We will introduce and analyze an iterative algorithm by relaxed extragradient-like viscosity method for finding a solution of a generalized equilibrium problem with constraints of several problems: a finite family of variational inclusions, a finite family of variational inequalities, and a fixed point problem in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.
Proof.
As
liminf
n
→
∞
γ
n
>
0
,
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1
, and
0
<
liminf
n
→
∞
r
n
≤
limsup
n
→
∞
r
n
<
2
ζ
, we may assume, without loss of generality, that
{
β
n
}
,
{
γ
n
}
⊂
[
a
,
a
^
]
⊂
(
0,1
)
,
{
r
n
}
⊂
[
c
,
c
^
]
⊂
(
0,2
ζ
)
and
β
n
+
γ
n
∥
V
∥
≤
1
for all
n
≥
1
. Put
(40)
Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
)
,
Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
)
,
for each
k
∈
{
1,2
,
…
,
M
}
and
i
∈
{
1,2
,
…
,
N
}
, and
Δ
0
=
Λ
0
=
I
, where
I
is the identity mapping on
H
. Moreover, set
v
n
=
Δ
M
u
n
and
w
n
=
Λ
N
u
n
.
Since
V
is a
γ
¯
-strongly positive bounded linear operator on
H
, we know that
(41)
∥
V
∥
=
sup
{
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≥
γ
¯
>
1
.
Taking into account that
β
n
+
γ
n
∥
V
∥
≤
1
for all
n
≥
1
, we have
(42)
〈
(
(
1
-
β
n
)
I
-
γ
n
V
)
u
,
u
〉
=
1
-
β
n
-
γ
n
〈
V
u
,
u
〉
≥
1
-
β
n
-
γ
n
∥
V
∥
≥
0
.
That is,
(
1
-
β
n
)
I
-
γ
n
V
is positive. It follows that
(43)
∥
(
1
-
β
n
)
I
-
γ
n
V
∥
=
sup
{
〈
(
(
1
-
β
n
)
I
-
γ
n
V
)
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
=
sup
{
1
-
β
n
-
γ
n
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≤
1
-
β
n
-
γ
n
γ
¯
.
In the meantime, it is not hard to find that
Δ
M
and
Λ
N
are nonexpansive. As a matter of fact, observe that, for all
x
,
y
∈
C
,
(44)
∥
Δ
M
x
-
Δ
M
y
∥
2
=
∥
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
x
-
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
y
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
x
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
y
∥
2
=
∥
(
Δ
M
-
1
x
-
Δ
M
-
1
y
)
-
ν
M
(
A
M
Δ
M
-
1
x
-
A
M
Δ
M
-
1
y
)
∥
2
≤
∥
Δ
M
-
1
x
-
Δ
M
-
1
y
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
x
-
A
M
Δ
M
-
1
y
∥
2
≤
∥
Δ
M
-
1
x
-
Δ
M
-
1
y
∥
2
⋮
≤
∥
Δ
0
x
-
Δ
0
y
∥
2
=
∥
x
-
y
∥
2
,
∥
Λ
N
x
-
Λ
N
y
∥
2
=
∥
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
x
-
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
y
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
x
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
y
∥
2
=
∥
(
Λ
N
-
1
x
-
Λ
N
-
1
y
)
-
μ
N
(
B
N
Λ
N
-
1
x
-
B
N
Λ
N
-
1
y
)
∥
2
≤
∥
Λ
N
-
1
x
-
Λ
N
-
1
y
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
x
-
B
N
Λ
N
-
1
y
∥
2
≤
∥
Λ
N
-
1
x
-
Λ
N
-
1
y
∥
2
⋮
≤
∥
Λ
0
x
-
Λ
0
y
∥
2
=
∥
x
-
y
∥
2
.
In addition, note that
(45)
〈
(
I
-
f
)
x
-
(
I
-
f
)
y
,
x
-
y
〉
=
∥
x
-
y
∥
2
-
〈
f
(
x
)
-
f
(
y
)
,
x
-
y
〉
≥
(
1
-
ρ
)
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
,
∥
(
I
-
f
)
x
-
(
I
-
f
)
y
∥
=
∥
x
-
y
∥
+
∥
f
(
x
)
-
f
(
y
)
∥
≤
(
1
+
ρ
)
∥
x
-
y
∥
,
∀
x
,
y
∈
H
.
That is,
I
-
f
is strongly monotone and Lipschitz continuous. So, there exists a unique solution
x
*
in
Ω
to the VIP
(46)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.
That is,
x
*
∈
VI
(
Ω
,
I
-
f
)
.
We divide the rest of the proof into several steps.
Step 1. We show that
{
x
n
}
is bounded. Indeed, take
p
∈
Ω
arbitrarily. Since
p
=
S
r
n
(
Θ
,
φ
)
(
p
-
r
n
A
p
)
,
A
is
ζ
-inverse strongly monotone, and
0
≤
r
n
≤
2
ζ
, we have, for any
n
≥
1
,
(47)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
(
x
n
-
p
)
-
r
n
(
A
x
n
-
A
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
2
r
n
〈
x
n
-
p
,
A
x
n
-
A
p
〉
+
r
n
2
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
2
r
n
ζ
∥
A
x
n
-
A
p
∥
2
+
r
n
2
∥
A
x
n
-
A
p
∥
2
=
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
.
