∥xn+1-xn∥)≤0.
By Lemma 2.2, we obtain
(3.14)lim n→∞ ∥zn-xn∥=0.
Hence,
(3.15)lim n→∞ ∥xn+1-xn∥=lim n→∞ 1+μ2∥zn-xn∥=0.
From (3.5), (3.7), Proposition 3.2, and the convexity of the norm, we deduce
(3.16)∥xn+1-x*∥2 ≤(1-μ)∥xn-x*∥2+μ∥yn-x*∥2 ≤(1-μ)∥xn-x*∥2+μ∥αn(-x*)+(1-αn)[(xn-λ1-αnA*(I-PQ)Axn) ≤(1-μ)∥xn-x*∥2+μ -(x*-λ1-αnA*(I-PQ)Ax*)]∥2 ≤μ[∥(I-λ1-αnA*(I-PQ)A)xn-(I-λ1-αnA*(I-PQ)A)x*∥2αn∥x*∥2+(1-αn) ×∥(I-λ1-αnA*(I-PQ)A)xn-(I-λ1-αnA*(I-PQ)A)x*∥2] +(1-μ)∥xn-x*∥2 ≤(1-αn)μ ×[∥xn-x*∥2+λ1-αn(λ1-αn-2∥A∥2)∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥2] +(1-μ)∥xn-x*∥2+αnμ∥x*∥2 ≤αn∥x*∥2+∥xn-x*∥2 +μa(b1-αn-2∥A∥2)∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥2.
Therefore, we have
(3.17)μa(2∥A∥2-b1-αn)∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥2 ≤αn∥x*∥2+∥xn-x*∥2-∥xn+1-x*∥2 ≤αn∥x*∥2+(∥xn-x*∥+∥xn+1-x*∥)×∥xn-xn+1∥.
Since αn→0 and ∥xn-xn+1∥→0 as n→∞, we obtain liminf n→∞μa((2/∥A∥2)- (b/(1-αn)))>0. Thus, we have
(3.18)lim n→∞ ∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥=0.

By the property (b) of the metric projection PC, we have
(3.19)∥yn-x*∥2 =∥PC[(1-αn)xn-λA*(I-PQ)Axn]-PC[x*-λA*(I-PQ)Ax*]∥2 ≤〈(1-αn)xn-λA*(I-PQ)Axn-(x*-λA*(I-PQ)Ax*),yn-x*〉 =12{∥xn-λA*(I-PQ)Axn-(x*-λA*(I-PQ)Ax*)-αnxn∥2+∥yn-x*∥2 -∥(1-αn)xn-λA*(I-PQ)Axn-(x*-λA*(I-PQ)Ax*)-(yn-x*)∥2} ≤12{∥(xn-λA*(I-PQ)Axn)-(x*-λA*(I-PQ)Ax*)∥2 +2αn∥xn∥∥xn-λA*(I-PQ)Axn-(x*-λA*(I-PQ)Ax*)-αnxn∥ +∥yn-x*∥2-∥(xn-yn)-λ(A*(I-PQ)Axn-A*(I-PQ)Ax*)-αnxn∥2} ≤12{∥(xn-λA*(I-PQ)Axn)-(x*-λA*(I-PQ)Ax*)∥2+αnM+∥yn-x*∥2 -∥(xn-yn)-λ(A*(I-PQ)Axn-A*(I-PQ)Ax*)-αnxn∥2} ≤12{∥xn-x*∥2+αnM+∥yn-x*∥2-∥xn-yn∥2+2αn〈xn,xn-yn〉 +2λ〈xn-yn,A*(I-PQ)Axn-A*(I-PQ)Ax*〉 -∥λ(A*(I-PQ)Axn-A*(I-PQ)Ax*)+αnxn∥2} ≤12{∥xn-x*∥2+αnM+∥yn-x*∥2-∥xn-yn∥2+2αn∥xn∥∥xn-yn∥ +2λ∥xn-yn∥∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥∥xn-x*∥2},
where M>0 is some constant such that
(3.20)sup n {2∥xn∥∥xn-λA*(I-PQ)Axn-(x*-λA*(I-PQ)Ax*)-αnxn∥}≤M.
It follows that
(3.21)∥yn-x*∥2≤∥xn-x*∥2+αnM-∥xn-yn∥2+2αn∥xn∥∥xn-yn∥+2λ∥xn-yn∥∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥,
and hence
(3.22)∥xn+1-x*∥2≤(1-μ)∥xn-x*∥2+μ∥yn-x*∥2≤∥xn-x*∥2+αnM-μ∥xn-yn∥2+2αn∥xn∥∥xn-yn∥+2λ∥xn-yn∥∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥,
which implies that
(3.23)μ∥xn-yn∥2 ≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+αnM+2αn∥xn∥∥xn-yn∥ +2λ∥xn-yn∥∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥.
Since αn→0, ∥xn-xn+1∥→0, and ∥A*(I-PQ)Axn-A*(I-PQ)Ax*∥→0, we derive
(3.24)lim n→∞ ∥xn-yn∥=0.

