# Algorithms of Common Solutions for Generalized Mixed Equilibria, Variational Inclusions, and Constrained Convex Minimization

Authors: Lu-Chuan Ceng; Suliman Al-Homidan
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H and let P
C be the metric projection of H onto C. Let S
:
C
→
C be a self-mapping on C. We denote by Fix (
S
) the set of fixed points of S and by R the set of all real numbers. A mapping A
:
C
→
H is called L-Lipschitz continuous if there exists a constant L
≥
0 such that
(1)
∥
A
x
-
A
y
∥
≤
L
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
In particular, if L
=
1, then A is called a nonexpansive mapping [1]; if L
∈
[
0,1
), then A is called a contraction.A mappingV is called strongly positive on H if there exists a constant μ
>
0 such that
(2)
〈
V
x
,
x
〉
≥
μ
∥
x
∥
2
,
∀
x
∈
H
.LetA
:
C
→
H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x
∈
C such that
(3)
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
The solution set of VIP (3) is denoted by VI (
C
,
A
).The VIP (3) was first discussed by Lions [2]. There are many applications of VIP (3) in various fields; see, for example, [3–6]. It is well known that if A is a strongly monotone and Lipschitz continuous mapping on C, then VIP (3) has a unique solution. In 1976, Korpelevič [7] proposed an iterative algorithm for solving the VIP (3) in Euclidean space R
n:
(4)
y
n
=
P
C
(
x
n
-
τ
A
x
n
)
,
x
n
+
1
=
P
C
(
x
n
-
τ
A
y
n
)
,
∀
n
≥
0
,
with τ
>
0 a given number, which is known as the extragradient method (see also [8]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [9–24] and references therein, to name but a few.Letφ
:
C
→
R be a real-valued function, A
:
H
→
H a nonlinear mapping, and Θ
:
C
×
C
→
R a bifunction. In 2008, Peng and Yao [12] introduced the following generalized mixed equilibrium problem (GMEP) of finding x
∈
C such that
(5)
Θ
(
x
,
y
)
+
φ
(
y
)
-
φ
(
x
)
+
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
We denote the set of solutions of GMEP (5) by GMEP (
Θ
,
φ
,
A
). The GMEP (5) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied; see, for example, [11, 14, 23, 25–28]. If φ
=
0 and A
=
0, then GMEP (5) reduces to the equilibrium problem (EP) which is to find x
∈
C such that
(6)
Θ
(
x
,
y
)
≥
0
,
∀
y
∈
C
.
It is considered and studied in [29]. The set of solutions of EP is denoted by EP (
Θ
). It is worth mentioning that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.Throughout this paper, it is assumed as in [12] that Θ
:
C
×
C
→
R is a bifunction satisfying conditions (A1)–(A4) and φ
:
C
→
R is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1)
Θ
(
x
,
x
)
=
0 for all x
∈
C;
(A2)
Θ is monotone; that is, Θ
(
x
,
y
)
+
Θ
(
y
,
x
)
≤
0 for any x
,
y
∈
C;
(A3)
Θ is upper-hemicontinuous; that is, for each x
,
y
,
z
∈
C,
(7)
limsup
⁡
t
→
0
+
Θ
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
Θ
(
x
,
y
)
;
(A4)
Θ
(
x
,
·
) is convex and lower semicontinuous for each x
∈
C;
(B1)
for eachx
∈
H and r
>
0, there exists a bounded subset D
x
⊂
C and y
x
∈
C such that, for any z
∈
C
∖
D
x,
(8)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
y
x
-
z
,
z
-
x
〉
<
0
;
(B2)
C is a bounded set.Next we list some elementary results for the MEP.Proposition 1 (see [26]).
Assume thatΘ
:
C
×
C
→
R satisfies (A1)–(A4) and let φ
:
C
→
R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r
>
0 and x
∈
H, define a mapping T
r
(
Θ
,
φ
)
:
H
→
C as follows:
(9)
T
r
(
Θ
,
φ
)
(
x
)
=
{
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
z
∈
C
:
Θ
(
z
,
y
)
+
φ
(
y
)
-
φ
(
z
)
T
r
(
Θ
,
φ
)
(
x
)
=
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
}
,
for all x
∈
H. Then the following hold:(i)
for each
x
∈
H
,
T
r
(
Θ
,
φ
)
(
x
)
is nonempty and single-valued;
(ii)
T
r
(
Θ
,
φ
)
is firmly nonexpansive; that is, for any
x
,
y
∈
H
,
(10)
∥
T
r
(
Θ
,
φ
)
x
-
T
r
(
Θ
,
φ
)
y
∥
2
≤
〈
T
r
(
Θ
,
φ
)
x
-
T
r
(
Θ
,
φ
)
y
,
x
-
y
〉
;
(iii)
Fix
(
T
r
(
Θ
,
φ
)
)
=
M
E
P
(
Θ
,
φ
)
;
(iv)
M
E
P
(
Θ
,
φ
)
is closed and convex;
(v)
∥
T
s
(
Θ
,
φ
)
x
-
T
t
(
Θ
,
φ
)
x
∥
2
≤
(
(
s
-
t
)
/
s
)
〈
T
s
(
Θ
,
φ
)
x
-
T
t
(
Θ
,
φ
)
x
,
T
s
(
Θ
,
φ
)
x
-
x
〉
for all
s
,
t
>
0
and
x
∈
H
.Letλ
n
,
1
,
λ
n
,
2
,
…
,
λ
n
,
N
∈
(
0,1
], n
≥
1. Given the nonexpansive mappings S
1
,
S
2
,
…
,
S
N on H, for each n
≥
1, the mappings U
n
,
1
,
U
n
,
2
,
…
,
U
n
,
N are defined by
(11)
U
n
,
1
=
λ
n
,
1
S
1
+
(
1
-
λ
n
,
1
)
I
,
U
n
,
2
=
λ
n
,
2
S
n
U
n
,
1
+
(
1
-
λ
n
,
2
)
I
,
U
n
,
n
-
1
=
λ
n
-
1
T
n
-
1
U
n
,
n
+
(
1
-
λ
n
-
1
)
I
,
⋮
U
n
,
N
-
1
=
λ
n
,
N
-
1
S
N
-
1
U
n
,
N
-
2
+
(
1
-
λ
n
,
N
-
1
)
I
,
W
n
:
=
U
n
,
N
=
λ
n
,
N
S
N
U
n
,
N
-
1
+
(
1
-
λ
n
,
N
)
I
.TheW
n is called the W-mapping generated by S
1
,
…
,
S
N and λ
n
,
1
,
λ
n
,
2
,
…
,
λ
n
,
N. Note that the nonexpansivity of S
i implies the nonexpansivity of  W
n.In 2012, combining the hybrid steepest-descent method in [30] and hybrid viscosity approximation method in [31], Ceng et al. [27] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of GMEP (5) and the set of fixed points of a finite family of nonexpansive mappings {
S
i
}
i
=
1
N.Theorem CGY (see [27, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ
:
C
×
C
→
R be a bifunction satisfying assumptions (A1)–(A4) and φ
:
C
→
R a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping A
:
H
→
H be δ-inverse-strongly monotone and {
S
i
}
i
=
1
N a finite family of nonexpansive mappings on H such that ∩
i
=
1
N
Fix
⁡
(
S
i
)
∩
G
M
E
P
(
Θ
,
φ
,
A
)
≠
∅. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with constants κ
,
η
>
0 and V
:
H
→
H a ρ-Lipschitzian mapping with constant ρ
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
ρ
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Suppose {
α
n
} and {
β
n
} are two sequences in (
0,1
), {
γ
n
} is a sequence in (
0,2
δ
], and {
λ
n
,
i
}
i
=
1
N is a sequence in [
a
,
b
] with 0
<
a
≤
b
<
1. For every n
≥
1, let W
n be the W-mapping generated by S
1
,
…
,
S
N and λ
n
,
1
,
λ
n
,
2
,
…
,
λ
n
,
N. Given x
1
∈
H arbitrarily, suppose that the sequences {
x
n
} and {
u
n
} are generated iteratively by
(12)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
x
n
+
1
=
α
n
γ
V
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
μ
F
)
W
n
u
n
,
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
μ
F
)
W
n
u
n
∀
n
≥
1
,
where the sequences {
α
n
}
,
{
β
n
}
,
{
r
n
} and the finite family of sequences {
λ
n
,
i
}
i
=
1
N satisfy the conditions:(i)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞;
(ii)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1;
(iii)
0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
δ and lim
⁡
n
→
∞
(
r
n
+
1
-
r
n
)
=
0;
(iv)
lim
⁡
n
→
∞
(
λ
n
+
1
,
i
-
λ
n
,
i
)
=
0 for all i
=
1,2
,
…
,
N.Then both {
x
n
} and {
u
n
} converge strongly to x
*
∈
∩
i
=
1
N
Fix
⁡
(
S
i
)
∩
G
M
E
P
(
Θ
,
φ
,
A
), where x
*
=
P
∩
i
=
1
N
Fix
(
S
i
)
∩
G
M
E
P
(
Θ
,
φ
,
A
)
(
I
-
μ
F
+
γ
f
)
x
* is a unique solution of the variational inequality problem (VIP):
(13)
〈
(
μ
F
-
γ
V
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
⋂
i
=
1
N
Fix
⁡
(
S
i
)
∩
G
M
E
P
(
Θ
,
φ
,
A
)
.LetB be a single-valued mapping of C into H and R a multivalued mapping with D
(
R
)
=
C. Consider the following variational inclusion: find a point x
∈
C such that
(14)
0
∈
B
x
+
R
x
.
We denote by I
(
B
,
R
) the solution set of the variational inclusion (14). In particular, if B
=
R
=
0, then I
(
B
,
R
)
=
C. If B
=
0, then problem (14) becomes the inclusion problem introduced by Rockafellar [32]. It is known that problem (14) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, and equilibria and game theory.In 1998, Huang [33] studied problem (14) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with D
(
R
)
=
C
=
H. Subsequently, Zeng et al. [34] further studied problem (14) in the case which is more general than Huang’s one [33]. Moreover, the authors [34] obtained the same strong convergence conclusion as in Huang’s result [33]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [35–39] and the references therein.Let{
T
n
}
n
=
1
∞ be an infinite family of nonexpansive self-mappings on C and {
λ
n
}
n
=
0
∞ a sequence of nonnegative numbers in [
0,1
]. For any n
≥
1, define a self-mapping W
n on C as follows:
(15)
U
n
,
n
+
1
=
I
,
U
n
,
n
=
λ
n
T
n
U
n
,
n
+
1
+
(
1
-
λ
n
)
I
,
U
n
,
n
-
1
=
λ
n
-
1
T
n
-
1
U
n
,
n
+
(
1
-
λ
n
-
1
)
I
,
⋮
U
n
,
k
=
λ
k
T
k
U
n
,
k
+
1
+
(
1
-
λ
k
)
I
,
U
n
,
k
-
1
=
λ
k
-
1
T
k
-
1
U
n
,
k
+
(
1
-
λ
k
-
1
)
I
,
⋮
U
n
,
2
=
λ
2
T
2
U
n
,
3
+
(
1
-
λ
2
)
I
,
W
n
=
U
n
,
1
=
λ
1
T
1
U
n
,
2
+
(
1
-
λ
1
)
I
.
Such a mapping W
n is called the W-mapping generated by T
n
,
T
n
-
1
,
…
,
T
1 and λ
n
,
λ
n
-
1
,
…
,
λ
1.WheneverC
=
H a real Hilbert space, Yao et al. [11] very recently introduced and analyzed an iterative algorithm for finding a common element of the set of solutions of GMEP (5), the set of solutions of the variational inclusion (14), and the set of fixed points of an infinite family of nonexpansive mappings.Theorem YCL (see [11, Theorem 3.2]).
