In this section, we will introduce and analyze an iterative algorithm for finding a solution of the THVI (24) with constraints of several problems: the finitely many GMEPs, the finitely many VIPs, GSVI (11), and CMP (14) in a real Hilbert space. This algorithm is based on the Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the averaged mapping approach to the GPA. We prove the strong convergence of the proposed algorithm to a unique solution of THVI (24) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with the same constraints.
Proof.
Since ∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism; see [44] (see also [31]). By Proposition 4(ii) we know that for θ>0, θ∇f is (1/θL)-ism. So by Proposition 4(iii) we deduce that I-θ∇f is (θL/2)-averaged. Now since the projection PC is (1/2)-averaged, it is easy to see from Proposition 5(iv) that the composite PC(I-θ∇f) is ((2+θL)/4)-averaged for θ∈(0,2/L). Hence we obtain that for each n≥1, PC(I-θn∇f) is ((2+θnL)/4)-averaged for each θn∈(0,2/L). Therefore, we can write
(55)PC(I-θn∇f)=2-θnL4I+2+θnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn≔sn(θn)=(2-θnL)/4∈(0,1/2) for each θn∈(0,2/L). It is clear that
(56)θn⟶2L⟺sn⟶0.
Put
(57)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1) ×(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn
for all k∈{1,2,…,M} and n≥1,
(58)Λni=PC(I-λi,nBi) ×PC(I-λi-1,nBi-1)⋯PC(I-λ1,nB1)
for all i∈{1,2,…,N}, Δn0=I and Λn0=I, where I is the identity mapping on H. Then we have un=ΔnMxn and vn=ΛnNun.
We divide the rest of the proof into several steps.
Step
1. Let us show that {xn} is bounded. Indeed, taking into account the assumption Ξ≠∅ in Problem 2, we know that Ω≠∅. By (H4), we may assume, without loss of generality, that αn≤sn for all n≥1. Taking p∈Ω arbitrarily. Then from (30) and Proposition 6(ii) we have
(59)∥un-p∥ =∥TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn -TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1p∥ ≤∥(I-rM,nBM)ΔnM-1xn-(I-rM,nBM)ΔnM-1p∥ ≤∥ΔnM-1xn-ΔnM-1p∥ ⋮ ≤∥Δn0xn-Δn0p∥ =∥xn-p∥.
Similarly, we have
(60)∥vn-p∥ =∥PC(I-λN,nAN)ΛnN-1un-PC(I-λN,nAN)ΛnN-1p∥ ≤∥(I-λN,nAN)ΛnN-1un-(I-λN,nAN)ΛnN-1p∥ ≤∥ΛnN-1un-ΛnN-1p∥ ⋮ ≤∥Λn0un-Λn0p∥ =∥un-p∥.
Combining (59) and (60), we have
(61)∥vn-p∥≤∥xn-p∥.
Since p=Gp=PC(I-ν1F1)PC(I-ν2F2)p,Fj is ζj-inverse-strongly monotone for j=1,2, and 0≤νj≤2ζj for j=1,2, we deduce that, for any n≥1,
(62)∥Gvn-p∥2 =∥PC(I-ν1F1)PC(I-ν2F2)vn -PC(I-ν1F1)PC(I-ν2F2)p∥2 ≤∥(I-ν1F1)PC(I-ν2F2)vn -(I-ν1F1)PC(I-ν2F2)p∥2 =∥[PC(I-ν2F2)vn-PC(I-ν2F2)p] -ν1[F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p]∥2 ≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2 +ν1(ν1-2ζ1)∥F1PC(I-ν2F2)vn -F1PC(I-ν2F2)p∥2 ≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2 ≤∥(I-ν2F2)vn-(I-ν2F2)p∥2 =∥(vn-p)-ν2(F2vn-F2p)∥2 ≤∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2 ≤∥vn-p∥2.
Utilizing Lemma 15 and the relation αn≤sn, from (54), (61), and (62), we obtain that
(63)∥yn-p∥ =∥αnγ(Svn-Sp)+(I-αnμF)WnGvn -(I-αnμF)p+αn(γS-μF)p∥ ≤αnγ∥Svn-Sp∥ +∥(I-αnμF)WnGvn-(I-αnμF)p∥ +αn∥(γS-μF)p∥ ≤αnγ∥vn-p∥+(1-αnτ)∥Gvn-p∥ +αn∥(γS-μF)p∥ ≤αnγ∥vn-p∥+(1-αnτ)∥vn-p∥ +αn∥(γS-μF)p∥ ≤αnγ∥xn-p∥+(1-αnτ)∥xn-p∥ +αn∥(γS-μF)p∥ =(1-αn(τ-γ))∥xn-p∥+αn∥(γS-μF)p∥ ≤∥xn-p∥+αn∥(γS-μF)p∥ ≤∥xn-p∥+sn∥(γS-μF)p∥,
and hence
(64)∥xn+1-p∥ =∥snγ(Vxn-Vp)+(I-snμF)Tnyn -(I-snμF)p+sn(γS-μF)p∥ ≤snγ∥Vxn-Vp∥ +∥(I-snμF)Tnyn-(I-snμF)p∥ +sn∥(γS-μF)p∥ ≤snγl∥xn-p∥+(1-snτ)∥yn-p∥ +sn∥(γS-μF)p∥ ≤snγl∥xn-p∥ +(1-snτ)(∥xn-p∥+sn∥(γS-μF)p∥) +sn∥(γS-μF)p∥ ≤snγl∥xn-p∥+(1-snτ)∥xn-p∥ +sn∥(γS-μF)p∥+sn∥(γS-μF)p∥ =(1-sn(τ-γl))∥xn-p∥ +sn(∥(γS-μF)p∥+∥(γS-μF)p∥) =(1-sn(τ-γl))∥xn-p∥ +sn(τ-γl)∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl ≤max{∥xn-p∥,∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl}.