Next we show that
(3.25)limsup n→∞ 〈x~,x~-yn〉≤0,
where x~=PΓ(0). To show it, we choose a subsequence {yni} of {yn} such that
(3.26)limsup n→∞ 〈x~,x~-yn〉=lim i→∞ 〈x~,x~-yni〉.
Since {xn} is bounded, we have that {yn} is also bounded. As {yni} is bounded, we have that a subsequence {ynij} of {yni} converges weakly to z.

Next we show that z∈Γ. We define a mapping T by
(3.27)Tv={A*(I-PQ)Av+NCv,v∈C,∅,v∉C.
Then T is maximal monotone. Let (v,w)∈G(T). Since w-A*(I-PQ)Av∈NCv and yn∈C, we have 〈v-yn,w-A*(I-PQ)Av〉≥0. On the other hand, from yn=PC[(1-αn)xn-λA*(I-PQ)Axn], we have
(3.28)〈v-yn,yn-(1-αn)xn+λA*(I-PQ)Axn〉≥0,
that is,
(3.29)〈v-yn,yn-xnλ+Axn+αnλxn〉≥0.
Therefore, we have
(3.30)〈v-yni,w〉≥〈v-yni,Av〉≥〈v-yni,A*(I-PQ)Av〉-〈v-yni,yni-xniλ+A*(I-PQ)Axni+αniλxni〉=〈v-yni,A*(I-PQ)Av-A*(I-PQ)Axni-yni-xniλ-αniλxni〉=〈v-yni,A*(I-PQ)Av-A*(I-PQ)Ayni〉-〈v-yni,yni-xniλ+αniλxni〉+〈v-yni,A*(I-PQ)Ayni-A*(I-PQ)Axni〉≥〈v-yni,A*(I-PQ)Ayni-A*(I-PQ)Axni〉-〈v-yni,yni-xniλ+αniλxni〉.
Noting that αni→0, ∥yni-xni∥→0, and A*(I-PQ)A is Lipschitz continuous, we deduce from above
(3.31)〈v-z,w〉≥0.
Since T is maximal monotone, we have z∈T-1(0) and hence z∈VI (C,A*(I-PQ)A)=Γ. Therefore,
(3.32)limsup n→∞〈x~,x~-yn〉=lim i→∞〈x~,x~-yni〉=〈x~,x~-z〉≤0.
By the property (b) of metric projection PC, we have
(3.33)∥yn-x~∥2 =∥PC[(1-αn)(xn-λ1-αnA*(I-PQ)Axn)] -PC[αnx~+(1-αn)(x~-λ1-αnA*(I-PQ)Ax~)]∥2 ≤〈αn(-x~)+(1-αn)[(xn-λ1-αnA*(I-PQ)Axn)-(x~-λ1-αnA*(I-PQ)Ax~)],yn-x~〉 ≤αn〈x~,x~-yn〉+(1-αn) ×∥(xn-λ1-αnA*(I-PQ)Axn)-(x~-λ1-αnA*(I-PQ)Ax~)∥∥yn-x~∥ ≤αn〈x~,x~-yn〉+(1-αn)∥xn-x~∥∥yn-x~∥ ≤αn〈x~,x~-yn〉+1-αn2(∥xn-x~∥2+∥yn-x~∥2).
Hence
(3.34)∥yn-x~∥2≤(1-αn)∥xn-x~∥2+2αn〈x~,x~-yn〉.
Therefore,
(3.35)∥xn+1-x~∥2≤(1-μ)∥xn-x~∥2+μ∥yn-x~∥2≤(1-μαn)∥xn-x~∥2+2μαn〈x~,x~-yn〉.
We apply Lemma 2.3 to the last inequality to deduce that xn→x~. This completes the proof.