Letφ
:
H
→
R be a lower semicontinuous and convex function and Θ
:
H
×
H
→
R a bifunction satisfying conditions (A1)–(A4) and (B1). Let V be a strongly positive bounded linear operator with coefficient μ
>
0 and R
:
H
→
2
H a maximal monotone mapping. Let the mappings A
,
B
:
H
→
H be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let f
:
H
→
H be a ρ-contraction. Let r
>
0, γ
>
0, and λ
>
0 be three constants such that r
<
2
α, λ
<
2
β, and 0
<
γ
<
μ
/
ρ. Let {
λ
}
n
=
1
∞ be a sequence of positive numbers in (
0
,
b
] for some b
∈
(
0,1
) and {
T
n
}
n
=
1
∞ an infinite family of nonexpansive self-mappings on H such that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
G
M
E
P
(
Θ
,
φ
,
A
)
∩
I
(
B
,
R
)
≠
∅. For arbitrarily given x
1
∈
H, let the sequence {
x
n
} be generated by
(16)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
y
-
u
n
,
A
x
n
〉
+
1
r
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
H
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
β
n
x
n
+
[
(
1
-
β
n
)
I
-
α
n
V
]
W
n
J
R
,
λ
(
u
n
-
λ
B
u
n
)
,
∀
n
≥
1
,
where {
α
n
}
,
{
β
n
} are two real sequences in [
0,1
] and W
n is the W-mapping defined by (15) (with X
=
H and C
=
H). Assume that the following conditions are satisfied:(C1)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞;
(C2)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1.Then the sequence {
x
n
} converges strongly to x
*
∈
Ω, where x
*
=
P
Ω
(
γ
f
(
x
*
)
+
(
I
-
V
)
x
*
) is a unique solution of the VIP:
(17)
〈
(
γ
f
-
V
)
x
*
,
y
-
x
*
〉
≤
0
,
∀
y
∈
Ω
.Letf
:
C
→
R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C
(18)
min
⁡
x
∈
C
f
(
x
)
(assuming the existence of minimizers). We denote by Γ the set of minimizers of CMP (18). It is well known that the gradient-projection algorithm (GPA) generates a sequence {
x
n
} determined by the gradient ∇
f and the metric projection P
C:
(19)
x
n
+
1
:
=
P
C
(
x
n
-
λ
∇
f
(
x
n
)
)
,
∀
n
≥
0
,
or more generally,
(20)
x
n
+
1
:
=
P
C
(
x
n
-
λ
n
∇
f
(
x
n
)
)
,
∀
n
≥
0
,
where, in both (19) and (20), the initial guess x
0 is taken from C arbitrarilyn and the parameters λ or λ
n are positive real numbers. The convergence of algorithms (19) and (20) depends on the behavior of the gradient ∇
f. As a matter of fact, it is known that if ∇
f is α-strongly monotone and L-Lipschitz continuous, then, for 0
<
λ
<
2
α
/
L
2, the operator P
C
(
I
-
λ
∇
f
) is a contraction; hence, the sequence {
x
n
} defined by the GPA (19) converges in norm to the unique solution of CMP (18). More generally, if the sequence {
λ
n
} is chosen to satisfy the property
(21)
0
<
liminf
⁡
n
→
∞
λ
n
≤
limsup
⁡
n
→
∞
λ
n
<
2
α
L
2
,
then the sequence {
x
n
} defined by the GPA (20) converges in norm to the unique minimizer of CMP (18). If the gradient ∇
f is only assumed to be Lipschitz continuous, then {
x
n
} can only be weakly convergent if H is infinite-dimensional (a counterexample is given in Section 5 of Xu [40]).Since the Lipschitz continuity of the gradient∇
f implies that it is actually (
1
/
L
)-inverse-strongly monotone (ism) [41], its complement can be an averaged mapping (i.e., it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping). Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Recently, Xu [40] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.Motivated and inspired by the above facts, we in this paper introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the CMP (18) for a convex functional f
:
C
→
R with L-Lipschitz continuous gradient ∇
f, the set of solutions of a finite family of GMEPs, and the set of solutions of a finite family of variational inclusions for maximal monotone and inverse-strong monotone mappings in a real Hilbert space. Under mild control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets. Our iterative algorithms are based on Korpelevich’s extragradient method, hybrid steepest-descent method in [30], viscosity approximation method, and averaged mapping approach to the GPA in [40]. The results obtained in this paper improve and extend the corresponding results announced by many others.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let C be a nonempty closed convex subset of H. We write x
n
⇀
x to indicate that the sequence {
x
n
} converges weakly to x and x
n
→
x to indicate that the sequence {
x
n
} converges strongly to x. Moreover, we use ω
w
(
x
n
) to denote the weak ω-limit set of the sequence {
x
n
}; that is,
(22)
ω
w
(
x
n
)
:
=
{
forsomesubsequence
{
x
n
i
}
of
{
x
n
}
x
∈
H
:
x
n
i
⇀
x
for
some
subsequence
{
x
n
i
}
of
{
x
n
}
}
.Recall that a mappingA
:
C
→
H is called(i)
monotone if(23)
〈
A
x
-
A
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
C
;
(ii)
η-strongly monotone if there exists a constant η
>
0 such that
(24)
〈
A
x
-
A
y
,
x
-
y
〉
≥
η
∥
x
-
y
∥
2
,
∀
x
,
y
∈
C
;
(iii)
α-inverse-strongly monotone if there exists a constant α
>
0 such that
(25)
〈
A
x
-
A
y
,
x
-
y
〉
≥
α
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.It is obvious that ifA is α-inverse-strongly monotone, then A is monotone and (
1
/
α
)-Lipschitz continuous.The metric (or nearest point) projection fromH onto C is the mapping P
C
:
H
→
C which assigns to each point x
∈
H the unique point P
C
x
∈
C satisfying the property
(26)
∥
x
-
P
C
x
∥
=
inf
⁡
y
∈
C
∥
x
-
y
∥
=
:
d
(
x
,
C
)
.Some important properties of projections are gathered in the following proposition.Proposition 2.
For givenx
∈
H and z
∈
C,(i)
z
=
P
C
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0
, for all
y
∈
C
;
(ii)
z
=
P
C
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2
, for all
y
∈
C
;
(iii)
〈
P
C
x
-
P
C
y
,
x
-
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2
, for all
y
∈
H
.
Consequently,P
C is nonexpansive and monotone.IfA is an α-inverse-strongly monotone mapping of C into H, then it is obvious that A is (
1
/
α
)-Lipschitz continuous. We also have that, for all u
,
v
∈
C and λ
>
0,
(27)
∥
(
I
-
λ
A
)
u
-
(
I
-
λ
A
)
v
∥
2
=
∥
(
u
-
v
)
-
λ
(
A
u
-
A
v
)
∥
2
=
∥
u
-
v
∥
2
-
2
λ
〈
A
u
-
A
v
,
u
-
v
〉
+
λ
2
∥
A
u
-
A
v
∥
2
≤
∥
u
-
v
∥
2
+
λ
(
λ
-
2
α
)
×
∥
A
u
-
A
v
∥
2
.
So, if λ
≤
2
α, then I
-
λ
A is a nonexpansive mapping from C to H.Definition 3.
A mappingT
:
H
→
H is said to be(a)
nonexpansive [1] if
(28)
∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
,
∀
x
,
y
∈
H
;
(b)
firmly nonexpansive if2
T
-
I is nonexpansive or, equivalently, if T is 1-inverse-strongly monotone (1-ism),
(29)
〈
x
-
y
,
T
x
-
T
y
〉
≥
∥
T
x
-
T
y
∥
2
,
∀
x
,
y
∈
H
;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(30)
T
=
1
2
(
I
+
S
)
,
where S
:
H
→
H is nonexpansive; projections are firmly nonexpansive.It can be easily seen that ifT is nonexpansive, then I
-
T is monotone. It is also easy to see that a projection P
C is 1-ism. Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.Definition 4.
A mappingT
:
H
→
H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,
(31)
T
≡
(
1
-
α
)
I
+
α
S
,
where α
∈
(
0,1
) and S
:
H
→
H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus, firmly nonexpansive mappings (in particular, projections) are (
1
/
2
)-averaged mappings.Proposition 5 (see [42]).
LetT
:
H
→
H be a given mapping.(i)
T
is nonexpansive if and only if the complement
I
-
T
is
(
1
/
2
)
-ism.
(ii)
If
T
is
ν
-ism, then, for
γ
>
0
,
γ
T
is
(
ν
/
γ
)
-ism.
(iii)
T
is averaged if and only if the complement
I
-
T
is
ν
-ism for some
ν
>
1
/
2
. Indeed, for
α
∈
(
0,1
)
,
T
is
α
-averaged if and only if
I
-
T
is
(
1
/
2
α
)
-ism.Proposition 6 (see [42, 43]).
LetS
,
T
,
V
:
H
→
H be given operators.(i)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is averaged and
V
is nonexpansive, then
T
is averaged.
(ii)
T
is firmly nonexpansive if and only if the complement
I
-
T
is firmly nonexpansive.
(iii)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is firmly nonexpansive and
V
is nonexpansive, then
T
is averaged.
(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings
{
T
i
}
i
=
1
N
is averaged, then so is the composite
T
1
⋯
T
N
. In particular, if
T
1
is
α
1
-averaged and
T
2
is
α
2
-averaged, where
α
1
,
α
2
∈
(
0,1
)
, then the composite
T
1
T
2
is
α
-averaged, where
α
=
α
1
+
α
2
-
α
1
α
2
.
(v)
If the mappings
{
T
i
}
i
=
1
N
are averaged and have a common fixed point, then
(32)
⋂
i
=
1
N
‍
Fix
⁡
(
T
i
)
=
Fix
⁡
(
T
1
⋯
T
N
)
.The notation Fix
⁡
(
T
) denotes the set of all fixed points of the mapping T; that is, Fix
⁡
(
T
)
=
{
x
∈
H
:
T
x
=
x
}.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 7.
LetX be a real inner product space. Then the following inequality holds:
(33)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
X
.Lemma 8.
LetA
:
C
→
H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2(i)) implies
(34)
u
∈
VI
⁡
(
C
,
A
)
⇔
u
=
P
C
(
u
-
λ
A
u
)
,
f
o
r
s
o
m
e
λ
>
0
.Lemma 9 (see [44, Demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with Fix
⁡
(
T
)
≠
∅. Then I
-
T is demiclosed. That is, whenever {
x
n
} is a sequence in C weakly converging to some x
∈
C and the sequence {
(
I
-
T
)
x
n
} strongly converges to some y, it follows that (
I
-
T
)
x
=
y. Here I is the identity operator of H.Lemma 10 (see [45]).
Let{
s
n
} be a sequence of nonnegative numbers satisfying the conditions
(35)
s
n
+
1
≤
(
1
-
α
n
)
s
n
+
α
n
β
n
,
∀
n
≥
1
,
where {
α
n
} and {
β
n
} are sequences of real numbers such that(i)
{
α
n
}
⊂
[
0,1
] and ∑
n
=
1
∞
‍
α
n
=
∞ or, equivalently,
(36)
∏
n
=
1
∞
‍
(
1
-
α
n
)
:
=
lim
⁡
n
→
∞
∏
k
=
1
n
‍
(
1
-
α
k
)
=
0
;
(ii)
limsup
⁡
n
→
∞
β
n
≤
0, or ∑
n
=
1
∞
‍
|
α
n
β
n
|
<
∞.Then lim
⁡
n
→
∞
s
n
=
0.Lemma 11 (see [46]).
Let{
x
n
} and {
z
n
} be bounded sequences in a Banach space X and {
β
n
} a sequence in [
0,1
] with
(37)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1
.
Suppose that x
n
+
1
=
(
1
-
β
n
)
z
n
+
β
n
x
n for each n
≥
1 and
(38)
limsup
⁡
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0
.
Then lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0.The following lemma can be easily proven and, therefore, we omit the proof.Lemma 12.
LetV
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0, and let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Then, for 0
≤
γ
l
<
μ
η,
(39)
〈
(
μ
F
-
γ
V
)
x
-
(
μ
F
-
γ
V
)
y
,
x
-
y
〉
≥
(
μ
η
-
γ
l
)
∥
x
-
y
∥
2
,
〈
(
μ
F
-
γ
V
)
x
-
(
μ
F
-
γ
V
)
y
,
x
-
y
〉
≥
(
μ
η
-
γ
l
)
∀
x
,
y
∈
H
.