Since
p
=
P
C
(
I
-
ν
k
A
k
)
p
,
Δ
k
p
=
p
, and
A
k
is
ζ
k
-inverse strongly monotone, where
ν
k
∈
(
0,2
ζ
k
)
,
k
∈
{
1,2
,
…
,
M
}
, by Proposition 3 we obtain that for each
n
≥
1
(48)
∥
v
n
-
p
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
p
∥
2
≤
∥
Δ
M
-
1
u
n
-
Δ
M
-
1
p
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
u
n
-
A
M
Δ
M
-
1
p
∥
2
≤
∥
Δ
M
-
1
u
n
-
Δ
M
-
1
p
∥
2
⋮
≤
∥
Δ
0
u
n
-
Δ
0
p
∥
2
=
∥
u
n
-
p
∥
2
.
Since
p
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p
,
Λ
i
p
=
p
, and
B
i
is
η
i
-inverse strongly monotone, where
μ
i
∈
(
0,2
η
i
)
,
i
∈
{
1,2
,
…
,
N
}
, by Lemma 14 we deduce that for each
n
≥
1
(49)
∥
w
n
-
p
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
p
∥
2
≤
∥
Λ
N
-
1
u
n
-
Λ
N
-
1
p
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
u
n
-
B
N
Λ
N
-
1
p
∥
2
≤
∥
Λ
N
-
1
u
n
-
Λ
N
-
1
p
∥
2
⋮
≤
∥
Λ
0
u
n
-
Λ
0
p
∥
2
=
∥
u
n
-
p
∥
2
.
Hence from (37)–(49), we have
(50)
∥
y
n
-
p
∥
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
Δ
M
u
n
-
p
)
dddddd
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
Λ
N
u
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
W
n
Δ
M
u
n
-
p
∥
+
∥
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
Λ
N
u
n
-
p
)
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
W
n
v
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
W
n
w
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
v
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
w
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
u
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
u
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
x
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
γ
n
(
γ
-
-
1
)
)
∥
x
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
γ
n
(
γ
-
-
1
)
)
∥
x
n
-
p
∥
+
γ
n
(
γ
-
-
1
)
∥
(
I
-
V
)
p
∥
γ
-
-
1
≤
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
-
-
1
}
.
Since
f
:
H
→
H
is a
ρ
-contraction with
ρ
∈
(
0,1
)
, from (37) and (50) we get
(51)
∥
x
n
+
1
-
p
∥
≤
α
n
∥
f
(
x
n
)
-
p
∥
+
(
1
-
α
n
)
∥
y
n
-
p
∥
≤
α
n
(
∥
f
(
x
n
)
-
f
(
p
)
∥
+
∥
f
(
p
)
-
p
∥
)
+
(
1
-
α
n
)
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
≤
α
n
(
ρ
∥
x
n
-
p
∥
+
∥
f
(
p
)
-
p
∥
)
+
(
1
-
α
n
)
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
≤
α
n
ρ
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
(
1
-
α
n
)
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
∥
f
(
p
)
-
p
∥
=
(
1
-
α
n
(
1
-
ρ
)
)
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
∥
(
I
-
f
)
p
∥
=
(
1
-
α
n
(
1
-
ρ
)
)
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
(
1
-
ρ
)
∥
(
I
-
f
)
p
∥
1
-
ρ
≤
max
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
,
∥
(
I
-
f
)
p
∥
1
-
ρ
}
.
By induction, we get
(52)
∥
x
n
-
p
∥
≤
max
{
∥
x
1
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
,
∥
(
I
-
f
)
p
∥
1
-
ρ
}
.
Therefore,
{
x
n
}
is bounded and so are the sequences
{
u
n
}
,
{
v
n
}
,
{
w
n
}
,
{
f
(
x
n
)
}
, and
{
W
n
w
n
}
.
Step
2. We show that
lim
n
→
∞
∥
x
n
-
x
n
+
1
∥
=
0
and
lim
n
→
∞
∥
x
n
-
y
n
∥
=
0
. Indeed, put
σ
n
=
(
1
-
α
n
)
β
n
,
for
all
n
≥
1
. Then it follows from conditions (iii) and (iv) that
(53)
β
n
≥
σ
n
=
(
1
-
α
n
)
β
n
≥
(
1
-
(
1
-
ρ
)
)
β
n
=
ρ
β
n
,
∀
n
≥
1
,
and hence
(54)
0
<
liminf
n
→
∞
σ
n
≤
limsup
n
→
∞
σ
n
<
1
.
Define
(55)
x
n
+
1
=
σ
n
x
n
+
(
1
-
σ
n
)
z
n
,
∀
n
≥
1
.