By induction, we get
(65)∥xn-p∥ ≤max{∥x1-p∥,∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl}, DQGQQQOPKRRRRRRRRDARRRRBMMMMM∀n≥1.
Hence {xn} is bounded and so are the sequences {un},{vn},{yn}.
Step
2. Let us show that limn→∞(∥xn+1-xn∥/αn)=0.
Indeed, taking into account the (1/L)-inverse-strong monotonicity of ∇f, we know that PC(I-θn∇f) is nonexpansive for θn∈(0,2/L). Hence it follows that for any given p∈Ω,
(66)∥PC(I-θn+1∇f)yn∥ ≤∥PC(I-θn+1∇f)yn-p∥+∥p∥ =∥PC(I-θn+1∇f)yn-PC(I-θn+1∇f)p∥+∥p∥ ≤∥yn-p∥+∥p∥ ≤∥yn∥+2∥p∥.
This together with the boundedness of {yn} implies that {PC(I-λn+1∇f)yn} is bounded. Also, observe that
(67)∥Tn+1yn-Tnyn∥ =∥4PC(I-θn+1∇f)-(2-θn+1L)I2+θn+1Lyn -4PC(I-θn∇f)-(2-θnL)I2+θnLyn∥ ≤∥4PC(I-θn+1∇f)2+θn+1Lyn-4PC(I-θn∇f)2+θnLyn∥ +∥2-θnL2+θnLyn-2-θn+1L2+θn+1Lyn∥ =∥×((2+θn+1L)(2+θnL))-1∥(4(2+θnL)PC(I-θn+1∇f)yn -4(2+θn+1L)PC(I-θn∇f)Gyn) ×((2+θn+1L)(2+θnL))-1∥ +4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥ =∥×((2+θn+1L)(2+θnL))-1∥((PC(I-θn+1∇f)yn-PC(I-θn∇f)yn))4L(θn-θn+1)PC(I-θn+1∇f)yn+4(2+θn+1L) ×(PC(I-θn+1∇f)yn-PC(I-θn∇f)yn)) ×((2+θn+1L)(2+θnL))-1∥ +4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥ ≤4L|θn-θn+1|∥PC(I-θn+1∇f)yn∥(2+θn+1L)(2+θnL) +(((2+θn+1L)(2+θnL))-1)(×∥PC(I-θn+1∇f)yn-PC(I-θn∇f)yn∥)4(2+θn+1L) ×∥PC(I-θn+1∇f)yn-PC(I-θn∇f)yn∥) ×((2+θn+1L)(2+θnL))-1) +4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥ ≤|θn+1-θn| ×[L∥PC(I-θn+1∇f)yn∥+4∥∇f(yn)∥+L∥yn∥] ≤M~|θn+1-θn|,
where supn≥1{L∥PC(I-θn+1∇f)yn∥+4∥∇f(yn)∥+L∥yn∥}≤M~ for some M~>0. So, by (67), we have that
(68)∥Tn+1yn+1-Tnyn∥ ≤∥Tn+1yn+1-Tn+1yn∥+∥Tn+1yn-Tnyn∥ ≤∥yn+1-yn∥+M~|θn+1-θn| ≤∥yn+1-yn∥+4M~L|sn+1-sn|.