That is, μ
F
-
γ
V is strongly monotone with constant μ
η
-
γ
l.LetC be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (
0,1
] and let μ
>
0. Associating with a nonexpansive mapping T
:
C
→
H, we define the mapping T
λ
:
C
→
H by
(40)
T
λ
x
:
=
T
x
-
λ
μ
F
(
T
x
)
,
∀
x
∈
C
,
where F
:
H
→
H is an operator such that, for some positive constants κ
,
η
>
0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:
(41)
∥
F
x
-
F
y
∥
≤
κ
∥
x
-
y
∥
,
〈
F
x
-
F
y
,
x
-
y
〉
≥
η
∥
x
-
y
∥
2
,
for all x
,
y
∈
H.Lemma 13 (see [45, Lemma 3.1]).
T
λ is a contraction provided 0
<
μ
<
2
η
/
κ
2; that is,
(42)
∥
T
λ
x
-
T
λ
y
∥
≤
(
1
-
λ
τ
)
∥
x
-
y
∥
,
∀
x
,
y
∈
C
,
where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
)
∈
(
0,1
].Recall that a set-valued mappingR
:
D
(
R
)
⊂
H
→
2
H is called monotone if for all x
,
y
∈
D
(
R
), f
∈
R
(
x
) and g
∈
R
(
y
) imply
(43)
〈
f
-
g
,
x
-
y
〉
≥
0
.
A set-valued mapping R is called maximal monotone if R is monotone and (
I
+
λ
R
)
D
(
R
)
=
H for each λ
>
0, where I is the identity mapping of H. We denote by G
(
R
) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (
x
,
f
)
∈
H
×
H, 〈
f
-
g
,
x
-
y
〉
≥
0 for every (
y
,
g
)
∈
G
(
R
) implies f
∈
R
(
x
).LetA
:
C
→
H be a monotone, k-Lipschitz continuous mapping and let N
C
v be the normal cone to C at v
∈
C; that is,
(44)
N
C
v
=
{
w
∈
H
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
C
}
.
Define
(45)
T
v
=
{
A
v
+
N
C
v
,
if
v
∈
C
,
∅
,
if
v
∉
C
.
Then, T is maximal monotone and 0
∈
T
v if and only if v
∈
V
I
(
C
,
A
); see [32].Assume thatR
:
D
(
R
)
⊂
H
→
2
H is a maximal monotone mapping. Then, for λ
>
0, associated with R, the resolvent operator J
R
,
λ can be defined as
(46)
J
R
,
λ
x
=
(
I
+
λ
R
)
-
1
x
,
∀
x
∈
H
.
In terms of Huang [33] (see also [34]), the following property holds for the resolvent operator J
R
,
λ
:
H
→
D
(
R
)
¯.Lemma 14.
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(47)
〈
J
R
,
λ
x
-
J
R
,
λ
y
,
x
-
y
〉
≥
∥
J
R
,
λ
x
-
J
R
,
λ
y
∥
2
,
∀
x
,
y
∈
H
.
Consequently,J
R
,
λ is nonexpansive and monotone.Lemma 15 (see [39]).
LetR be a maximal monotone mapping with D
(
R
)
=
C. Then, for any given λ
>
0, u
∈
C is a solution of problem (14) if and only if u
∈
C satisfies
(48)
u
=
J
R
,
λ
(
u
-
λ
B
u
)
.Lemma 16 (see [34]).
LetR be a maximal monotone mapping with D
(
R
)
=
C and let B
:
C
→
H be a strongly monotone, continuous, and single-valued mapping. Then, for each z
∈
H, the equation z
∈
(
B
+
λ
R
)
x has a unique solution x
λ for λ
>
0.Lemma 17 (see [39]).
LetR be a maximal monotone mapping with D
(
R
)
=
C and let B
:
C
→
H be a monotone, continuous, and single-valued mapping. Then (
I
+
λ
(
R
+
B
)
)
C
=
H for each λ
>
0. In this case, R
+
B is maximal monotone.


## 3. Implicit Iterative Algorithm and Its Convergence Criteria

We now state and prove the first main result of this paper.Theorem 18.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let M
,
N be two integers. Let Θ
k be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
k
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function, where k
∈
{
1,2
,
…
,
M
}. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
H
→
H and B
i
:
C
→
H be μ
k-inverse-strongly monotone and η
i-inverse-strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
}, i
∈
{
1,2
,
…
,
N
}. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
∩
k
=
1
M
G
M
E
P
(
Θ
k
,
φ
k
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. Let {
x
n
} be a sequence generated by
(49)
u
n
=
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
T
r
M
-
1
,
n
(
Θ
M
-
1
,
φ
M
-
1
)
×
(
I
-
r
M
-
1
,
n
A
M
-
1
)
⋯
T
r
1
,
n
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
v
n
=
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
J
R
N
-
1
,
λ
N
-
1
,
n
×
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
x
n
=
s
n
γ
V
x
n
+
(
I
-
s
n
μ
F
)
T
n
v
n
,
x
n
+
(
I
-
s
n
μ
F
)
T
n
v
n
,
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)
). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), for all i
∈
{
1,2
,
…
,
N
};
(iii)
{
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
), for all k
∈
{
1,2
,
…
,
M
}.Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of the VIP:
(50)
〈
(
μ
F
-
γ
V
)
q
,
p
-
q
〉
≥
0
,
∀
p
∈
Ω
.
Equivalently, q
=
P
Ω
(
I
-
μ
F
+
γ
V
)
q.Proof.
First of all, let us show that the sequence{
x
n
} is well defined. Indeed, since ∇
f is L-Lipschitzian, it follows that ∇
f is 1
/
L-ism; see [41]. By Proposition 5(ii) we know that, for λ
>
0, λ
∇
f is (
1
/
λ
L
)-ism. So by Proposition 5(iii) we deduce that I
-
λ
∇
f is  (
λ
L
/
2
)-averaged. Now since the projection P
C is  (
1
/
2
)-averaged, it is easy to see from Proposition 6(iv) that the composite P
C
(
I
-
λ
∇
f
) is (
2
+
λ
L
)
/
4-averaged for λ
∈
(
0
,
2
/
L
). Hence, we obtain that, for each n
≥
1, P
C
(
I
-
λ
n
∇
f
) is  (
(
2
+
λ
n
L
)
/
4
)-averaged for each λ
n
∈
(
0
,
2
/
L
). Therefore, we can write
(51)
P
C
(
I
-
λ
n
∇
f
)
=
2
-
λ
n
L
4
I
+
2
+
λ
n
L
4
T
n
=
s
n
I
+
(
1
-
s
n
)
T
n
,
where T
n is nonexpansive and s
n
:
=
s
n
(
λ
n
)
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
). It is clear that
(52)
λ
n
⟶
2
L
⇔
s
n
⟶
0
.
Put(53)
Δ
n
k
=
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
T
r
k
-
1
,
n
(
Θ
k
-
1
,
φ
k
-
1
)
×
(
I
-
r
k
-
1
,
n
A
k
-
1
)
⋯
T
r
1
,
n
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
for all k
∈
{
1,2
,
…
,
M
} and n
≥
1,
(54)
Λ
n
i
=
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
J
R
i
-
1
,
λ
i
-
1
,
n
×
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
,
for all i
∈
{
1,2
,
…
,
N
} and n
≥
1, and Δ
n
0
=
Λ
n
0
=
I, where I is the identity mapping on H. Then we have that u
n
=
Δ
n
M
x
n and v
n
=
Λ
n
N
u
n.
Consider the following mappingG
n on H defined by
(55)
G
n
x
=
s
n
γ
V
x
+
(
I
-
s
n
μ
F
)
T
n
Λ
n
N
Δ
n
M
x
,
∀
x
∈
H
,
n
≥
1
,
where s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
). By Proposition 1(ii) and Lemma 13 we obtain from (27) that for all x
,
y
∈
H
(56)
∥
G
n
x
-
G
n
y
∥
≤
s
n
γ
∥
V
x
-
V
y
∥
+
∥
(
I
-
s
n
μ
F
)
T
n
Λ
n
N
Δ
n
M
x
≤
s
n
γ
∥
V
x
-
V
y
∥
+
0
-
(
I
-
s
n
μ
F
)
T
n
Λ
n
N
Δ
n
M
y
∥
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Λ
n
N
Δ
n
M
x
-
Λ
n
N
Δ
n
M
y
∥
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
Δ
n
M
x
≤
s
n
γ
∥
V
x
-
V
y
∥
+
000000
-
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
Δ
n
M
y
∥
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Λ
n
N
-
1
Δ
n
M
x
-
Λ
n
N
-
1
Δ
n
M
y
∥
⋮
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Λ
n
0
Δ
n
M
x
-
Λ
n
0
Δ
n
M
y
∥
=
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Δ
n
M
x
-
Δ
n
M
y
∥
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
≤
s
n
γ
∥
V
x
-
V
y
∥
+
0000000
-
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
y
∥
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Δ
n
M
-
1
x
-
Δ
n
M
-
1
y
∥
⋮
≤
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
Δ
n
0
x
-
Δ
n
0
y
∥
=
s
n
γ
l
∥
x
-
y
∥
+
(
1
-
s
n
τ
)
∥
x
-
y
∥
=
(
1
-
s
n
(
τ
-
γ
l
)
)
∥
x
-
y
∥
.
Since 0
<
1
-
s
n
(
τ
-
γ
l
)
<
1, G
n
:
H
→
H is a contraction. Therefore, by the Banach contraction principle, G
n has a unique fixed point x
n
∈
H, which uniquely solves the fixed point equation:
(57)
x
n
=
s
n
γ
V
x
n
+
(
I
-
s
n
μ
F
)
T
n
Λ
n
N
Δ
n
M
x
n
.
This shows that the sequence {
x
n
} is defined well.
Note that0
≤
γ
l
<
τ and μ
η
≥
τ
⇔
κ
≥
η. Hence, by Lemma 12 we know that
(58)
〈
(
μ
F
-
γ
V
)
x
-
(
μ
F
-
γ
V
)
y
,
x
-
y
〉
≥
(
μ
η
-
γ
l
)
∥
x
-
y
∥
2
,
〈
(
μ
F
-
γ
V
)
x
-
(
μ
F
-
γ
V
)
y
,
x
-
y
〉
≥
(
μ
η
-
)
0
∀
x
,
y
∈
H
.
That is, μ
F
-
γ
V is strongly monotone for 0
≤
γ
l
<
τ
≤
μ
η. Moreover, it is clear that μ
F
-
γ
V is Lipschitz continuous. So the VIP (50) has only one solution. Below we use q
∈
Ω to denote the unique solution of the VIP (50).
Now, let us show that{
x
n
} is bounded. In fact, take p
∈
Ω arbitrarily. Then from (27) and Proposition 1(ii) we have
(59)
∥
u
n
-
p
∥
=
∥
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
n
=
=
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
p
∥
≤
∥
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
n
=
=
-
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
p
∥
≤
∥
Δ
n
M
-
1
x
n
-
Δ
n
M
-
1
p
∥
⋮
≤
∥
Δ
n
0
x
n
-
Δ
n
0
p
∥
=
∥
x
n
-
p
∥
.
Similarly, we have
(60)
∥
v
n
-
p
∥
=
∥
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
=
=
-
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
p
∥
≤
∥
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
-
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
p
∥
≤
∥
Λ
n
N
-
1
u
n
-
Λ
n
N
-
1
p
∥
⋮
≤
∥
Λ
n
0
u
n
-
Λ
n
0
p
∥
=
∥
u
n
-
p
∥
.
Combining (59) and (60), we have
(61)
∥
v
n
-
p
∥
≤
∥
x
n
-
p
∥
.
Since
(62)
p
=
P
C
(
I
-
λ
n
∇
f
)
p
=
s
n
p
+
(
1
-
s
n
)
T
n
p
,
∀
λ
n
∈
(
0
,
2
L
)
,
where s
n
:
=
s
n
(
λ
n
)
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
), it is clear that T
n
p
=
p for each λ
n
∈
(
0
,
2
/
L
). Thus, utilizing Lemma 13 and the nonexpansivity of T
n, we obtain from (61) that
(63)
∥
x
n
-
p
∥
=
∥
s
n
(
γ
V
x
n
-
μ
F
p
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
≤
∥
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
+
s
n
∥
γ
V
x
n
-
μ
F
p
∥
≤
(
1
-
s
n
τ
)
∥
v
n
-
p
∥
+
s
n
(
γ
∥
V
x
n
-
V
p
∥
+
∥
γ
V
p
-
μ
F
p
∥
)
≤
(
1
-
s
n
τ
)
∥
x
n
-
p
∥
+
s
n
(
γ
l
∥
x
n
-
p
∥
+
∥
γ
V
p
-
μ
F
p
∥
)
=
(
1
-
s
n
(
τ
-
γ
l
)
)
∥
x
n
-
p
∥
+
s
n
∥
γ
V
p
-
μ
F
p
∥
.