Observe that
(56)
z
n
+
1
-
z
n
=
x
n
+
2
-
σ
n
+
1
x
n
+
1
1
-
σ
n
+
1
-
x
n
+
1
-
σ
n
x
n
1
-
σ
n
=
α
n
+
1
f
(
x
n
+
1
)
+
(
1
-
α
n
+
1
)
y
n
+
1
-
σ
n
+
1
x
n
+
1
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
-
σ
n
x
n
1
-
σ
n
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
-
(
(
1
-
α
n
)
[
β
n
x
n
+
γ
n
W
n
v
n
h
h
h
h
h
h
h
h
h
h
h
h
g
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
w
n
]
-
σ
n
x
n
)
×
(
1
-
σ
n
)
-
1
+
(
(
1
-
α
n
+
1
)
[
β
n
+
1
x
n
+
1
+
γ
n
+
1
W
n
+
1
v
n
+
1
hhhhhhhhhhhhhh
+
[
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
]
W
n
+
1
w
n
+
1
]
-
σ
n
+
1
x
n
+
1
β
n
+
1
)
×
(
1
-
σ
n
+
1
)
-
1
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
+
1
-
α
n
+
1
1
-
σ
n
+
1
×
(
γ
n
+
1
W
n
+
1
v
n
+
1
+
(
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
)
W
n
+
1
w
n
+
1
)
-
1
-
α
n
1
-
σ
n
(
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
)
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
γ
n
+
1
W
n
+
1
v
n
+
1
+
(
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
)
W
n
+
1
w
n
+
1
1
-
β
n
+
1
222222222
-
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
]
+
[
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
-
(
1
-
α
n
)
(
1
-
β
n
)
1
-
σ
n
]
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
(
f
(
x
n
+
1
)
-
f
(
x
n
)
)
+
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
f
(
x
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
γ
n
+
1
1
-
β
n
+
1
W
n
+
1
v
n
+
1
-
γ
n
1
-
β
n
W
n
v
n
+
W
n
+
1
w
n
+
1
fffffffffffff
-
W
n
w
n
+
γ
n
1
-
β
n
V
W
n
w
n
-
γ
n
+
1
1
-
β
n
+
1
V
W
n
+
1
w
n
+
1
]
-
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
(
f
(
x
n
+
1
)
-
f
(
x
n
)
)
+
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
f
(
x
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
(
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
)
(
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
)
ggggggg
+
(
W
n
+
1
v
n
+
1
-
W
n
v
n
)
γ
n
1
-
β
n
ggggggg
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
(
W
n
+
1
w
n
+
1
-
W
n
w
n
)
1
-
β
n
]
-
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
,
and hence
(57)
∥
z
n
+
1
-
z
n
∥
≤
α
n
+
1
1
-
σ
n
+
1
∥
f
(
x
n
+
1
)
-
f
(
x
n
)
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
dddddddd
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
dddddddd
+
∥
(
(
1
-
β
n
)
I
-
γ
n
V
)
(
W
n
+
1
w
n
+
1
-
W
n
w
n
)
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
γ
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
ρ
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
gggggggg
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
gggggggg
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
×
γ
n
∥
W
n
v
n
∥
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
.
From (9), since
W
n
,
T
n
, and
U
n
,
i
are all nonexpansive, we have
(58)
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
≤
∥
W
n
+
1
v
n
+
1
-
W
n
+
1
v
n
∥
+
∥
W
n
+
1
w
n
-
W
n
v
n
∥
=
∥
W
n
+
1
v
n
+
1
-
W
n
+
1
v
n
∥
+
∥
λ
1
T
1
U
n
+
1,2
v
n
-
λ
1
T
1
U
n
,
2
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
∥
U
n
+
1,2
v
n
-
U
n
,
2
v
n
∥
=
∥
v
n
+
1
-
v
n
∥
+
λ
1
∥
λ
2
T
2
U
n
+
1,3
v
n
-
λ
2
T
2
U
n
,
3
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
λ
2
∥
U
n
+
1,3
v
n
-
U
n
,
3
v
n
∥
⋮
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
λ
2
⋯
λ
n
∥
U
n
+
1
,
n
+
1
v
n
-
U
n
,
n
+
1
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
M
~
∏
i
=
1
n
λ
i
,
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
≤
∥
W
n
+
1
w
n
+
1
-
W
n
+
1
w
n
∥
+
∥
W
n
+
1
w
n
-
W
n
w
n
∥
=
∥
W
n
+
1
w
n
+
1
-
W
n
+
1
w
n
∥
+
∥
λ
1
T
1
U
n
+
1,2
w
n
-
λ
1
T
1
U
n
,
2
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
∥
U
n
+
1,2
w
n
-
U
n
,
2
w
n
∥
=
∥
w
n
+
1
-
w
n
∥
+
λ
1
∥
λ
2
T
2
U
n
+
1,3
w
n
-
λ
2
T
2
U
n
,
3
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
λ
2
∥
U
n
+
1,3
w
n
-
U
n
,
3
w
n
∥
⋮
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
λ
2
⋯
λ
n
∥
U
n
+
1
,
n
+
1
w
n
-
U
n
,
n
+
1
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
λ
i
,
where
sup
n
≥
1
{
∥
U
n
+
1
,
n
+
1
v
n
∥
+
∥
U
n
,
n
+
1
v
n
∥
}
≤
M
~
and
sup
n
≥
1
{
∥
U
n
+
1
,
n
+
1
w
n
∥
+
∥
U
n
,
n
+
1
w
n
∥
}
≤
M
~
for some
M
~
>
0
.