Note that
(69)∥vn+1-vn∥ =∥Λn+1Nun+1-ΛnNun∥ =∥PC(I-λN,n+1AN)Λn+1N-1un+1 -PC(I-λN,nAN)ΛnN-1un∥ ≤∥PC(I-λN,n+1AN)Λn+1N-1un+1 -PC(I-λN,nAN)Λn+1N-1un+1∥ +∥PC(I-λN,nAN)Λn+1N-1un+1 -PC(I-λN,nAN)ΛnN-1un∥ ≤∥(I-λN,n+1AN)Λn+1N-1un+1 -(I-λN,nAN)Λn+1N-1un+1∥ +∥(I-λN,nAN)Λn+1N-1un+1 -(I-λN,nAN)ΛnN-1un∥ ≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥ +∥Λn+1N-1un+1-ΛnN-1un∥ ≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥ +|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥ +∥Λn+1N-2un+1-ΛnN-2un∥ ⋮ ≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥ +|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥ +⋯+|λ1,n+1-λ1,n|∥A1Λn+10un+1∥ +∥Λn+10un+1-Λn0un∥ ≤M~0∑i=1N|λi,n+1-λi,n|+∥un+1-un∥,
where supn≥1{∑i=1N∥AiΛn+1i-1un+1∥}≤M~0 for some M~0>0. Also, utilizing Proposition 6(ii), (v) we deduce that
(70)∥un+1-un∥ =∥Δn+1Mxn+1-ΔnMxn∥ =∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥ ≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥ +∥TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1 -TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥ ≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥ +∥TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥ +∥(I-rM,nBM)Δn+1M-1xn+1 -(I-rM,nBM)ΔnM-1xn∥ ≤|rM,n+1-rM,n|rM,n+1 ×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -(I-rM,n+1BM)Δn+1M-1xn+1∥ +|rM,n+1-rM,n|∥BMΔn+1M-1xn+1∥ +∥Δn+1M-1xn+1-ΔnM-1xn∥ =|rM,n+1-rM,n| ×[∥BMΔn+1M-1xn+1∥+1rM,n+1 ×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -(I-rM,n+1BM)Δn+1M-1xn+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥[∥BMΔn+1M-1xn+1∥+1rM,n+1] +∥Δn+1M-1xn+1-ΔnM-1xn∥ ⋮ ≤|rM,n+1-rM,n| ×[∥BMΔn+1M-1xn+1∥+1rM,n+1 ×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1 -(I-rM,n+1BM)Δn+1M-1xn+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥[∥BMΔn+1M-1xn+1∥+1rM,n+1] +⋯+|r1,n+1-r1,n| ×[∥B1Δn+10xn+1∥+1r1,n+1 ×∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1 -(I-r1,n+1B1)Δn+10xn+1∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1∥[∥B1Δn+10xn+1∥+1r1,n+1] +∥Δn+10xn+1-Δn0xn∥ ≤M~1∑k=1M|rk,n+1-rk,n|+∥xn+1-xn∥,
where M~1>0 is a constant such that for each n≥1(71)∑k=1M[∥BkΔn+1k-1xn+1∥+1rk,n+1 ×∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1 -(I-rk,n+1Bk)Δn+1k-1xn+1∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1∥[∥BkΔn+1k-1xn+1∥+1rk,n+1]≤M~1.
Also, from (54) we have
(72)yn+1=αn+1γSvn+1+(I-αn+1μF)Wn+1Gvn+1,yn=αnγSvn+(I-αnμF)WnGvn, ∀n≥1.
Simple calculation shows that
(73)yn+1-yn=αn+1γ(Svn+1-Svn) +(αn+1-αn)(γSvn-μFWnGvn) +(I-αn+1μF)Wn+1Gvn+1 -(I-αn+1μF)WnGvn.
In the meantime, from (41), since Wn, Sn, and Un,i are all nonexpansive, we have
(74)∥Wn+1Gvn+1-WnGvn∥ ≤∥Wn+1Gvn+1-Wn+1Gvn∥ +∥Wn+1Gvn-WnGvn∥ ≤∥vn+1-vn∥+∥Wn+1Gvn-WnGvn∥ =∥vn+1-vn∥ +∥λ1T1Un+1,2Gvn-λ1T1Un,2Gvn∥ ≤∥vn+1-vn∥+λ1∥Un+1,2Gvn-Un,2Gvn∥ =∥vn+1-vn∥ +λ1∥λ2T2Un+1,3Gvn-λ2T2Un,3Gvn∥ ≤∥vn+1-vn∥+λ1λ2∥Un+1,3Gvn-Un,3Gvn∥ ⋮ ≤∥vn+1-vn∥+λ1λ2⋯λn ×∥Un+1,n+1Gvn-Un,n+1Gvn∥ ≤∥vn+1-vn∥+M~2∏i=1nλi,
where M~2 is a constant such that ∥Un+1,n+1Gvn∥+∥Un,n+1Gvn∥≤M~2 for each n≥1. Therefore, by utilizing Lemma 15, from (69)–(74) and {λn}⊂(0,b]⊂(0,1) it follows that
(75)∥yn+1-yn∥ ≤αn+1γ∥Svn+1-Svn∥ +|αn+1-αn|∥γSvn-μFWnGvn∥ +∥(I-αn+1μF)Wn+1Gvn+1 -(I-αn+1μF)WnGvn∥ ≤αn+1γ∥vn+1-vn∥+|αn+1-αn| ×∥γSvn-μFWnGvn∥ +(1-αn+1τ)∥Wn+1Gvn+1-WnGvn∥ ≤αn+1γ∥vn+1-vn∥+|αn+1-αn| ×∥γSvn-μFWnGvn∥ +(1-αn+1τ)[∥vn+1-vn∥+M~2∏i=1nλi] ≤(1-αn+1(τ-γ))∥vn+1-vn∥ +|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi ≤(1-αn+1(τ-γ)) ×[M~0∑i=1N|λi,n+1-λi,n|+∥un+1-un∥] +|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi ≤(1-αn+1(τ-γ)) ×[M~0∑i=1N|λi,n+1-λi,n| +M~1∑k=1M|rk,n+1-rk,n|+∥xn+1-xn∥] +|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi ≤(1-αn+1(τ-γ))∥xn+1-xn∥ +M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n| +|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi.
On the other hand, from (54) we have
(76)xn+2=sn+1γVxn+1+(I-sn+1μF)Tn+1yn+1,xn+1=snγVxn+(I-snμF)Tnyn, ∀n≥1.
The simple calculations show that
(77)xn+2-xn+1=(I-sn+1μF)Tn+1yn+1 -(I-sn+1μF)Tnyn +(sn+1-sn)(γVxn-μFTnyn) +sn+1γ(Vxn+1-Vxn).