This implies that ∥
x
n
-
p
∥
≤
∥
γ
V
p
-
μ
F
p
∥
/
(
τ
-
γ
l
). Hence, {
x
n
} is bounded. So, according to (59) and (61) we know that {
u
n
}, {
v
n
}, {
T
n
v
n
}, {
V
x
n
}, and {
F
T
n
v
n
} are bounded.
Next let us show that∥
u
n
-
x
n
∥
→
0, ∥
v
n
-
u
n
∥
→
0, and ∥
x
n
-
T
n
x
n
∥
→
0 as n
→
∞.
Indeed, from (27) it follows that for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}
(64)
∥
v
n
-
p
∥
2
=
∥
Λ
n
N
u
n
-
p
∥
2
≤
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
,
∥
u
n
-
p
∥
2
=
∥
Δ
n
M
x
n
-
p
∥
2
≤
∥
Δ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
T
r
k
,
n
(
Θ
k
,
φ
k
)
∥
2
≤
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
.
Thus, utilizing Lemma 7, from (49) and (64) we have
(65)
∥
x
n
-
p
∥
2
=
∥
s
n
(
γ
V
x
n
-
μ
F
p
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
2
≤
∥
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
2
+
2
s
n
〈
γ
V
x
n
-
μ
F
p
,
x
n
-
p
〉
≤
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
2
s
n
〈
γ
V
x
n
-
μ
F
p
,
x
n
-
p
〉
=
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
2
s
n
γ
〈
V
x
n
-
V
p
,
x
n
-
p
〉
+
2
s
n
〈
γ
V
p
-
μ
F
p
,
x
n
-
p
〉
≤
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
=
[
1
-
2
s
n
(
τ
-
γ
l
)
+
s
n
2
τ
2
]
∥
x
n
-
p
∥
2
-
(
1
-
s
n
τ
)
2
[
+
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
∥
x
n
-
p
∥
2
+
s
n
2
τ
2
∥
x
n
-
p
∥
2
-
(
1
-
s
n
τ
)
2
×
[
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
,
which implies that
(66)
(
1
-
s
n
τ
)
2
[
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
]
≤
s
n
2
τ
2
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
.
Since {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) and {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
) for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}, from s
n
→
0 we conclude immediately that
(67)
lim
⁡
n
→
∞
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
=
0
,
lim
⁡
n
→
∞
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
=
0
,
for all k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}.
Furthermore, by Proposition1(ii) we obtain that for each k
∈
{
1,2
,
…
,
M
}
(68)
∥
Δ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
〈
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
,
Δ
n
k
x
n
-
p
〉
=
1
2
(
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
+
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
-
(
Δ
n
k
x
n
-
p
)
∥
2
)
≤
1
2
(
∥
Δ
n
k
-
1
x
n
-
p
∥
2
+
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
-
r
k
,
n
(
A
k
Δ
n
k
-
1
x
n
-
A
k
p
)
∥
2
)
,
which implies that
(69)
∥
Δ
n
k
x
n
-
p
∥
2
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
-
r
k
,
n
(
A
k
Δ
n
k
-
1
x
n
-
A
k
p
)
∥
2
=
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
-
r
k
,
n
2
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
2
r
k
,
n
〈
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
,
A
k
Δ
n
k
-
1
x
n
-
A
k
p
〉
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
≤
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
.
Also, by Lemma 14, we obtain that for each i
∈
{
1,2
,
…
,
N
}
(70)
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
〈
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
,
Λ
n
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
-
(
Λ
n
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies
(71)
∥
Λ
n
i
u
n
-
p
∥
2
≤
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
-
λ
i
,
n
2
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
2
λ
i
,
n
〈
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
,
B
i
Λ
n
i
-
1
u
n
-
B
i
p
〉
≤
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
.
Thus, utilizing Lemma 7, from (49), (69), and (71) we have
(72)
∥
x
n
-
p
∥
2
=
∥
s
n
(
γ
V
x
n
-
μ
F
p
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
2
≤
∥
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
2
+
2
s
n
〈
γ
V
x
n
-
μ
F
p
,
x
n
-
p
〉
≤
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
=
(
1
-
s
n
τ
)
2
∥
Λ
n
N
u
n
-
p
∥
2
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
∥
Λ
n
i
u
n
-
p
∥
2
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
=
(
1
-
s
n
τ
)
2
[
∥
Δ
n
M
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
Δ
n
M
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
[
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
]
+
2
s
n
γ
l
∥
x
n
-
p
∥
2
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
(
1
-
2
s
n
(
τ
-
γ
l
)
+
s
n
2
τ
2
)
∥
x
n
-
p
∥
2
-
(
1
-
s
n
τ
)
2
(
+
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
)
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
≤
∥
x
n
-
p
∥
2
+
s
n
2
τ
2
∥
x
n
-
p
∥
2
-
(
1
-
s
n
τ
)
2
×
(
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
)
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
.
It immediately follows that
(73)
(
1
-
s
n
τ
)
2
(
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
)
≤
s
n
2
τ
2
∥
x
n
-
p
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
s
n
∥
γ
V
p
-
μ
F
p
∥
∥
x
n
-
p
∥
.
Since {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) and {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
) for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}, from (67) and s
n
→
0 we deduce that
(74)
lim
⁡
n
→
∞
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
=
0
,
lim
⁡
n
→
∞
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
=
0
,
for all k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Hence, we get
(75)
∥
x
n
-
u
n
∥
=
∥
Δ
n
0
x
n
-
Δ
n
M
x
n
∥
≤
∥
Δ
n
0
x
n
-
Δ
n
1
x
n
∥
+
∥
Δ
n
1
x
n
-
Δ
n
2
x
n
∥
+
⋯
+
∥
Δ
n
M
-
1
x
n
-
Δ
n
M
x
n
∥
⟶
0
as
n
⟶
∞
,
(76)
∥
u
n
-
v
n
∥
=
∥
Λ
n
0
u
n
-
Λ
n
N
u
n
∥
≤
∥
Λ
n
0
u
n
-
Λ
n
1
u
n
∥
+
∥
Λ
n
1
u
n
-
Λ
n
2
u
n
∥
+
⋯
+
∥
Λ
n
N
-
1
u
n
-
Λ
n
N
u
n
∥
⟶
0
as
n
⟶
∞
.
So, taking into account that ∥
x
n
-
v
n
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
v
n
∥, we have
(77)
lim
⁡
n
→
∞
∥
x
n
-
v
n
∥
=
0
.
Thus, from (77) and s
n
→
0 we have
(78)
∥
v
n
-
T
n
v
n
∥
≤
∥
v
n
-
x
n
∥
+
∥
x
n
-
T
n
v
n
∥
=
∥
v
n
-
x
n
∥
+
s
n
∥
γ
V
x
n
-
μ
F
T
n
v
n
∥
⟶
0
as
n
⟶
∞
.
Now we show that ∥
x
n
-
T
n
x
n
∥
→
0 as n
→
∞. In fact, from the nonexpansivity of T
n, we have
(79)
∥
x
n
-
T
n
x
n
∥
≤
∥
x
n
-
v
n
∥
+
∥
v
n
-
T
n
v
n
∥
+
∥
T
n
v
n
-
T
n
x
n
∥
≤
2
∥
x
n
-
v
n
∥
+
∥
v
n
-
T
n
v
n
∥
.
By (77) and (78), we get
(80)
lim
⁡
n
→
∞
∥
x
n
-
T
n
x
n
∥
=
0
.
From (78) it is easy to see that
(81)
lim
⁡
n
→
∞
∥
x
n
-
T
n
v
n
∥
=
0
.
Observe that
(82)
∥
P
C
(
I
-
λ
n
∇
f
)
v
n
-
v
n
∥
=
∥
s
n
v
n
+
(
1
-
s
n
)
T
n
v
n
-
v
n
∥
=
(
1
-
s
n
)
∥
T
n
v
n
-
v
n
∥
≤
∥
T
n
v
n
-
v
n
∥
,
where s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
). Hence, we have
(83)
∥
P
C
(
I
-
2
L
∇
f
)
v
n
-
v
n
∥
≤
∥
P
C
(
I
-
2
L
∇
f
)
v
n
-
P
C
(
I
-
λ
n
∇
f
)
v
n
∥
+
∥
P
C
(
I
-
λ
n
∇
f
)
v
n
-
v
n
∥
≤
∥
(
I
-
2
L
∇
f
)
v
n
-
(
I
-
λ
n
∇
f
)
v
n
∥
+
∥
P
C
(
I
-
λ
n
∇
f
)
v
n
-
v
n
∥
≤
(
2
L
-
λ
n
)
∥
∇
f
(
v
n
)
∥
+
∥
T
n
v
n
-
v
n
∥
.
From the boundedness of {
v
n
}
,
s
n
→
0
(
⇔
λ
n
→
2
/
L
) and ∥
T
n
v
n
-
v
n
∥
→
0 (due to (78)), it follows that
(84)
lim
⁡
n
→
∞
∥
v
n
-
P
C
(
I
-
2
L
∇
f
)
v
n
∥
=
0
.
Further, we show thatω
w
(
x
n
)
⊂
Ω. Indeed, since {
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} which converges weakly to some w. Note that lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0 (due to (75)). Hence, x
n
i
⇀
w. Since C is closed and convex, C is weakly closed. So, we have w
∈
C. From (74) and (75), we have that Δ
n
i
k
x
n
i
⇀
w, Λ
n
i
m
u
n
i
⇀
w, u
n
i
⇀
w, v
n
i
⇀
w, where k
∈
{
1,2
,
…
,
M
}, m
∈
{
1,2
,
…
,
N
}. First, 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
n
m
u
n
=
J
R
m
,
λ
m
,
n
(
I
-
λ
m
,
n
B
m
)
Λ
n
m
-
1
u
n, n
≥
1, m
∈
{
1,2
,
…
,
N
}, we have
(85)
Λ
n
m
-
1
u
n
-
λ
m
,
n
B
m
Λ
n
m
-
1
u
n
∈
(
I
+
λ
m
,
n
R
m
)
Λ
n
m
u
n
;
that is,
(86)
1
λ
m
,
n
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
-
λ
m
,
n
B
m
Λ
n
m
-
1
u
n
)
∈
R
m
Λ
n
m
u
n
.
In terms of the monotonicity of R
m, we get
(87)
〈
v
-
Λ
n
m
u
n
,
g
-
B
m
v
-
1
λ
m
,
n
×
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
-
λ
m
,
n
B
m
Λ
n
m
-
1
u
n
)
v
-
Λ
n
m
u
n
,
g
-
B
m
v
-
1
λ
m
,
n
〉
≥
0
and hence
(88)
〈
v
-
Λ
n
m
u
n
,
g
〉
≥
〈
v
-
Λ
n
m
u
n
,
B
m
v
+
1
λ
m
,
n
×
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
-
λ
m
,
n
B
m
Λ
n
m
-
1
u
n
)
v
-
Λ
n
m
u
n
,
B
m
v
+
1
λ
m
,
n
〉
=
〈
B
m
Λ
n
m
-
1
u
n
+
1
λ
m
,
n
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
)
v
-
Λ
n
m
u
n
,
B
m
v
-
B
m
Λ
n
m
u
n
+
B
m
Λ
n
m
u
n
-
B
m
Λ
n
m
-
1
u
n
+
1
λ
m
,
n
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
)
〉
≥
〈
v
-
Λ
n
m
u
n
,
B
m
Λ
n
m
u
n
-
B
m
Λ
n
m
-
1
u
n
〉
+
〈
v
-
Λ
n
m
u
n
,
1
λ
m
,
n
(
Λ
n
m
-
1
u
n
-
Λ
n
m
u
n
)
〉
.