On the other hand, we estimate
∥
v
n
+
1
-
v
n
∥
and
∥
w
n
+
1
-
w
n
∥
. First observe that
(59)
∥
v
n
+
1
-
v
n
∥
2
=
∥
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
+
1
-
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
+
1
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
∥
2
=
∥
(
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
)
d
d
d
-
ν
M
(
A
M
Δ
M
-
1
u
n
+
1
-
A
M
Δ
M
-
1
u
n
)
∥
2
≤
∥
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
u
n
+
1
-
A
M
Δ
M
-
1
u
n
∥
2
≤
∥
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
∥
2
⋮
≤
∥
Δ
0
u
n
+
1
-
Δ
0
u
n
∥
2
=
∥
u
n
+
1
-
u
n
∥
2
.
Utilizing Remark 5 and Lemma 14, we have
(60)
∥
w
n
+
1
-
w
n
∥
2
=
∥
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
+
1
d
d
d
-
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
+
1
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
∥
2
=
∥
(
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
)
d
d
d
-
μ
N
(
B
N
Λ
N
-
1
u
n
+
1
-
B
N
Λ
N
-
1
u
n
)
∥
2
≤
∥
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
u
n
+
1
-
B
N
Λ
N
-
1
u
n
∥
2
≤
∥
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
∥
2
⋮
≤
∥
Λ
0
u
n
+
1
-
Λ
0
u
n
∥
2
=
∥
u
n
+
1
-
u
n
∥
2
,
∥
(
I
-
r
n
+
1
A
)
x
n
+
1
-
(
I
-
r
n
A
)
x
n
∥
=
∥
x
n
+
1
-
x
n
-
r
n
+
1
(
A
x
n
+
1
-
A
x
n
)
+
(
r
n
-
r
n
+
1
)
A
x
n
∥
≤
∥
x
n
+
1
-
x
n
-
r
n
+
1
(
A
x
n
+
1
-
A
x
n
)
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
,
∥
u
n
+
1
-
u
n
∥
=
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
=
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
222222
+
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
(
I
-
r
n
+
1
A
)
x
n
+
1
-
(
I
-
r
n
A
)
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
+
|
r
n
+
1
-
r
n
|
r
n
+
1
·
ν
σ
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
×
(
∥
A
x
n
∥
+
ν
c
σ
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
)
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
,
where
sup
n
≥
1
{
∥
A
x
n
∥
+
(
ν
/
c
σ
)
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
}
≤
M
~
1
for some
M
~
1
>
0
. So, combining (57)–(60) we get
(61)
∥
z
n
+
1
-
z
n
∥
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
(
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
(
∥
v
n
+
1
-
v
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
(
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
λ
i
)
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
∏
i
=
1
n
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
1
-
β
n
∥
u
n
+
1
-
u
n
∥
ddddddd
+
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
1
-
β
n
M
~
∏
i
=
1
n
λ
i
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
(
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
)
+
M
~
∏
i
=
1
n
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
+
M
~
∏
i
=
1
n
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
a
^
≤
∥
x
n
+
1
-
x
n
∥
+
M
~
2
(
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
+
|
r
n
+
1
-
r
n
|
+
b
n
)
,
where
sup
n
≥
1
{
∥
f
(
x
n
)
∥
+
(
(
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
)
/
(
1
-
a
^
)
)
+
∥
W
n
v
n
-
V
W
n
w
n
∥
+
M
~
1
+
M
~
}
≤
M
~
2
for some
M
~
2
>
0
. Thus, from (61),
b
∈
(
0,1
)
, and conditions (v)-(vi) it follows that
(62)
limsup
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0
.
Since
x
n
+
1
=
σ
n
x
n
+
(
1
-
σ
n
)
z
n
for all
n
≥
1
, by Lemma 12 we obtain from
0
<
liminf
n
→
∞
σ
n
≤
limsup
n
→
∞
σ
n
<
1
that
(63)
lim
n
→
∞
∥
z
n
-
x
n
∥
=
0
,
which immediately yields
(64)
lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
n
→
∞
(
1
-
σ
n
)
∥
z
n
-
x
n
∥
=
0
.
Note that
(65)
ρ
∥
y
n
-
x
n
∥
=
(
1
-
(
1
-
ρ
)
)
∥
y
n
-
x
n
∥
≤
(
1
-
α
n
)
∥
y
n
-
x
n
∥
=
∥
x
n
+
1
-
x
n
-
α
n
(
f
(
x
n
)
-
x
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
α
n
(
f
(
x
n
)
-
x
n
)
∥
.
Consequently, it follows from (64) and
α
n
(
f
(
x
n
)
-
x
n
)
→
0
that
(66)
lim
n
→
∞
∥
y
n
-
x
n
∥
=
0
.
Step
3. We prove
lim
n
→
∞
∥
x
n
-
u
n
∥
=
0
.
Indeed, for any
p
∈
Ω
, we find that
(67)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
x
n
-
p
-
r
n
(
A
x
n
-
A
p
)
∥
2
≤
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
.