Utilizing Lemma 15 we deduce from (68), (75), and (77) that
(78)∥xn+2-xn+1∥ ≤∥(I-sn+1μF)Tn+1yn+1-(I-sn+1μF)Tnyn∥ +|sn+1-sn|∥γVxn-μFTnyn∥ +sn+1γ∥Vxn+1-Vxn∥ ≤(1-sn+1τ)∥Tn+1yn+1-Tnyn∥ +|sn+1-sn|∥γVxn-μFTnyn∥ +sn+1γl∥xn+1-xn∥ ≤(1-sn+1τ)[∥yn+1-yn∥+4M~L|sn+1-sn|] +|sn+1-sn|∥γVxn-μFTnyn∥ +sn+1γl∥xn+1-xn∥ ≤(1-sn+1τ) ×[M~2∏i=1nλi+4M~L|sn+1-sn|](1-αn+1(τ-γ))∥xn+1-xn∥ +M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n| +|αn+1-αn|∥γSvn-μFWnGvn∥ +M~2∏i=1nλi+4M~L|sn+1-sn|] +|sn+1-sn|∥γVxn-μFTnyn∥ +sn+1γl∥xn+1-xn∥ ≤(1-sn+1τ)∥xn+1-xn∥ +M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n| +|αn+1-αn|∥γSvn-μFWnGvn∥ +M~2bn+(4M~L+∥γVxn-μFTnyn∥)|sn+1-sn| +sn+1γl∥xn+1-xn∥ =(1-sn+1(τ-γl))∥xn+1-xn∥ +M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n| +|αn+1-αn|∥γSvn-μFWnGvn∥ +M~2bn+(4M~L+∥γVxn-μFTnyn∥)|sn+1-sn| ≤(1-sn+1(τ-γl))∥xn+1-xn∥ +M~3(∑i=1N|λi,n+1-λi,n|+∑k=1M|rk,n+1-rk,n| +|αn+1-αn|+bn+|sn+1-sn|(∑i=1N|λi,n+1-λi,n|+∑k=1M|rk,n+1-rk,n|),
where M~3>0 is a constant such that for each n≥1(79)M~0+M~1+M~2+4M~L+∥γVxn-μFTnyn∥ +∥γSvn-μFWnGvn∥≤M~3.
Therefore,
(80)∥xn+1-xn∥αn ≤(1-sn(τ-γl))∥xn-xn-1∥αn +M~3(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn +|αn-αn-1|αn+bn-1αn+|sn-sn-1|αn(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn) =(1-sn(τ-γl))∥xn-xn-1∥αn-1 +(1-sn(τ-γl))∥xn-xn-1∥(1αn-1αn-1) +M~3(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn +|αn-αn-1|αn+bn-1αn+|sn-sn-1|αn(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn) ≤(1-(τ-γl)sn)∥xn-xn-1∥αn-1+(τ-γl)sn·1τ-γl ×{|αn-αn-1|snαn+bn-1snαn+|sn-sn-1|snαn(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn)}∥xn-xn-1∥1sn|1αn-1αn-1| +M~3(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn +|αn-αn-1|snαn+bn-1snαn+|sn-sn-1|snαn(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn)} ≤(1-(τ-γl)sn)∥xn-xn-1∥αn-1+(τ-γl)sn·M~4τ-γl ×{1sn|1-αn-1αn|+1αn|1-sn-1sn|+bn-1snαn}1sn|1αn-1αn-1| +∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn +1sn|1-αn-1αn|+1αn|1-sn-1sn|+bn-1snαn},
where M~3+∥xn-xn-1∥≤M~4, for all n>1 for some M~4>0. From (H2), (H3), (H5), and (H6), it follows that ∑n=1∞(τ-γl)sn=∞ and
(81)limn→∞M~4τ-γl{1sn|1αn-1αn-1|+∑i=1N|λi,n-λi,n-1|snαn +∑k=1M|rk,n-rk,n-1|snαn+1sn|1-αn-1αn| +1αn|1-sn-1sn|+bn-1snαn{1sn|1αn-1αn-1|+∑i=1N|λi,n-λi,n-1|snαn}=0.
Thus, applying Lemma 16 to (80), we immediately conclude that
(82)limn→∞∥xn+1-xn∥αn=0.
So, from limn→∞αn=0 it follows that
(83)limn→∞∥xn+1-xn∥=0.
Step
3. We prove that ωw(xn)⊂Ω provided limn→∞∥xn-yn∥=0.
Indeed, first of all, let us show that ∥yn-PC(I-(2/L)∇f)yn∥→0, ∥xn-un∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0, and ∥vn-Wvn∥→0 as n→∞. As a matter of fact, utilizing Lemmas 7 and 15 we obtain from (54), (61), and (62) that
(84)∥yn-p∥2 =∥αnγ(Svn-Sp)+(I-αnμF)WnGvn -(I-αnμF)p+αn(γS-μF)p∥2 ≤∥αnγ(Svn-Sp) +(I-αnμF)WnGvn-(I-αnμF)p∥2 +2αn〈(γS-μF)p,yn-p〉 ≤[αnγ∥Svn-Sp∥ +∥(I-αnμF)WnGvn-(I-αnμF)p∥]2 +2αn〈(γV-μF)p,yn-p〉 ≤[αnγ∥vn-p∥+(1-αnτ)∥Gvn-p∥]2 +2αn〈(γS-μF)p,yn-p〉 ≤[αnτ∥vn-p∥+(1-αnτ)∥Gvn-p∥]2 +2αn〈(γS-μF)p,yn-p〉 ≤αnτ∥vn-p∥2+(1-αnτ)∥Gvn-p∥2 +2αn〈(γS-μF)p,yn-p〉 ≤αnτ∥vn-p∥2+(1-αnτ)∥vn-p∥2 +2αn〈(γS-μF)p,yn-p〉 =∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉 ≤∥xn-p∥2+2αn〈(γS-μF)p,yn-p〉.