In particular,
(89)
〈
v
-
Λ
n
i
m
u
n
i
,
g
〉
≥
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
Λ
n
i
m
u
n
i
-
B
m
Λ
n
i
m
-
1
u
n
i
〉
+
〈
v
-
Λ
n
i
m
u
n
i
,
1
λ
m
,
n
i
(
Λ
n
i
m
-
1
u
n
i
-
Λ
n
i
m
u
n
i
)
〉
.
Since ∥
Λ
n
m
u
n
-
Λ
n
m
-
1
u
n
∥
→
0 (due to (74)) and ∥
B
m
Λ
n
m
u
n
-
B
m
Λ
n
m
-
1
u
n
∥
→
0 (due to the Lipschitz continuity of B
m), we conclude from Λ
n
i
m
u
n
i
⇀
w and condition (ii) that
(90)
lim
⁡
i
→
∞
〈
v
-
Λ
n
i
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 prove that w
∈
∩
k
=
1
M
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
). Since Δ
n
k
x
n
=
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n, n
≥
1, k
∈
{
1,2
,
…
,
M
}, we have
(91)
Θ
k
(
Δ
n
k
x
n
,
y
)
+
φ
k
(
y
)
-
φ
k
(
Δ
n
k
x
n
)
+
〈
A
k
Δ
n
k
-
1
x
n
,
y
-
Δ
n
k
x
n
〉
+
1
r
k
,
n
〈
y
-
Δ
n
k
x
n
,
Δ
n
k
x
n
-
Δ
n
k
-
1
x
n
〉
≥
0
.
By (A2), we have
(92)
φ
k
(
y
)
-
φ
k
(
Δ
n
k
x
n
)
+
〈
A
k
Δ
n
k
-
1
x
n
,
y
-
Δ
n
k
x
n
〉
+
1
r
k
,
n
〈
y
-
Δ
n
k
x
n
,
Δ
n
k
x
n
-
Δ
n
k
-
1
x
n
〉
≥
Θ
k
(
y
,
Δ
n
k
x
n
)
.
Let z
t
=
t
y
+
(
1
-
t
)
w for all t
∈
(
0,1
] and y
∈
C. This implies that z
t
∈
C. Then, we have
(93)
〈
z
t
-
Δ
n
k
x
n
,
A
k
z
t
〉
≥
φ
k
(
Δ
n
k
x
n
)
-
φ
k
(
z
t
)
+
〈
z
t
-
Δ
n
k
x
n
,
A
k
z
t
〉
-
〈
z
t
-
Δ
n
k
x
n
,
A
k
Δ
n
k
-
1
x
n
〉
-
〈
z
t
-
Δ
n
k
x
n
,
Δ
n
k
x
n
-
Δ
n
k
-
1
x
n
r
k
,
n
〉
+
Θ
k
(
z
t
,
Δ
n
k
x
n
)
=
φ
k
(
Δ
n
k
x
n
)
-
φ
k
(
z
t
)
+
〈
z
t
-
Δ
n
k
x
n
,
A
k
z
t
-
A
k
Δ
n
k
x
n
〉
+
〈
z
t
-
Δ
n
k
x
n
,
A
k
Δ
n
k
x
n
-
A
k
Δ
n
k
-
1
x
n
〉
-
〈
z
t
-
Δ
n
k
x
n
,
Δ
n
k
x
n
-
Δ
n
k
-
1
x
n
r
k
,
n
〉
+
Θ
k
(
z
t
,
Δ
n
k
x
n
)
.
By (74), we have ∥
A
k
Δ
n
k
x
n
-
A
k
Δ
n
k
-
1
x
n
∥
→
0 (due to the Lipschitz continuity of A
k). Furthermore, by the monotonicity of A
k, we obtain 〈
z
t
-
Δ
n
k
x
n
,
A
k
z
t
-
A
k
Δ
n
k
x
n
〉
≥
0. Then, by (A4) we obtain
(94)
〈
z
t
-
w
,
A
k
z
t
〉
≥
φ
k
(
w
)
-
φ
k
(
z
t
)
+
Θ
k
(
z
t
,
w
)
.
Utilizing (A1), (A4), and (94), we obtain
(95)
0
=
Θ
k
(
z
t
,
z
t
)
+
φ
k
(
z
t
)
-
φ
k
(
z
t
)
≤
t
Θ
k
(
z
t
,
y
)
+
(
1
-
t
)
Θ
k
(
z
t
,
w
)
+
t
φ
k
(
y
)
+
(
1
-
t
)
φ
k
(
w
)
-
φ
k
(
z
t
)
≤
t
[
Θ
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
]
+
(
1
-
t
)
〈
z
t
-
w
,
A
k
z
t
〉
=
t
[
Θ
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
]
+
(
1
-
t
)
t
〈
y
-
w
,
A
k
z
t
〉
,
and hence
(96)
0
≤
Θ
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
+
(
1
-
t
)
〈
y
-
w
,
A
k
z
t
〉
.
Letting t
→
0, we have, for each y
∈
C,
(97)
0
≤
Θ
k
(
w
,
y
)
+
φ
k
(
y
)
-
φ
k
(
w
)
+
〈
y
-
w
,
A
k
w
〉
.
This implies that w
∈
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
) and hence w
∈
∩
k
=
1
M
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
). Further, let us show that w
∈
Γ. As a matter of fact, from (84), v
n
i
⇀
w, and Lemma 9, we conclude that
(98)
w
=
P
C
(
I
-
2
L
∇
f
)
w
.
So, w
∈
VI
⁡
(
C
,
∇
f
)
=
Γ. Therefore, w
∈
∩
k
=
1
M
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
∩
Γ
=
:
Ω. This shows that ω
w
(
x
n
)
⊂
Ω.
Finally, let us show thatx
n
→
q as n
→
∞ where q is the unique solution of the VIP (50). Indeed, we note that, for w
∈
Ω with x
n
i
⇀
w,
(99)
x
n
-
w
=
s
n
(
γ
V
x
n
-
μ
F
w
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
w
.
By (61) and Lemma 13, we obtain that
(100)
∥
x
n
-
w
∥
2
=
s
n
〈
γ
V
x
n
-
μ
F
w
,
x
n
-
w
〉
+
〈
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
w
,
x
n
-
w
〉
=
s
n
〈
γ
V
x
n
-
μ
F
w
,
x
n
-
w
〉
+
∥
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
w
∥
∥
x
n
-
w
∥
≤
s
n
〈
γ
V
x
n
-
μ
F
w
,
x
n
-
w
〉
+
(
1
-
s
n
τ
)
∥
v
n
-
w
∥
∥
x
n
-
w
∥
≤
s
n
〈
γ
V
x
n
-
μ
F
w
,
x
n
-
w
〉
+
(
1
-
s
n
τ
)
∥
x
n
-
w
∥
2
.
Hence, it follows that
(101)
∥
x
n
-
w
∥
2
≤
1
τ
〈
γ
V
x
n
-
μ
F
w
,
x
n
-
w
〉
=
1
τ
(
γ
〈
V
x
n
-
V
w
,
x
n
-
w
〉
=
1
τ
0
.
.
+
〈
γ
V
w
-
μ
F
w
,
x
n
-
w
〉
)
≤
1
τ
(
γ
l
∥
x
n
-
w
∥
2
+
〈
γ
V
w
-
μ
F
w
,
x
n
-
w
〉
)
,
which hence leads to
(102)
∥
x
n
-
w
∥
2
≤
〈
γ
V
w
-
μ
F
w
,
x
n
-
w
〉
τ
-
γ
l
.
In particular, we have
(103)
∥
x
n
i
-
w
∥
2
≤
〈
γ
V
w
-
μ
F
w
,
x
n
i
-
w
〉
τ
-
γ
l
.
Since x
n
i
⇀
w, it follows from (103) that x
n
i
→
w as i
→
∞.
Now we show thatw solves the VIP (50). Since x
n
=
s
n
γ
V
x
n
+
(
I
-
s
n
μ
F
)
T
n
v
n, we have
(104)
(
μ
F
-
γ
V
)
x
n
=
-
1
s
n
(
(
I
-
s
n
μ
F
)
x
n
-
(
I
-
s
n
μ
F
)
T
n
v
n
)
.
It follows that, for each p
∈
Ω,
(105)
〈
(
μ
F
-
γ
V
)
x
n
,
x
n
-
p
〉
=
-
1
s
n
〈
(
I
-
s
n
μ
F
)
x
n
-
(
I
-
s
n
μ
F
)
T
n
v
n
,
x
n
-
p
〉
=
-
1
s
n
〈
(
I
-
s
n
μ
F
)
x
n
-
(
I
-
s
n
μ
F
)
T
n
v
n
,
x
n
-
p
〉
=
-
1
s
n
〈
(
I
-
s
n
μ
F
)
x
n
-
(
I
-
s
n
μ
F
)
T
n
Λ
n
N
Δ
n
M
x
n
,
x
n
-
p
〉
=
-
1
s
n
〈
(
I
-
T
n
Λ
n
N
Δ
n
M
)
x
n
-
(
I
-
T
n
Λ
n
N
Δ
n
M
)
p
,
x
n
-
p
〉
+
〈
μ
F
x
n
-
μ
F
T
n
Λ
n
N
Δ
n
M
x
n
,
x
n
-
p
〉
≤
〈
μ
F
x
n
-
μ
F
T
n
Λ
n
N
Δ
n
M
x
n
,
x
n
-
p
〉
,
since I
-
T
n
Λ
n
N
Δ
n
M is monotone (i.e., 〈
(
I
-
T
n
Λ
n
N
Δ
n
M
)
x
-
(
I
-
T
n
Λ
n
N
Δ
n
M
)
y
,
x
-
y
〉
≥
0 for all x
,
y
∈
H. This is due to the nonexpansivity of T
n
Λ
n
N
Δ
n
M). Since ∥
x
n
-
T
n
v
n
∥
=
∥
(
I
-
T
n
Λ
n
N
Δ
n
M
)
x
n
∥
→
0 as n
→
∞, by replacing n in (105) with n
i and letting i
→
∞, we get
(106)
〈
(
μ
F
-
γ
V
)
w
,
w
-
p
〉
=
lim
⁡
i
→
∞
〈
(
μ
F
-
γ
V
)
x
n
i
,
x
n
i
-
p
〉
≤
lim
⁡
i
→
∞
〈
μ
F
x
n
i
-
μ
F
T
n
i
v
n
i
,
x
n
i
-
p
〉
=
0
.
That is, w
∈
Ω is a solution of VIP (50).
Finally, in terms of the uniqueness of solutions of VIP (50), we deduce that w
=
q and x
n
i
→
q as n
→
∞. So, every weak convergence subsequence of {
x
n
} converges strongly to the unique solution q of VIP (50). Therefore, {
x
n
} converges strongly to the unique solution q of VIP (50). In addition, the VIP (50) can be rewritten as
(107)
〈
(
I
-
μ
F
+
γ
V
)
q
-
q
,
q
-
p
〉
≥
0
,
∀
p
∈
Ω
.
By Proposition 2(i), this is equivalent to the fixed point equation:
(108)
P
Ω
(
I
-
μ
F
+
γ
V
)
q
=
q
.