From (37), (48), (49), and (67), we obtain
(68)
∥
y
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
22222
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
22222
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
W
n
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
×
∥
1
1
-
β
n
-
γ
n
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
[
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
×
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which immediately yields
(69)
(
1
-
β
n
-
γ
n
∥
V
∥
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting
p
=
x
*
we have
(70)
(
1
-
β
n
-
γ
n
∥
V
∥
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
x
*
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since
limsup
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1
,
0
<
liminf
n
→
∞
r
n
≤
limsup
n
→
∞
r
n
<
2
ζ
, and
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
, we obtain from (66) and the boundedness of
{
x
n
}
and
{
y
n
}
that
(71)
lim
n
→
∞
∥
A
x
n
-
A
x
*
∥
=
0
.
Furthermore, from the firm nonexpansivity of
S
r
n
(
Θ
,
φ
)
, we have
(72)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
〈
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
,
u
n
-
p
〉
=
1
2
[
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
-
(
u
n
-
p
)
∥
2
]
≤
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
n
-
u
n
-
r
n
(
A
x
n
-
A
p
)
∥
2
]
=
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
222222
+
2
r
n
〈
A
x
n
-
A
p
,
x
n
-
u
n
〉
-
r
n
2
∥
A
x
n
-
A
p
∥
2
]
,
which implies that
(73)
∥
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
.
From (68) and (73), we have
(74)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
[
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
55555
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which immediately yields
(75)
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting
p
=
x
*
we have
(76)
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
x
n
-
u
n
∥
2
≤
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
=
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
.
Since
limsup
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1
and
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
, we deduce from (66) and (71) and the boundedness of
{
u
n
}
,
{
x
n
}
, and
{
y
n
}
that
(77)
lim
n
→
∞
∥
x
n
-
u
n
∥
=
0
.
Step 4. We prove that
lim
n
→
∞
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
=
0
and
lim
n
→
∞
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
=
0
for each
k
∈
{
1,2
,
…
,
M
}
and
i
∈
{
1,2
,
…
,
N
}
.
Indeed, let us show that
lim
n
→
∞
∥
A
k
Δ
k
u
n
-
A
k
x
*
∥
=
0
and
lim
n
→
∞
∥
B
i
Λ
i
u
n
-
B
i
x
*
∥
=
0
for each
k
∈
{
1,2
,
…
,
M
}
and
i
∈
{
1,2
,
…
,
N
}
. Observe that, for any
p
∈
Ω
,
(78)
∥
Δ
k
u
n
-
p
∥
2
=
∥
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
P
C
(
I
-
ν
k
A
k
)
p
∥
2
≤
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
∥
2
≤
∥
Δ
k
-
1
u
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
≤
∥
u
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
,
∥
Λ
i
u
n
-
p
∥
2
=
∥
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p
∥
2
≤
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
∥
2
≤
∥
Λ
i
-
1
u
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
.
From (68) and (78) we have
(79)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
Δ
k
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
Λ
i
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
[
∥
x
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
]
+
(
1
-
β
n
-
γ
n
γ
¯
)
×
[
∥
x
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
+
γ
n
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
+
γ
n
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which leads to
(80)
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting
p
=
x
*
we have
(81)
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
2
≤
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since
0
<
liminf
n
→
∞
γ
n
≤
limsup
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1
and
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
, we obtain from (66),
ν
k
∈
(
0,2
ζ
k
)
,
μ
i
∈
(
0,2
η
i
)
,
k
∈
{
1,2
,
…
,
M
}
,
i
∈
{
1,2
,
…
,
N
}
, and the boundedness of
{
x
n
}
and
{
y
n
}
that
(82)
lim
n
→
∞
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
=
0
,
lim
n
→
∞
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
=
0
,
for each
k
∈
{
1,2
,
…
,
M
}
and
i
∈
{
1,2
,
…
,
N
}
.
Furthermore, by Proposition 3(iii) and Lemma 7(a), we obtain
(83)
∥
Δ
k
u
n
-
p
∥
2
=
∥
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
P
C
(
I
-
ν
k
A
k
)
p
∥
2
≤
〈
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
,
Δ
k
u
n
-
p
〉
=
1
2
(
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
∥
2
5555
5
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
5555
5
5
5
5
5
-
(
I
-
ν
k
A
k
)
p
-
(
Δ
k
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Δ
k
-
1
u
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
,
which implies that
(84)
∥
Δ
k
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
ν
k
2
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
2
ν
k
〈
Δ
k
-
1
u
n
-
Δ
k
u
n
,
A
k
Δ
k
-
1
u
n
-
A
k
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
.
By Lemmas 7(a) and 14, we get
(85)
∥
Λ
i
u
n
-
p
∥
2
=
∥
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p
∥
2
≤
〈
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
,
Λ
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
-
(
Λ
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Λ
i
-
1
u
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies that
(86)
∥
Λ
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
-
μ
i
2
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
μ
i
〈
Λ
i
-
1
u
n
-
Λ
i
u
n
,
B
i
Λ
i
-
1
u
n
-
B
i
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
.