Note that xn+1=snγVxn+(I-snμF)Tnyn. Hence we have
(85)xn+1-yn=sn(γVxn-μFTnyn)+Tnyn-yn,
which yields
(86)∥Tnyn-yn∥ ≤∥xn+1-yn-sn(γVxn-μFTnyn)∥ ≤∥xn+1-yn∥+sn∥γVxn-μFTnyn∥ ≤∥xn+1-xn∥+∥xn-yn∥+sn∥γVxn-μFTnyn∥.
Since limn→∞sn=0 and limn→∞∥xn-xn+1∥=0, from the assumption limn→∞∥xn-yn∥=0 and the boundedness of {xn}, {yn}, we obtain
(87)limn→∞∥yn-Tnyn∥=0.
It is clear that
(88)∥PC(I-θn∇f)yn-yn∥=∥snyn+(1-sn)Tnyn-yn∥=(1-sn)∥Tnyn-yn∥≤∥Tnyn-yn∥,
where sn=((2-θnL)/4)∈(0,1/2) for each θn∈(0,2/L). Hence we have
(89)∥PC(I-2L∇f)yn-yn∥ ≤∥PC(I-2L∇f)yn-PC(I-θn∇f)yn∥ +∥PC(I-θn∇f)yn-yn∥ ≤∥(I-2L∇f)yn-(I-θn∇f)yn∥ +∥PC(I-θn∇f)yn-yn∥ ≤(2L-θn)∥∇f(yn)∥+∥Tnyn-yn∥.
From the boundedness of {yn}, sn→0 (⇔θn→2/L) and ∥Tnyn-yn∥→0 (due to (87)), it follows that
(90)limn→∞∥yn-PC(I-2L∇f)yn∥=0.
Also, from (30) it follows that for all i∈{1,2,…,N} and k∈{1,2,…,M}(91)∥vn-p∥2=∥ΛnNun-p∥2≤∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2≤∥Λni-1un-p∥2 +λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2,∥un-p∥2=∥ΔnMxn-p∥2≤∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn -Trk,n(Θk,φk)(I-rk,nBk)p∥2≤∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2≤∥Δnk-1xn-p∥2 +rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2 +rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2.
So, from (84) and (91) it follows that
(92)∥yn-p∥2 ≤∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉 ≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2 +2αn〈(γS-μF)p,yn-p〉 ≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2 +λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2 +2αn∥(γS-μF)p∥∥yn-p∥,
which hence leads to
(93)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2 +λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2 ≤∥xn-p∥2-∥yn-p∥2 +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-yn∥(∥xn-p∥+∥yn-p∥) +2αn∥(γS-μF)p∥∥yn-p∥.
Since limn→∞αn=0, {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk) for all i∈{1,2,…,N} and k∈{1,2,…,M}, by the assumption limn→∞∥xn-yn∥=0 and the boundedness of {xn}, {yn}, we conclude immediately that
(94)limn→∞∥AiΛni-1un-Aip∥=0,limn→∞∥BkΔnk-1xn-Bkp∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}.
Furthermore, by Proposition 6(ii) we obtain that for each k∈{1,2,…,M}(95)∥Δnkxn-p∥2 =∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2 ≤〈(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p,Δnkxn-p〉 =12(∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2 +∥Δnkxn-p∥2-∥(I-rk,nBk)Δnk-1xn -(I-rk,nBk)p-(Δnkxn-p)∥2) ≤12(∥Δnk-1xn-p∥2+∥Δnkxn-p∥2 -∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(96)∥Δnkxn-p∥2 ≤∥Δnk-1xn-p∥2 -∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2 =∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2 -rk,n2∥BkΔnk-1xn-Bkp∥2 +2rk,n〈Δnk-1xn-Δnkxn,BkΔnk-1xn-Bkp〉 ≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2 +2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥ ≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2 +2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥.
Also, by Proposition 1(iii), we obtain that for each i∈{1,2,…,N}(97)∥Λniun-p∥2 =∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2 ≤〈(I-λi,nAi)Λni-1un-(I-λi,nAi)p,Λniun-p〉 =12(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2 +∥Λniun-p∥2 -∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p -(Λniun-p)∥2(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2) ≤12(∥Λni-1un-p∥2+∥Λniun-p∥2 -∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2) ≤12(-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)∥un-p∥2+∥Λniun-p∥2 -∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2),
which implies
(98)∥Λniun-p∥2 ≤∥un-p∥2 -∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2 =∥un-p∥2 -∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2 +2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉 ≤∥un-p∥2-∥Λni-1un-Λniun∥2 +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥.