This completes the proof.Corollary 19.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let Θ be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H and B
i
:
C
→
H be ζ-inverse-strongly monotone and η
i-inverse-strongly monotone, respectively, for i
=
1,2. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
G
M
E
P
(
Θ
,
φ
,
A
)
∩
I
(
B
1
,
R
1
)
∩
I
(
B
2
,
R
2
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. Let {
x
n
} be a sequence generated by
(109)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
v
n
=
J
R
2
,
λ
2
,
n
(
I
-
λ
2
,
n
B
2
)
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
x
n
=
s
n
γ
V
x
n
+
(
I
-
s
n
μ
F
)
T
n
v
n
,
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) for i
=
1,2;
(iii)
{
r
n
}
⊂
[
e
,
f
]
⊂
(
0,2
ζ
).Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of the VIP:
(110)
〈
(
μ
F
-
γ
V
)
q
,
p
-
q
〉
≥
0
,
∀
p
∈
Ω
.Corollary 20.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let Θ be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function. Let R
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H and B
:
C
→
H be ζ-inverse-strongly monotone and ξ-inverse-strongly monotone, respectively. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
G
M
E
P
(
Θ
,
φ
,
A
)
∩
I
(
B
,
R
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. Let {
x
n
} be a sequence generated by
(111)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
x
n
=
s
n
γ
V
x
n
+
(
I
-
s
n
μ
F
)
T
n
J
R
,
ρ
n
(
u
n
-
ρ
n
B
u
n
)
,
x
n
+
(
I
-
s
n
μ
F
)
T
n
J
R
,
ρ
n
(
u
n
-
ρ
n
B
u
n
)
,
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
ρ
n
}
⊂
[
a
,
b
]
⊂
(
0,2
ξ
);
(iii)
{
r
n
}
⊂
[
e
,
f
]
⊂
(
0,2
ζ
). Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of the VIP:
(112)
〈
(
μ
F
-
γ
V
)
q
,
p
-
q
〉
≥
0
,
∀
p
∈
Ω
.


## 4. Explicit Iterative Algorithm and Its Convergence Criteria

We next state and prove the second main result of this paper.Theorem 21.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let M
,
N be two integers. Let Θ
k be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
k
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function, where k
∈
{
1,2
,
…
,
M
}. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
H
→
H and B
i
:
C
→
H be μ
k-inverse-strongly monotone and η
i-inverse-strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
}, i
∈
{
1,2
,
…
,
N
}. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
∩
k
=
1
M
G
M
E
P
(
Θ
k
,
φ
k
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. For arbitrarily given x
1
∈
H, let {
x
n
} be a sequence generated by
(113)
u
n
=
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
T
r
M
-
1
,
n
(
Θ
M
-
1
,
φ
M
-
1
)
×
(
I
-
r
M
-
1
,
n
A
M
-
1
)
⋯
T
r
1
,
n
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
v
n
=
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
J
R
N
-
1
,
λ
N
-
1
,
n
×
(
I
-
λ
N
-
1
,
n
A
N
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
x
n
+
1
=
s
n
γ
V
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
,
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), and lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
β
n
}
⊂
(
0,1
) and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1;
(iii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) and lim
⁡
n
→
∞
|
λ
i
,
n
+
1
-
λ
i
,
n
|
=
0 for all i
∈
{
1,2
,
…
,
N
};
(iv)
{
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
) and lim
⁡
n
→
∞
|
r
k
,
n
+
1
-
r
k
,
n
|
=
0 for all k
∈
{
1,2
,
…
,
M
}.Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of VIP (50).Proof.
First of all, repeating the same arguments as in Theorem18, we can write
(114)
P
C
(
I
-
λ
n
∇
f
)
=
2
-
λ
n
L
4
I
+
2
+
λ
n
L
4
T
n
=
s
n
I
+
(
1
-
s
n
)
T
n
,
where T
n is nonexpansive and s
n
:
=
s
n
(
λ
n
)
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
). It is clear that
(115)
λ
n
⟶
2
L
⇔
s
n
⟶
0
.
Put(116)
Δ
n
k
=
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
T
r
k
-
1
,
n
(
Θ
k
-
1
,
φ
k
-
1
)
×
(
I
-
r
k
-
1
,
n
A
k
-
1
)
⋯
T
r
1
,
n
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
for all k
∈
{
1,2
,
…
,
M
} and n
≥
1,
(117)
Λ
n
i
=
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
J
R
i
-
1
,
λ
i
-
1
,
n
×
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
for all i
∈
{
1,2
,
…
,
N
} and n
≥
1, and Δ
n
0
=
Λ
n
0
=
I, where I is the identity mapping on H. Then we have that u
n
=
Δ
n
M
x
n and v
n
=
Λ
n
N
u
n. In addition, taking into consideration conditions (i) and (ii), we may assume, without loss of generality, that s
n
≤
1
-
β
n for all n
≥
1.
We divide the remainder of the proof into several steps.
Step 1
. Let us show that ∥
x
n
-
p
∥
≤
max
⁡
{
∥
x
1
-
p
∥
,
∥
γ
V
p
-
μ
F
p
∥
/
(
τ
-
γ
l
)
} for all n
≥
1 and p
∈
Ω. Indeed, take p
∈
Ω arbitrarily. Repeating the same arguments as those of (59)–(61) in the proof of Theorem 18, we obtain
(118)
∥
u
n
-
p
∥
≤
∥
x
n
-
p
∥
,
∥
v
n
-
p
∥
≤
∥
u
n
-
p
∥
,
∥
v
n
-
p
∥
≤
∥
x
n
-
p
∥
.
Then from (118), T
n
p
=
p, and Lemma 13, we have
(119)
∥
x
n
+
1
-
p
∥
=
∥
s
n
(
γ
V
x
n
-
μ
F
p
)
+
β
n
(
x
n
-
p
)
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
-
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
p
∥
≤
s
n
∥
γ
V
x
n
-
μ
F
p
∥
+
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
∥
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
p
∥
≤
s
n
(
∥
γ
V
x
n
-
γ
V
p
∥
+
∥
γ
V
p
-
μ
F
p
∥
)
+
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
(
1
-
s
n
τ
1
-
β
n
)
∥
v
n
-
p
∥
≤
s
n
(
∥
γ
V
x
n
-
γ
V
p
∥
+
∥
γ
V
p
-
μ
F
p
∥
)
+
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
s
n
τ
)
∥
x
n
-
p
∥
≤
s
n
γ
l
∥
x
n
-
p
∥
+
s
n
∥
γ
V
p
-
μ
F
p
∥
+
(
1
-
s
n
τ
)
∥
x
n
-
p
∥
=
(
1
-
s
n
(
τ
-
γ
l
)
)
∥
x
n
-
p
∥
+
s
n
(
τ
-
γ
l
)
∥
γ
V
p
-
μ
F
p
∥
τ
-
γ
l
≤
max
⁡
⁡
{
∥
x
n
-
p
∥
,
∥
γ
V
p
-
μ
F
p
∥
τ
-
γ
l
}
.
By induction, we have
(120)
∥
x
n
-
p
∥
≤
max
⁡
{
∥
x
1
-
p
∥
,
∥
γ
V
p
-
μ
F
p
∥
τ
-
γ
l
}
,
∀
n
≥
1
.
Hence, {
x
n
} is bounded. According to (118), {
u
n
}, {
v
n
}, {
T
n
v
n
}, {
V
x
n
}, and {
F
T
n
v
n
} are also bounded.
Step
2. Let us show that ∥
x
n
+
1
-
x
n
∥
→
0 as n
→
∞. To this end, define
(121)
x
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
z
n
,
∀
n
≥
1
.
Observe that, from the definition of z
n,
(122)
z
n
+
1
-
z
n
=
x
n
+
2
-
β
n
+
1
x
n
+
1
1
-
β
n
+
1
-
x
n
+
1
-
β
n
x
n
1
-
β
n
=
s
n
+
1
γ
V
x
n
+
1
+
(
(
1
-
β
n
+
1
)
I
-
s
n
+
1
μ
F
)
T
n
+
1
v
n
+
1
1
-
β
n
+
1
-
s
n
γ
V
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
1
-
β
n
=
s
n
+
1
1
-
β
n
+
1
γ
V
x
n
+
1
-
s
n
1
-
β
n
γ
V
x
n
+
T
n
+
1
v
n
+
1
-
T
n
v
n
+
s
n
1
-
β
n
μ
F
T
n
v
n
-
s
n
+
1
1
-
β
n
+
1
μ
F
T
n
+
1
v
n
+
1
=
s
n
+
1
1
-
β
n
+
1
(
γ
V
x
n
+
1
-
μ
F
T
n
+
1
v
n
+
1
)
+
s
n
1
-
β
n
(
μ
F
T
n
v
n
-
γ
V
x
n
)
+
T
n
+
1
v
n
+
1
-
T
n
v
n
.
Thus, it follows that
(123)
∥
z
n
+
1
-
z
n
∥
≤
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
∥
T
n
+
1
v
n
+
1
-
T
n
v
n
∥
.
On the other hand, since ∇
f is  (
1
/
L
)-ism, P
C
(
I
-
λ
n
∇
f
) is nonexpansive for λ
n
∈
(
0
,
2
/
L
). So, it follows that, for any given p
∈
Ω,
(124)
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
∥
≤
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
-
p
∥
+
∥
p
∥
=
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
-
P
C
(
I
-
λ
n
+
1
∇
f
)
p
∥
+
∥
p
∥
≤
∥
v
n
-
p
∥
+
∥
p
∥
≤
∥
v
n
∥
+
2
∥
p
∥
.
This together with the boundedness of {
v
n
} implies that {
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
} is bounded. Also, observe that
(125)
∥
T
n
+
1
v
n
-
T
n
v
n
∥
=
∥
4
P
C
(
I
-
λ
n
+
1
∇
f
)
-
(
2
-
λ
n
+
1
L
)
I
2
+
λ
n
+
1
L
v
n
-
4
P
C
(
I
-
λ
n
∇
f
)
-
(
2
-
λ
n
L
)
I
2
+
λ
n
L
v
n
∥
≤
∥
4
P
C
(
I
-
λ
n
+
1
∇
f
)
2
+
λ
n
+
1
L
v
n
-
4
P
C
(
I
-
λ
n
∇
f
)
2
+
λ
n
L
v
n
∥
+
∥
2
-
λ
n
L
2
+
λ
n
L
v
n
-
2
-
λ
n
+
1
L
2
+
λ
n
+
1
L
v
n
∥
=
∥
×
(
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
)
-
1
(
4
(
2
+
λ
n
L
)
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
-
4
(
2
+
λ
n
+
1
L
)
P
C
(
I
-
λ
n
∇
f
)
v
n
)
×
(
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
)
-
1
∥
+
4
L
|
λ
n
+
1
-
λ
n
|
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
∥
v
n
∥
=
∥
×
(
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
)
-
1
(
4
L
(
λ
n
-
λ
n
+
1
)
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
+
4
(
2
+
λ
n
+
1
L
)
×
(
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
-
P
C
(
I
-
λ
n
∇
f
)
v
n
)
)
×
(
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
)
-
1
∥
+
4
L
|
λ
n
+
1
-
λ
n
|
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
∥
v
n
∥
≤
4
L
|
λ
n
-
λ
n
+
1
|
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
∥
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
+
(
4
(
2
+
λ
n
+
1
L
)
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
-
P
C
(
I
-
λ
n
∇
f
)
v
n
∥
)
×
(
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
)
-
1
+
4
L
|
λ
n
+
1
-
λ
n
|
(
2
+
λ
n
+
1
L
)
(
2
+
λ
n
L
)
∥
v
n
∥
≤
|
λ
n
+
1
-
λ
n
|
[
L
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
∥
+
4
∥
∇
f
(
v
n
)
∥
+
L
∥
v
n
∥
]
≤
M
~
|
λ
n
+
1
-
λ
n
|
,
where sup
⁡
n
≥
1
{
L
∥
P
C
(
I
-
λ
n
+
1
∇
f
)
v
n
∥
+
4
∥
∇
f
(
v
n
)
∥
+
L
∥
v
n
∥
}
≤
M
~ for some M
~
>
0. So, by (125), we have that
(126)
∥
T
n
+
1
v
n
+
1
-
T
n
v
n
∥
≤
∥
T
n
+
1
v
n
+
1
-
T
n
+
1
v
n
∥
+
∥
T
n
+
1
v
n
-
T
n
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
M
~
|
λ
n
+
1
-
λ
n
|
≤
∥
v
n
+
1
-
v
n
∥
+
4
M
~
L
(
s
n
+
1
+
s
n
)
.