From (68), (84), and (86) we have
(87)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
[
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
2222222
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
]
+
(
1
-
β
n
-
γ
n
γ
¯
)
×
[
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
22222
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which hence implies that
(88)
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting
p
=
x
*
we have
(89)
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since
0
<
liminf
n
→
∞
γ
n
≤
limsup
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1
and
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
, we obtain from (66) and (82) and the boundedness of
{
u
n
}
,
{
x
n
}
, and
{
y
n
}
that
(90)
lim
n
→
∞
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
=
0
,
lim
n
→
∞
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
=
0
,
for each
k
∈
{
1,2
,
…
,
M
}
and
i
∈
{
1,2
,
…
,
N
}
. Consequently, from (90) it follows that
(91)
∥
u
n
-
v
n
∥
=
∥
Δ
0
u
n
-
Δ
M
u
n
∥
≤
∥
Δ
0
u
n
-
Δ
1
u
n
∥
+
∥
Δ
1
u
n
-
Δ
2
u
n
∥
+
⋯
+
∥
Δ
M
-
1
u
n
-
Δ
M
u
n
∥
⟶
0
as
n
⟶
∞
,
(92)
∥
u
n
-
w
n
∥
=
∥
Λ
0
u
n
-
Λ
N
u
n
∥
≤
∥
Λ
0
u
n
-
Λ
1
u
n
∥
+
∥
Λ
1
u
n
-
Λ
2
u
n
∥
+
⋯
+
∥
Λ
N
-
1
u
n
-
Λ
N
u
n
∥
⟶
0
as
n
⟶
∞
.
By (77), (91), and (92), we have
(93)
lim
n
→
∞
∥
x
n
-
v
n
∥
=
0
,
lim
n
→
∞
∥
x
n
-
w
n
∥
=
0
.
Step 5. We show that
lim
n
→
∞
∥
v
n
-
W
v
n
∥
=
0
. Indeed, utilizing Lemma 7(b), from (37), (48), (49), and (67) we obtain that, for any
p
∈
Ω
,
(94)
∥
y
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
5555
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
5555
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
W
n
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
×
∥
1
1
-
β
n
-
γ
n
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
u
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
u
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which implies that
(95)
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting
p
=
x
*
we have
(96)
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since
liminf
n
→
∞
β
n
>
0
,
liminf
n
→
∞
γ
n
>
0
, and
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
, we deduce from (66) and the boundedness of
{
x
n
}
and
{
y
n
}
that
(97)
lim
n
→
∞
∥
x
n
-
W
n
v
n
∥
=
0
.
Also, observe that
(98)
∥
v
n
-
W
n
v
n
∥
≤
∥
v
n
-
u
n
∥
+
∥
u
n
-
x
n
∥
+
∥
x
n
-
W
n
v
n
∥
.
Thus, from (77), (91), and (97) it follows that
(99)
lim
n
→
∞
∥
v
n
-
W
n
v
n
∥
=
0
.
Moreover, note that
(100)
∥
v
n
-
W
v
n
∥
≤
∥
v
n
-
W
n
v
n
∥
+
∥
W
n
v
n
-
W
v
n
∥
.
From (99), [23, Remark 3.2], and the boundedness of
{
v
n
}
we immediately obtain
(101)
lim
n
→
∞
∥
v
n
-
W
v
n
∥
=
0
.
Step 6. We show that
limsup
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
≤
0
.
Indeed, we observe that there exists a subsequence
{
x
n
i
}
of
{
x
n
}
such that
(102)
limsup
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
=
lim
i
→
∞
〈
(
f
-
I
)
x
*
,
x
n
i
-
x
*
〉
.
Since
{
x
n
i
}
is bounded, there exists a subsequence
{
x
n
i
j
}
of
{
x
n
i
}
which converges weakly to some
w
. Without loss of generality, we may assume that
x
n
i
⇀
w
. From (77), (90), and (93), we have that
u
n
i
⇀
w
,
v
n
i
⇀
w
,
Δ
k
u
n
i
⇀
w
, and
Λ
m
u
n
i
⇀
w
, where
k
∈
{
1,2
,
…
,
M
}
and
m
∈
{
1,2
,
…
,
N
}
. By (101) we have that
∥
v
n
-
W
v
n
∥
→
0
as
n
→
∞
. Then, by Lemma 10 we obtain
w
∈
Fix
(
W
)
=
∩
n
=
1
∞
Fix
(
T
n
)
(due to Lemma 9). Next we prove that
w
∈
∩
k
=
1
M
VI
(
C
,
A
k
)
. Let
(103)
T
~
k
v
=
{
A
k
v
+
N
C
v
,
v
∈
C
,
∅
,
v
∉
C
,
where
k
∈
{
1,2
,
…
,
M
}
. Let
(
v
,
u
)
∈
G
(
T
~
k
)
. Since
u
-
A
k
v
∈
N
C
v
and
Δ
k
u
n
∈
C
, we have
(104)
〈
v
-
Δ
k
u
n
,
u
-
A
k
v
〉
≥
0
.
Also, from
Δ
k
u
n
=
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
and
v
∈
C
, we have
(105)
〈
v
-
Δ
k
u
n
,
Δ
k
u
n
-
(
Δ
k
-
1
u
n
-
ν
k
A
k
Δ
k
-
1
u
n
)
〉
≥
0
,
and hence
(106)
〈
v
-
Δ
k
u
n
,
Δ
k
u
n
-
Δ
k
-
1
u
n
ν
k
+
A
k
Δ
k
-
1
u
n
〉
≥
0
.