Thus, from (84), (96), and (98), we have
(99)∥yn-p∥2 ≤∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉 ≤∥Λniun-p∥2+2αn〈(γS-μF)p,yn-p〉 ≤∥un-p∥2-∥Λni-1un-Λniun∥2 +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥ +2αn〈(γS-μF)p,yn-p〉 ≤∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2 +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥ +2αn〈(γS-μF)p,yn-p〉 ≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2 +2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥ -∥Λni-1un-Λniun∥2 +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥ +2αn∥(γS-μF)p∥∥yn-p∥,
which yields
(100)∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2 ≤∥xn-p∥2-∥yn-p∥2 +2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥ +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥ +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-yn∥(∥xn-p∥+∥yn-p∥) +2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥ +2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥ +2αn∥(γS-μF)p∥∥yn-p∥.
Since limn→∞αn=0, {xn}, {yn}, and {un} are bounded. For all i∈{1,2,…,N} and k∈{1,2,…,M}, we have {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk), then by (94) and the assumption limn→∞∥xn-yn∥=0, we conclude immediately that
(101)limn→∞∥Λni-1un-Λniun∥,limn→∞∥Δnk-1xn-Δnkxn∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}. Note that
(102)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥ +∥Δn1xn-Δn2xn∥+⋯ +∥ΔnM-1xn-ΔnMxn∥,∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥ +∥Λn1un-Λn2un∥+⋯ +∥ΛnN-1un-ΛnNun∥.
Thus, from (101) we have
(103)limn→∞∥xn-un∥=0, limn→∞∥un-vn∥=0.
It is easy to see that as n→∞(104)∥xn-vn∥≤∥xn-un∥+∥un-vn∥⟶0.
On the other hand, for simplicity, we write p~=PC(I-ν2F2)p, v~n=PC(I-ν2F2)vn, and wn=Gvn=PC(I-ν1F1)v~n for all n≥1. Then
(105)p=Gp=PC(I-ν1F1)p~=PC(I-ν1F1)PC(I-ν2F2)p.
We now show that limn→∞∥Gvn-vn∥=0; that is, limn→∞∥wn-vn∥=0. As a matter of fact, for p∈Ω, it follows from (61), (62), and (84) that
(106)∥yn-p∥2 ≤αnτ∥vn-p∥2+(1-αnτ)∥Gvn-p∥2 +2αn〈(γS-μF)p,yn-p〉 ≤αnτ∥vn-p∥2+(1-αnτ)∥wn-p∥2 +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2 +ν1(ν1-2ζ1)∥F1v~n-F1p~∥2] +2αn∥(γS-μF)p∥∥yn-p∥ =∥vn-p∥2+(1-αnτ) ×[ν2(ν2-2ζ2)∥F2vn-F2p∥2 +ν1(ν1-2ζ1)∥F1v~n-F1p~∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-p∥2+(1-αnτ) ×[ν2(ν2-2ζ2)∥F2vn-F2p∥2 +ν1(ν1-2ζ1)∥F1v~n-F1p~∥2] +2αn∥(γS-μF)p∥∥yn-p∥,
which immediately yields
(107)(1-αnτ)[ν2(2ζ2-ν2)∥F2vn-F2p∥2 +ν1(2ζ1-ν1)∥F1v~n-F1p~∥2] ≤∥xn-p∥2-∥yn-p∥2 +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-yn∥(∥xn-p∥+∥yn-p∥) +2αn∥(γS-μF)p∥∥yn-p∥.
Since limn→∞αn=0, νj∈(0,2ζj) for j=1,2 and {xn} and {yn} are bounded, by the assumption limn→∞∥xn-yn∥=0, we get
(108)limn→∞∥F2vn-F2p∥=0, limn→∞∥F1v~n-F1p~∥=0.
Also, in terms of the firm nonexpansivity of PC and the ζj-inverse-strong monotonicity of Fj for j=1,2, we obtain from νj∈(0,2ζj), j=1,2 and (67) that
(109)∥v~n-p~∥2 =∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2 ≤〈(I-ν2F2)vn-(I-ν2F2)p,v~n-p~〉 =12[∥(I-ν2F2)vn-(I-ν2F2)p∥2+∥v~n-p~∥2 -∥(I-ν2F2)vn-(I-ν2F2)p-(v~n-p~)∥2] ≤12[∥vn-p∥2+∥v~n-p~∥2 -∥(vn-v~n)-ν2(F2vn-F2p)-(p-p~)∥2] =12[∥vn-p∥2+∥v~n-p~∥2 -∥(vn-v~n)-(p-p~)∥2 +2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉 -ν22∥F2vn-F2p∥2],∥wn-p∥2 =∥PC(I-ν1F1)v~n-PC(I-ν1F1)p~∥2 ≤〈(I-ν1F1)v~n-(I-ν1F1)p~,wn-p〉 =12[∥(I-ν1F1)v~n-(I-ν1F1)p~∥2+∥wn-p∥2 -∥(I-ν1F1)v~n-(I-ν1F1)p~-(wn-p)∥2] ≤12[∥v~n-p~∥2+∥wn-p∥2 -∥(v~n-wn)+(p-p~)∥2 +2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉 -ν12∥F1v~n-F1p~∥2] ≤12[∥vn-p∥2+∥wn-p∥2 -∥(v~n-wn)+(p-p~)∥2 +2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉[∥vn-p∥2+∥wn-p∥2].