Note that
(127)
∥
v
n
+
1
-
v
n
∥
=
∥
Λ
n
+
1
N
u
n
+
1
-
Λ
n
N
u
n
∥
=
∥
J
R
N
,
λ
N
,
n
+
1
(
I
-
λ
N
,
n
+
1
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
-
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
∥
≤
∥
J
R
N
,
λ
N
,
n
+
1
(
I
-
λ
N
,
n
+
1
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
-
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
∥
+
∥
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
-
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
∥
≤
∥
(
I
-
λ
N
,
n
+
1
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
-
(
I
-
λ
N
,
n
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
∥
+
∥
(
I
-
λ
N
,
n
B
N
)
Λ
n
+
1
N
-
1
u
n
+
1
-
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
∥
≤
|
λ
N
,
n
+
1
-
λ
N
,
n
|
∥
B
N
Λ
n
+
1
N
-
1
u
n
+
1
∥
+
∥
Λ
n
+
1
N
-
1
u
n
+
1
-
Λ
n
N
-
1
u
n
∥
≤
|
λ
N
,
n
+
1
-
λ
N
,
n
|
∥
B
N
Λ
n
+
1
N
-
1
u
n
+
1
∥
+
|
λ
N
-
1
,
n
+
1
-
λ
N
-
1
,
n
|
∥
B
N
-
1
Λ
n
+
1
N
-
2
u
n
+
1
∥
+
∥
Λ
n
+
1
N
-
2
u
n
+
1
-
Λ
n
N
-
2
u
n
∥
⋮
≤
|
λ
N
,
n
+
1
-
λ
N
,
n
|
∥
B
N
Λ
n
+
1
N
-
1
u
n
+
1
∥
+
|
λ
N
-
1
,
n
+
1
-
λ
N
-
1
,
n
|
∥
B
N
-
1
Λ
n
+
1
N
-
2
u
n
+
1
∥
+
⋯
+
|
λ
1
,
n
+
1
-
λ
1
,
n
|
∥
B
1
Λ
n
+
1
0
u
n
+
1
∥
+
∥
Λ
n
+
1
0
u
n
+
1
-
Λ
n
0
u
n
∥
≤
M
~
0
∑
i
=
1
N
‍
|
λ
i
,
n
+
1
-
λ
i
,
n
|
+
∥
u
n
+
1
-
u
n
∥
,
where sup
⁡
n
≥
1
{
∑
i
=
1
N
‍
∥
B
i
Λ
n
+
1
i
-
1
u
n
+
1
∥
}
≤
M
~
0 for some M
~
0
>
0. Also, utilizing Proposition 1(v) we deduce that
(128)
∥
u
n
+
1
-
u
n
∥
=
∥
Δ
n
+
1
M
x
n
+
1
-
Δ
n
M
x
n
∥
=
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
n
∥
≤
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
∥
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
n
∥
≤
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
∥
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
∥
(
I
-
r
M
,
n
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
A
M
)
Δ
n
M
-
1
x
n
∥
≤
|
r
M
,
n
+
1
-
r
M
,
n
|
r
M
,
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
|
r
M
,
n
+
1
-
r
M
,
n
|
∥
A
M
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
∥
Δ
n
+
1
M
-
1
x
n
+
1
-
Δ
n
M
-
1
x
n
∥
=
|
r
M
,
n
+
1
-
r
M
,
n
|
×
[
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
[
∥
A
M
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
1
r
M
,
n
+
1
]
∥
A
M
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
1
r
M
,
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
[
∥
A
M
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
1
r
M
,
n
+
1
]
+
∥
Δ
n
+
1
M
-
1
x
n
+
1
-
Δ
n
M
-
1
x
n
∥
⋮
≤
|
r
M
,
n
+
1
-
r
M
,
n
|
×
[
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
]
∥
A
M
Δ
n
+
1
M
-
1
x
n
+
1
∥
+
1
r
M
,
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
T
r
M
,
n
+
1
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
-
(
I
-
r
M
,
n
+
1
A
M
)
Δ
n
+
1
M
-
1
x
n
+
1
∥
]
+
⋯
+
|
r
1
,
n
+
1
-
r
1
,
n
|
×
[
-
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
]
-
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
T
r
1
,
n
+
1
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
-
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
T
r
1
,
n
+
1
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
∥
A
1
Δ
n
+
1
0
x
n
+
1
∥
+
1
r
1
,
n
+
1
∥
-
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
T
r
1
,
n
+
1
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
T
r
1
,
n
+
1
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
-
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
T
r
1
,
n
+
1
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
+
1
A
1
)
Δ
n
+
1
0
x
n
+
1
∥
]
+
∥
Δ
n
+
1
0
x
n
+
1
-
Δ
n
0
x
n
∥
≤
M
~
1
∑
k
=
1
M
‍
|
r
k
,
n
+
1
-
r
k
,
n
|
+
∥
x
n
+
1
-
x
n
∥
,
where M
~
1
>
0 is a constant such that for each n
≥
1
(129)
∑
k
=
1
M
‍
[
∥
A
k
Δ
n
+
1
k
-
1
x
n
+
1
∥
+
1
r
k
,
n
+
1
×
∥
T
r
k
,
n
+
1
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
+
1
A
k
)
Δ
n
+
1
k
-
1
x
n
+
1
-
(
I
-
r
k
,
n
+
1
A
k
)
Δ
n
+
1
k
-
1
x
n
+
1
T
r
k
,
n
+
1
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
+
1
A
k
)
Δ
n
+
1
k
-
1
x
n
+
1
∥
∥
A
k
Δ
n
+
1
k
-
1
x
n
+
1
∥
+
1
r
k
,
n
+
1
]
≤
M
~
1
.
Combining (123)–(128), we get
(130)
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
≤
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
∥
T
n
+
1
v
n
+
1
-
T
n
v
n
∥
-
∥
x
n
+
1
-
x
n
∥
≤
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
∥
v
n
+
1
-
v
n
∥
+
4
M
~
L
(
s
n
+
1
+
s
n
)
-
∥
x
n
+
1
-
x
n
∥
≤
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
M
~
0
∑
i
=
1
N
‍
|
λ
i
,
n
+
1
-
λ
i
,
n
|
+
∥
u
n
+
1
-
u
n
∥
+
4
M
~
L
(
s
n
+
1
+
s
n
)
-
∥
x
n
+
1
-
x
n
∥
≤
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
M
~
0
∑
i
=
1
N
‍
|
λ
i
,
n
+
1
-
λ
i
,
n
|
+
M
~
1
∑
k
=
1
M
‍
|
r
k
,
n
+
1
-
r
k
,
n
|
+
∥
x
n
+
1
-
x
n
∥
+
4
M
~
L
(
s
n
+
1
+
s
n
)
-
∥
x
n
+
1
-
x
n
∥
=
s
n
+
1
1
-
β
n
+
1
(
γ
∥
V
x
n
+
1
∥
+
μ
∥
F
T
n
+
1
v
n
+
1
∥
)
+
s
n
1
-
β
n
(
μ
∥
F
T
n
v
n
∥
+
γ
∥
V
x
n
∥
)
+
M
~
0
∑
i
=
1
N
‍
|
λ
i
,
n
+
1
-
λ
i
,
n
|
+
M
~
1
∑
k
=
1
M
‍
|
r
k
,
n
+
1
-
r
k
,
n
|
+
4
M
~
L
(
s
n
+
1
+
s
n
)
.
Thus, it follows from (130) and conditions (i)–(iv) that
(131)
limsup
⁡
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0
.
Hence, by Lemma 11 we have
(132)
lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0
.
Consequently,
(133)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
⁡
n
→
∞
(
1
-
β
n
)
∥
z
n
-
x
n
∥
=
0
,
and, by (126)–(128),
(134)
lim
⁡
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0
,
lim
⁡
n
→
∞
∥
v
n
+
1
-
v
n
∥
=
0
,
lim
⁡
n
→
∞
∥
T
n
+
1
v
n
+
1
-
T
n
v
n
∥
=
0
.
Step
3. Let us show that ∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
→
0 and ∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
→
0 for all k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}.
Indeed, since(135)
x
n
+
1
=
s
n
γ
V
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
,
we have
(136)
∥
x
n
-
T
n
v
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
T
n
v
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
s
n
∥
γ
V
x
n
-
μ
F
T
n
v
n
∥
+
β
n
∥
x
n
-
T
n
v
n
∥
;
that is,
(137)
∥
x
n
-
T
n
v
n
∥
≤
1
1
-
β
n
∥
x
n
-
x
n
+
1
∥
+
s
n
1
-
β
n
(
γ
∥
V
x
n
∥
+
μ
∥
F
T
n
v
n
∥
)
.
So, from s
n
→
0, ∥
x
n
+
1
-
x
n
∥
→
0, and condition (ii), it follows that
(138)
lim
⁡
n
→
∞
∥
x
n
-
T
n
v
n
∥
=
0
.
Also, from (27) it follows that for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}
(139)
∥
v
n
-
p
∥
2
=
∥
Λ
n
N
u
n
-
p
∥
2
≤
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
,
∥
u
n
-
p
∥
2
=
∥
Δ
n
M
x
n
-
p
∥
2
≤
∥
Δ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
.
Furthermore, utilizing Lemma 7, we deduce from (113) that
(140)
∥
x
n
+
1
-
p
∥
2
=
∥
s
n
(
γ
V
x
n
-
μ
F
p
)
+
β
n
(
x
n
-
T
n
v
n
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
p
∥
2
≤
∥
β
n
(
x
n
-
T
n
v
n
)
+
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
p
∥
2
+
2
s
n
〈
γ
V
x
n
-
μ
F
p
,
x
n
+
1
-
p
〉
≤
[
β
n
∥
x
n
-
T
n
v
n
∥
+
∥
(
I
-
s
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
μ
F
)
T
n
p
∥
]
2
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
[
β
n
∥
x
n
-
T
n
v
n
∥
+
(
1
-
s
n
τ
)
∥
v
n
-
p
∥
]
2
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
=
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
.
From (139)–(140), it follows that
(141)
∥
x
n
+
1
-
p
∥
2
≤
∥
v
n
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
,
and so
(142)
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
×
∥
x
n
-
x
n
+
1
∥
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
.
Since {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) and {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
) for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}, by (133), (138), and (142) we conclude immediately that
(143)
lim
⁡
n
→
∞
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
=
0
,
lim
⁡
n
→
∞
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
=
0
,
for all k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}.
Step
4. Let us show that ∥
x
n
-
u
n
∥
→
0, ∥
u
n
-
v
n
∥
→
0, and ∥
v
n
-
T
n
v
n
∥
→
0 as n
→
∞.
Indeed, by Proposition1(iii) we obtain that for each k
∈
{
1,2
,
…
,
M
}
(144)
∥
Δ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
T
r
k
,
n
(
Θ
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
〈
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
,
Δ
n
k
x
n
-
p
〉
=
1
2
(
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
+
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
(
I
-
r
k
,
n
A
k
)
Δ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
-
(
Δ
n
k
x
n
-
p
)
∥
2
)
≤
1
2
(
∥
Δ
n
k
-
1
x
n
-
p
∥
2
+
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
-
r
k
,
n
(
A
k
Δ
n
k
-
1
x
n
-
A
k
p
)
∥
2
)
,
which implies that
(145)
∥
Δ
n
k
x
n
-
p
∥
2
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
(
A
k
Δ
n
k
-
1
x
n
-
A
k
p
)
-
r
k
,
n
(
A
k
Δ
n
k
-
1
x
n
-
A
k
p
)
∥
2
=
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
-
r
k
,
n
2
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
2
+
2
r
k
,
n
〈
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
,
A
k
Δ
n
k
-
1
x
n
-
A
k
p
〉
≤
∥
Δ
n
k
-
1
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
≤
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
.
Also, by Lemma 14, we obtain that for each i
∈
{
1,2
,
…
,
N
}
(146)
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
〈
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
,
Λ
n
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
-
(
Λ
n
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies
(147)
∥
Λ
n
i
u
n
-
p
∥
2
≤
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
-
λ
i
,
n
2
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
2
λ
i
,
n
〈
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
,
B
i
Λ
n
i
-
1
u
n
-
B
i
p
〉
≤
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
.