Therefore we have
(107)
〈
v
-
Δ
k
u
n
i
,
u
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
v
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
v
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
+
A
k
Δ
k
-
1
u
n
i
〉
=
〈
v
-
Δ
k
u
n
i
,
A
k
v
-
A
k
Δ
k
u
n
i
〉
+
〈
v
-
Δ
k
u
n
i
,
A
k
Δ
k
u
n
i
-
A
k
Δ
k
-
1
u
n
i
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
Δ
k
u
n
i
-
A
k
Δ
k
-
1
u
n
i
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
〉
.
From (90) and since
A
k
is uniformly continuous, we obtain that
lim
n
→
∞
∥
A
k
Δ
k
u
n
-
A
k
Δ
k
-
1
u
n
∥
=
0
. From
Δ
k
u
n
i
⇀
w
,
ν
k
∈
(
0,2
ζ
k
)
,
for
all
k
∈
{
1,2
,
…
,
M
}
and (90), we have
(108)
〈
v
-
w
,
u
〉
≥
0
.
Since
T
~
k
is maximal monotone, we have
w
∈
T
~
k
-
1
0
and hence
w
∈
VI
(
C
,
A
k
)
,
k
=
1,2
,
…
,
M
, which implies
w
∈
∩
k
=
1
M
VI
(
C
,
A
k
)
. Next, we prove that
w
∈
∩
m
=
1
N
I
(
B
m
,
R
m
)
. As a matter of fact, since
B
m
is
η
m
-inverse strongly monotone,
B
m
is a monotone and Lipschitz continuous mapping. It follows from Lemma 17 that
R
m
+
B
m
is maximal monotone. Let
(
v
,
g
)
∈
G
(
R
m
+
B
m
)
; that is,
g
-
B
m
v
∈
R
m
v
. Again, since
Λ
m
u
n
=
J
R
m
,
μ
m
(
I
-
μ
m
B
m
)
Λ
m
-
1
u
n
,
n
≥
1
,
m
∈
{
1,2
,
…
,
N
}
, we have
(109)
Λ
m
-
1
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
∈
(
I
+
μ
m
R
m
)
Λ
m
u
n
.
That is,
(110)
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
∈
R
m
Λ
m
u
n
.
In terms of the monotonicity of
R
m
, we get
(111)
〈
1
μ
m
v
-
Λ
m
u
n
,
g
-
B
m
v
-
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
〉
≥
0
and hence
(112)
〈
v
-
Λ
m
u
n
,
g
〉
≥
〈
1
μ
m
v
-
Λ
m
u
n
,
22222
B
m
v
+
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
〉
=
〈
1
μ
m
v
-
Λ
m
u
n
,
B
m
v
-
B
m
Λ
m
u
n
+
B
m
Λ
m
u
n
22222
-
B
m
Λ
m
-
1
u
n
+
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
)
〉
≥
〈
v
-
Λ
m
u
n
,
B
m
Λ
m
u
n
-
B
m
Λ
m
-
1
u
n
〉
+
〈
v
-
Λ
m
u
n
,
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
)
〉
.
In particular,
(113)
〈
v
-
Λ
m
u
n
i
,
g
〉
≥
〈
v
-
Λ
m
u
n
i
,
B
m
Λ
m
u
n
i
-
B
m
Λ
m
-
1
u
n
i
〉
+
〈
v
-
Λ
m
u
n
i
,
1
μ
m
(
Λ
m
-
1
u
n
i
-
Λ
m
u
n
i
)
〉
.
Since
∥
Λ
m
u
n
-
Λ
m
-
1
u
n
∥
→
0
(due to (90)) and
∥
B
m
Λ
m
u
n
-
B
m
Λ
m
-
1
u
n
∥
→
0
(due to the Lipschitz continuity of
B
m
), we conclude from
Λ
m
u
n
i
⇀
w
and
μ
m
∈
(
0,2
η
m
)
,
m
∈
{
1,2
,
…
,
N
}
, that
(114)
lim
i
→
∞
〈
v
-
Λ
m
u
n
i
,
g
〉
=
〈
v
-
w
,
g
〉
≥
0
.
It follows from the maximal monotonicity of
B
m
+
R
m
that
0
∈
(
R
m
+
B
m
)
w
; that is,
w
∈
I
(
B
m
,
R
m
)
. Therefore,
w
∈
∩
m
=
1
N
I
(
B
m
,
R
m
)
.
Next, we show that
w
∈
GMEP
(
Θ
,
φ
,
A
)
. In fact, from
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
, we know that
(115)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
.
From (H2) it follows that
(116)
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
Θ
(
y
,
u
n
)
,
∀
y
∈
C
.
Replacing
n
by
n
i
, we have
(117)
φ
(
y
)
-
φ
(
u
n
i
)
+
〈
A
x
n
i
,
y
-
u
n
i
〉
+
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
y
-
u
n
i
〉
≥
Θ
(
y
,
u
n
i
)
,
22222222222222222222222222222222
22222
∀
y
∈
C
.