Thus, we have
(110)∥v~n-p~∥2≤∥vn-p∥2 -∥(vn-v~n)-(p-p~)∥2 +2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉 -ν22∥F2vn-F2p∥2,(111)∥wn-p∥2≤∥vn-p∥2 -∥(v~n-wn)+(p-p~)∥2 +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥.
Consequently, from (61), (106), and (110) it follows that
(112)∥yn-p∥2 ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ)∥v~n-p~∥2 +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2 +2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉 -ν22∥F2vn-F2p∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2 +2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥vn-p∥2-(1-αnτ)∥(vn-v~n)-(p-p~)∥2 +2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥ +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-p∥2-(1-αnτ)∥(vn-v~n)-(p-p~)∥2 +2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥ +2αn∥(γS-μF)p∥∥yn-p∥,
which hence leads to
(113)(1-αnτ)∥(vn-v~n)-(p-p~)∥2 ≤∥xn-p∥2-∥yn-p∥2 +2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥ +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-yn∥(∥xn-p∥+∥yn-p∥) +2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥ +2αn∥(γS-μF)p∥∥yn-p∥.
Since limn→∞αn=0, {xn}, {yn}, {vn}, and {v~n} are bounded sequences, by the assumption limn→∞∥xn-yn∥=0, we conclude from (108) that
(114)limn→∞∥(vn-v~n)-(p-p~)∥=0.
Furthermore, from (62), (106), and (111) it follows that
(115)∥yn-p∥2 ≤αnτ∥vn-p∥2+(1-αnτ)∥wn-p∥2 +2αn∥(γS-μF)p∥∥yn-p∥ ≤αnτ∥vn-p∥2+(1-αnτ) ×[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2 +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2] +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥vn-p∥2-(1-αnτ)∥(v~n-wn)+(p-p~)∥2 +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥ +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-p∥2-(1-αnτ)∥(v~n-wn)+(p-p~)∥2 +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥ +2αn∥(γS-μF)p∥∥yn-p∥,
which hence yields
(116)(1-αnτ)∥(v~n-wn)+(p-p~)∥2 ≤∥xn-p∥2-∥yn-p∥2 +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥ +2αn∥(γS-μF)p∥∥yn-p∥ ≤∥xn-yn∥(∥xn-p∥+∥yn-p∥) +2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥ +2αn∥(γS-μF)p∥∥yn-p∥.
Since limn→∞αn=0 and {xn}, {yn}, {wn}, and {v~n} are bounded sequences, by the assumption limn→∞∥xn-yn∥=0, we conclude from (108) that
(117)limn→∞∥(v~n-wn)+(p-p~)∥=0.
Note that
(118)∥vn-wn∥≤∥(vn-v~n)-(p-p~)∥ +∥(v~n-wn)+(p-p~)∥.
Hence from (114) and (117) we get
(119)limn→∞∥vn-Gvn∥=limn→∞∥vn-wn∥=0.
Also, observe that
(120)yn=αnγSvn+(I-αnμF)WnGvn.
Hence we get
(121)yn-WnGvn=αn(γSvn-μFWnGvn).
So, from limn→∞αn=0 and the boundedness of {vn} we deduce that
(122)limn→∞∥yn-WnGvn∥=0.
In addition, it is readily found that
(123)∥Wnvn-vn∥ ≤∥Wnvn-WnGvn∥+∥WnGvn-vn∥ ≤∥vn-Gvn∥+∥WnGvn-vn∥ ≤∥vn-Gvn∥+∥WnGvn-yn∥+∥yn-vn∥ ≤∥vn-Gvn∥+∥WnGvn-yn∥ +∥yn-xn∥+∥xn-vn∥.
Thus, by the assumption limn→∞∥xn-yn∥=0, from (103) and (119)–(123) we have
(124)limn→∞∥Wnvn-vn∥=0.
Taking into account that ∥vn-Wvn∥≤∥vn-Wnvn∥+∥Wnvn-Wvn∥, we obtain from ∥vn-Wnvn∥→0 and Remark 12 that
(125)limn→∞∥vn-Wvn∥=0.
Next, let us show that ωw(xn)⊂Ω. In fact, since H is reflexive and {xn} is bounded, there exists at least a weak convergence subsequence of {xn}. Hence it is known that ωw(xn)≠∅. Now, take an arbitrary w∈ωw(xn). Then there exists a subsequence {xni} of {xn} such that xni⇀w. From (101) and (103) and the assumption limn→∞∥xn-yn∥=0, we have that yni⇀w, uni⇀w, vni⇀w, Δnikxni⇀w, and Λnimuni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. Utilizing Lemma 9, we deduce from xni⇀w, vni⇀w, (90), (119), and (125) that w∈Fix(PC(I-(2/L)∇f))=VI(C,∇f)=Γ, w∈GSVI(G), and w∈Fix(W)= ∩n=1∞Fix(Sn) (due to Lemma 13). Thus, we get w∈ ∩n=1∞Fix(Sn)∩GSVI(G)∩Γ. Next we prove that w∈ ∩m=1NVI(C,Am). Let
(126)T~mv={Amv+NCv,v∈C,∅,v∉C,
where m∈{1,2,…,N}. Let (v,u)∈G(T~m). Since u-Amv∈NCv and Λnmun∈C, we have
(127)〈v-Λnmun,u-Amv〉≥0.