Thus, from (140), (145), and (147), we have
(148)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
s
n
τ
)
2
∥
v
n
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
=
(
1
-
s
n
τ
)
2
∥
Λ
n
N
u
n
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
1
-
s
n
τ
)
2
∥
Λ
n
i
u
n
-
p
∥
2
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
u
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
×
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
=
(
1
-
s
n
τ
)
2
[
∥
Δ
n
M
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
Δ
n
M
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
×
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
Δ
n
k
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
×
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
1
-
s
n
τ
)
2
[
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
×
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
∥
x
n
-
p
∥
2
-
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
]
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
∥
x
n
-
p
∥
2
-
(
1
-
s
n
τ
)
2
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
-
(
1
-
s
n
τ
)
2
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
×
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
;
that is,
(149)
(
1
-
s
n
τ
)
2
[
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
2
+
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
]
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
2
r
k
,
n
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
∥
A
k
Δ
n
k
-
1
x
n
-
A
k
p
∥
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
β
n
2
∥
x
n
-
T
n
v
n
∥
2
+
2
(
1
-
s
n
τ
)
β
n
∥
v
n
-
p
∥
×
∥
x
n
-
T
n
v
n
∥
+
2
s
n
∥
γ
V
x
n
-
μ
F
p
∥
∥
x
n
+
1
-
p
∥
.
So, from s
n
→
0, (133), (138), and (143), we immediately get
(150)
lim
⁡
n
→
∞
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
=
0
,
lim
⁡
n
→
∞
∥
Δ
n
k
-
1
x
n
-
Δ
n
k
x
n
∥
=
0
,
for all i
∈
{
1,2
,
…
,
N
} and k
∈
{
1,2
,
…
,
M
}. Note that
(151)
∥
x
n
-
u
n
∥
=
∥
Δ
n
0
x
n
-
Δ
n
M
x
n
∥
≤
∥
Δ
n
0
x
n
-
Δ
n
1
x
n
∥
+
∥
Δ
n
1
x
n
-
Δ
n
2
x
n
∥
+
⋯
+
∥
Δ
n
M
-
1
x
n
-
Δ
n
M
x
n
∥
,
∥
u
n
-
v
n
∥
=
∥
Λ
n
0
u
n
-
Λ
n
N
u
n
∥
≤
∥
Λ
n
0
u
n
-
Λ
n
1
u
n
∥
+
∥
Λ
n
1
u
n
-
Λ
n
2
u
n
∥
+
⋯
+
∥
Λ
n
N
-
1
u
n
-
Λ
n
N
u
n
∥
.
Thus, from (150) we have
(152)
lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0
,
lim
⁡
n
→
∞
∥
u
n
-
v
n
∥
=
0
.
It is easy to see that as n
→
∞
(153)
∥
x
n
-
v
n
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
v
n
∥
⟶
0
.
Also, observe that
(154)
∥
T
n
v
n
-
v
n
∥
≤
∥
T
n
v
n
-
x
n
∥
+
∥
x
n
-
v
n
∥
.
Hence, we have from (138)
(155)
lim
⁡
n
→
∞
∥
T
n
v
n
-
v
n
∥
=
0
.
Step
5. Let us show that limsup
⁡
n
→
∞
〈
(
μ
F
-
γ
V
)
q
,
q
-
x
n
〉
≤
0, where q
∈
Ω is the same as in Theorem 18; that is, q
∈
Ω is a unique solution of VIP (50). To show this inequality, we choose a subsequence {
x
n
i
} of {
x
n
} such that
(156)
limsup
⁡
n
→
∞
〈
(
μ
F
-
γ
V
)
q
,
q
-
x
n
〉
=
lim
⁡
i
→
∞
〈
(
μ
F
-
γ
V
)
q
,
q
-
x
n
i
〉
.
Since {
x
n
} is bounded, there exists a subsequence {
x
n
i
j
} of {
x
n
i
} which converges weakly to w. Without loss of generality, we may assume that x
n
i
⇀
w. From Step 4, we have that Δ
n
i
k
x
n
i
⇀
w, Λ
n
i
m
u
n
i
⇀
w, u
n
i
⇀
w, and v
n
i
⇀
w, where k
∈
{
1,2
,
…
,
M
}, m
∈
{
1,2
,
…
,
N
}. Since v
n
-
T
n
v
n
→
0 by Step 4, by the same arguments as in the proof of Theorem 18, we get w
∈
Ω. Since q
=
P
Ω
(
I
-
μ
F
+
γ
V
)
q, it follows that
(157)
limsup
⁡
n
→
∞
〈
(
μ
F
-
γ
V
)
q
,
q
-
x
n
〉
=
lim
⁡
i
→
∞
〈
(
μ
F
-
γ
V
)
q
,
q
-
x
n
i
〉
=
〈
(
μ
F
-
γ
V
)
q
,
q
-
w
〉
≤
0
.
Step
6. Let us show that lim
⁡
n
→
∞
∥
x
n
-
q
∥
=
0, where q
∈
Ω is the same as in Theorem 18; that is, q
∈
Ω is a unique solution of VIP (50). From (113), we know that
(158)
x
n
+
1
-
q
=
s
n
(
γ
V
x
n
-
μ
F
q
)
+
β
n
(
T
n
v
n
-
q
)
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
-
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
q
.
Applying Lemmas 7 and 13 and noticing T
n
q
=
q and ∥
v
n
-
q
∥
≤
∥
x
n
-
q
∥ for all n
≥
1, we have
(159)
∥
x
n
+
1
-
q
∥
2
≤
∥
β
n
(
T
n
v
n
-
q
)
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
-
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
q
∥
2
+
2
s
n
〈
γ
V
x
n
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
[
β
n
∥
T
n
v
n
-
q
∥
+
∥
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
-
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
q
∥
β
n
∥
T
n
v
n
-
q
∥
+
∥
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
]
2
+
2
s
n
〈
γ
V
x
n
-
γ
V
q
,
x
n
+
1
-
q
〉
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
=
[
×
∥
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
q
∥
]
β
n
∥
T
n
v
n
-
q
∥
+
(
1
-
β
n
)
×
∥
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
v
n
-
(
I
-
s
n
1
-
β
n
μ
F
)
T
n
q
∥
]
2
+
2
s
n
〈
γ
V
x
n
-
γ
V
q
,
x
n
+
1
-
q
〉
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
[
β
n
∥
v
n
-
q
∥
+
(
1
-
β
n
)
(
1
-
s
n
τ
1
-
β
n
)
∥
v
n
-
q
∥
2
]
2
+
2
s
n
〈
γ
V
x
n
-
γ
V
q
,
x
n
+
1
-
q
〉
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
=
[
β
n
∥
v
n
-
q
∥
+
(
1
-
β
n
-
s
n
τ
)
∥
v
n
-
q
∥
]
2
+
2
s
n
〈
γ
V
x
n
-
γ
V
q
,
x
n
+
1
-
q
〉
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
[
β
n
∥
x
n
-
q
∥
+
(
1
-
β
n
-
s
n
τ
)
∥
x
n
-
q
∥
]
2
+
2
s
n
〈
γ
V
x
n
-
γ
V
q
,
x
n
+
1
-
q
〉
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
(
1
-
s
n
τ
)
2
∥
x
n
-
q
∥
2
+
2
s
n
γ
l
∥
x
n
-
q
∥
×
∥
x
n
+
1
-
q
∥
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
(
1
-
s
n
τ
)
2
∥
x
n
-
q
∥
2
+
s
n
γ
l
(
∥
x
n
-
q
∥
2
+
∥
x
n
+
1
-
q
∥
2
)
+
2
s
n
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
.
This implies that
(160)
∥
x
n
+
1
-
q
∥
2
≤
1
-
2
τ
s
n
+
τ
2
s
n
2
+
s
n
γ
l
1
-
s
n
γ
l
∥
x
n
-
q
∥
2
+
2
s
n
1
-
s
n
γ
l
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
=
(
1
-
2
(
τ
-
γ
l
)
s
n
1
-
s
n
γ
l
)
∥
x
n
-
q
∥
2
+
τ
2
s
n
2
1
-
s
n
γ
l
∥
x
n
-
q
∥
2
+
2
s
n
1
-
s
n
γ
l
〈
γ
V
q
-
μ
F
q
,
x
n
+
1
-
q
〉
≤
(
1
-
2
(
τ
-
γ
l
)
1
-
s
n
γ
l
s
n
)
∥
x
n
-
q
∥
2
+
2
(
τ
-
γ
l
)
s
n
1
-
s
n
γ
l
×
(
τ
2
s
n
2
(
τ
-
γ
l
)
M
~
2
+
1
τ
-
γ
l
〈
μ
F
q
-
γ
V
q
,
q
-
x
n
+
1
〉
τ
2
s
n
2
(
τ
-
γ
l
)
M
~
2
)
=
(
1
-
σ
n
)
∥
x
n
-
q
∥
2
+
σ
n
δ
n
,
where M
~
2
=
sup
⁡
n
≥
1
∥
x
n
-
q
∥
2, σ
n
=
(
2
(
τ
-
γ
l
)
/
(
1
-
s
n
γ
l
)
)
s
n, and
(161)
δ
n
=
τ
2
s
n
2
(
τ
-
γ
l
)
M
~
2
+
1
τ
-
γ
l
〈
μ
F
q
-
γ
V
q
,
q
-
x
n
+
1
〉
.
From condition (i) and Step 5, it is easy to see that σ
n
→
0, ∑
n
=
0
∞
‍
σ
n
=
∞ and limsup
⁡
n
→
∞
δ
n
≤
0. Hence, by Lemma 10, we conclude that x
n
→
q as n
→
∞. This completes the proof.Corollary 22.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let Θ be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H and B
i
:
C
→
H be ζ-inverse-strongly monotone and η
i-inverse-strongly monotone, respectively, for i
=
1,2. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
G
M
E
P
(
Θ
,
φ
,
A
)
∩
I
(
B
1
,
R
1
)
∩
I
(
B
2
,
R
2
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. For arbitrarily given x
1
∈
H, let {
x
n
} be a sequence generated by
(162)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
v
n
=
J
R
2
,
λ
2
,
n
(
I
-
λ
2
,
n
B
2
)
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
x
n
+
1
=
s
n
γ
V
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
,
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
v
n
,
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), and lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
β
n
}
⊂
(
0,1
) and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1;
(iii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) and lim
⁡
n
→
∞
|
λ
i
,
n
+
1
-
λ
i
,
n
|
=
0 for i
=
1,2;
(iv)
{
r
n
}
⊂
[
e
,
f
]
⊂
(
0,2
ζ
) and lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0.Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of VIP (110).Corollary 23.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f
:
C
→
R be a convex functional with L-Lipschitz continuous gradient ∇
f. Let Θ be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function. Let R
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H and B
:
C
→
H be ζ-inverse-strongly monotone and ξ-inverse-strongly monotone, respectively. Let F
:
H
→
H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ
,
η
>
0. Let V
:
H
→
H be an l-Lipschitzian mapping with constant l
≥
0. Let 0
<
μ
<
2
η
/
κ
2 and 0
≤
γ
l
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Assume that Ω
:
=
G
M
E
P
(
Θ
,
φ
,
A
)
∩
I
(
B
,
R
)
∩
Γ
≠
∅ and that either (B1) or (B2) holds. For arbitrarily given x
1
∈
H, let {
x
n
} be a sequence generated by
(163)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
x
n
+
1
=
s
n
γ
V
x
n
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
s
n
μ
F
)
T
n
J
R
,
ρ
n
(
u
n
-
ρ
n
B
u
n
)
,
∀
n
≥
1
,
where P
C
(
I
-
λ
n
∇
f
)
=
s
n
I
+
(
1
-
s
n
)
T
n (here T
n is nonexpansive and s
n
=
(
2
-
λ
n
L
)
/
4
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
)). Assume that the following conditions hold:(i)
s
n
∈
(
0
,
1
/
2
) for each λ
n
∈
(
0
,
2
/
L
), and lim
⁡
n
→
∞
s
n
=
0
(
⇔
lim
⁡
n
→
∞
λ
n
=
2
/
L
);
(ii)
{
β
n
}
⊂
(
0,1
) and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1;
(iii)
{
ρ
n
}
⊂
[
a
,
b
]
⊂
(
0,2
ξ
) and lim
⁡
n
→
∞
|
ρ
n
+
1
-
ρ
n
|
=
0;
(iv)
{
r
n
}
⊂
[
e
,
f
]
⊂
(
0,2
ζ
) and lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0.Then {
x
n
} converges strongly as λ
n
→
2
/
L
(
⇔
s
n
→
0
) to a point q
∈
Ω, which is a unique solution of VIP (112).



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*Source: 132053-2014-01-23.xml*