Put
u
t
=
t
y
+
(
1
-
t
)
w
for all
t
∈
(
0,1
]
and
y
∈
C
. Then, from (117) we have
(118)
〈
u
t
-
u
n
i
,
A
u
t
〉
≥
〈
u
t
-
u
n
i
,
A
u
t
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
u
t
-
u
n
i
,
A
x
n
i
〉
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
≥
〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
+
〈
u
t
-
u
n
i
,
A
u
n
i
-
A
x
n
i
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
.
Since
∥
u
n
i
-
x
n
i
∥
→
0
as
i
→
∞
, we deduce from the Lipschitz continuity of
A
and
K
′
that
∥
A
u
n
i
-
A
x
n
i
∥
→
0
and
∥
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
∥
→
0
as
i
→
∞
. Further, from the monotonicity of
A
, we have
〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
≥
0
. So, from (H4), the weakly lower semicontinuity of
φ
,
(
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
)
/
r
n
i
→
0
, and
u
n
i
⇀
w
, we have
(119)
〈
u
t
-
w
,
A
u
t
〉
≥
-
φ
(
u
t
)
+
φ
(
w
)
+
Θ
(
u
t
,
w
)
,
as
i
⟶
∞
.
From (H1), (H4), and (119) we also have
(120)
0
=
Θ
(
u
t
,
u
t
)
+
φ
(
u
t
)
-
φ
(
u
t
)
≤
t
Θ
(
u
t
,
y
)
+
(
1
-
t
)
Θ
(
u
t
,
w
)
+
t
φ
(
y
)
+
(
1
-
t
)
φ
(
w
)
-
φ
(
u
t
)
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
[
Θ
(
u
t
,
w
)
+
φ
(
w
)
-
φ
(
w
)
-
φ
(
u
t
)
]
≤
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
〈
u
t
-
w
,
A
u
t
〉
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
t
〈
y
-
w
,
A
u
t
〉
,
and hence
(121)
0
≤
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
+
(
1
-
t
)
〈
y
-
w
,
A
u
t
〉
.
Letting
t
→
0
, we have, for each
y
∈
C
,
(122)
0
≤
Θ
(
w
,
y
)
+
φ
(
y
)
-
φ
(
w
)
+
〈
A
w
,
y
-
w
〉
.
This implies that
w
∈
GMEP
(
Θ
,
φ
,
A
)
. Therefore,
w
∈
∩
n
=
1
∞
Fix
(
T
n
)
∩
GMEP
(
Θ
,
φ
,
A
)
∩
∩
k
=
1
M
VI
(
A
k
,
C
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
=
Ω
. This shows that
ω
w
(
x
n
)
⊂
Ω
. Consequently, from (102) and
x
*
∈
VI
(
Ω
,
I
-
f
)
, we have
(123)
limsup
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
=
〈
(
f
-
I
)
x
*
,
w
-
x
*
〉
≤
0
.
Step 7. Finally, we show that
x
n
→
x
*
∈
Ω
as
n
→
∞
.
Indeed, in terms of (68) we get
(124)
∥
y
n
-
x
*
∥
2
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
,
which, together with (37), implies that
(125)
∥
x
n
+
1
-
x
*
∥
2
=
∥
α
n
(
f
(
x
n
)
-
f
(
x
*
)
)
+
(
1
-
α
n
)
(
y
n
-
x
*
)
+
α
n
(
f
(
x
*
)
-
x
*
)
∥
2
≤
∥
α
n
(
f
(
x
n
)
-
f
(
x
*
)
)
+
(
1
-
α
n
)
(
y
n
-
x
*
)
∥
2
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
∥
f
(
x
n
)
-
f
(
x
*
)
∥
2
+
(
1
-
α
n
)
∥
y
n
-
x
*
∥
2
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
×
[
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
]
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
=
(
1
-
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
(
1
-
α
n
)
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
=
(
1
-
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
×
[
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
]
=
(
1
-
δ
n
)
∥
x
n
-
x
*
∥
2
+
σ
n
δ
n
,
where
δ
n
=
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
and
(126)
σ
n
=
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
γ
¯
-
1
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
.
Since
{
ρ
n
}
⊂
[
0,1
-
ρ
]
,
{
γ
n
}
⊂
[
a
,
a
^
]
⊂
(
0,1
)
, and
∑
n
=
1
∞
α
n
=
∞
, we deduce that
∑
n
=
1
∞
δ
n
=
∑
n
=
1
∞
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
=
∞
and
2
α
n
/
(
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
≤
2
/
ρ
a
(
γ
¯
-
1
)
. Note that
(127)
limsup
n
→
∞
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
=
limsup
n
→
∞
[
〈
(
I
-
V
)
x
*
,
y
n
-
x
n
〉
+
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
]
≤
0
.
Hence from (123) and Lemma 18 it follows that
(128)
limsup
n
→
∞
σ
n
=
limsup
n
→
∞
[
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
5555555555
×
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
]
≤
limsup
n
→
∞
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
γ
¯
-
1
+
limsup
n
→
∞
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
0
.
Applying Lemma 13 to (125), we infer that the sequence
{
x
n
}
converges strongly to
x
*
. This completes the proof.