On the other hand, from Λnmun=PC(I-λm,nAm)Λnm-1un and v∈C, we have
(128)〈v-Λnmun,Λnmun-(Λnm-1un-λm,nAmΛnm-1un)〉≥0,
and hence
(129)〈v-Λnmun,Λnmun-Λnm-1unλm,n+AmΛnm-1un〉≥0.
Therefore we have
(130)〈v-Λnimuni,u〉 ≥〈v-Λnimuni,Amv〉 ≥〈v-Λnimuni,Amv〉 -〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni+AmΛnim-1uni〉 =〈v-Λnimuni,Amv-AmΛnimuni〉 +〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉 -〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉 ≥〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉 -〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉.
From (101) and since Am is Lipschitz continuous, we obtain that limn→∞∥AmΛnmun-AmΛnm-1un∥=0. From Λnimuni⇀w, {λi,n}⊂[ai,bi]⊂(0,2ηi), for all i∈{1,2,…,N} and (101), we have
(131)〈v-w,u〉≥0.
Since T~m is maximal monotone, we have w∈T~m-10 and hence w∈VI(C,Am), m=1,2,…,N, which implies w∈ ∩m=1NVI(C,Am). Next we prove that w∈ ∩k=1MGMEP(Θk,φk,Bk). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(132)Θk(Δnkxn,y)+φk(y)-φk(Δnkxn) +〈BkΔnk-1xn,y-Δnkxn〉 +1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥0.
By (A2), we have
(133)φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉 +1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉 ≥Θk(y,Δnkxn).
Let zt=ty+(1-t)w for all t∈(0,1] and y∈C. This implies that zt∈C. Then, we have
(134)〈zt-Δnkxn,Bkzt〉 ≥φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt〉 -〈zt-Δnkxn,BkΔnk-1xn〉 -〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn) =φk(Δnkxn)-φk(zt) +〈zt-Δnkxn,Bkzt-BkΔnkxn〉 +〈zt-Δnkxn,BkΔnkxn-BkΔnk-1xn〉 -〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn).
By (101), we have ∥BkΔnkxn-BkΔnk-1xn∥→0 as n→∞. Furthermore, by the monotonicity of Bk, we obtain 〈zt-Δnkxn,Bkzt-BkΔnkxn〉≥0. Then, by (A4) we obtain
(135)〈zt-w,Bkzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and (135), we obtain
(136)0=Θk(zt,zt)+φk(zt)-φk(zt)≤tΘk(zt,y)+(1-t)Θk(zt,w) +tφk(y)+(1-t)φk(w)-φk(zt)≤t[Θk(zt,y)+φk(y)-φk(zt)] +(1-t)〈zt-w,Bkzt〉=t[Θk(zt,y)+φk(y)-φk(zt)] +(1-t)t〈y-w,Bkzt〉,
and hence
(137)0≤Θk(zt,y)+φk(y)-φk(zt) +(1-t)〈y-w,Bkzt〉.
Letting t→0, we have, for each y∈C,
(138)0≤Θk(w,y)+φk(y)-φk(w)+〈y-w,Bkw〉.
This implies that w∈GMEP(Θk,φk,Bk) and hence w∈ ∩k=1MGMEP(Θk,φk,Bk). Consequently, w∈(∩n=1∞Fix(Sn))∩(∩k=1MGMEP(Θk,φk,Bk))∩(∩i=1NVI(C,Ai))∩GSVI(G)∩Γ≕Ω. (This shows that ωw(xn)⊂Ω.)
Step
4. We prove that ωw(xn)⊂Ξ provided that ∥xn-yn∥=o(αn) additionally.
Indeed, let w∈ωw(xn) be the same as mentioned in Step 3. Then we get xni⇀w. In addition, from (84) we have that for every p∈Ω(139)∥yn-p∥2≤∥xn-p∥2 +2αn〈(γS-μF)p,yn-p〉,
which immediately implies that
(140)2〈(γS-μF)p,p-yn〉 ≤1αn(∥xn-p∥2-∥yn-p∥2) ≤∥xn-yn∥αn(∥xn-p∥+∥yn-p∥).
This together with ∥xn-yn∥=o(αn) leads to
(141)limsup n→∞〈(γS-μF)p,p-yn〉≤0.
Observe that
(142)limsup n→∞〈(γS-μF)p,p-xn〉 =limsup n→∞(〈(γS-μF)p,yn-xn〉 +〈(γS-μF)p,p-yn〉) =limsup n→∞〈(γS-μF)p,p-yn〉≤0.
So, it follows from xni⇀w that
(143)〈(γS-μF)p,p-w〉≤0, ∀p∈Ω.
Also, note that 0<γ≤τ and
(144)μη≥τ⟺μη≥1-1-μ(2η-μκ2)⟺1-μ(2η-μκ2)≥1-μη⟺1-2μη+μ2κ2≥1-2μη+μ2η2⟺κ2≥η2⟺κ≥η.
It is clear that
(145)〈(μF-γS)x-(μF-γS)y,x-y〉 ≥(μη-γ)∥x-y∥2, ∀x,y∈H.
Hence, it follows from 0<γ≤τ≤μη that μF-γS is monotone. Since w∈ωw(xn)⊂Ω, by Minty’s lemma [39] we have
(146)〈(γS-μF)w,p-w〉≤0, ∀p∈Ω;
that is, w∈Ξ. Therefore, ωw(xn)⊂Ξ. This completes the proof.