# Solving Split Variational Inclusion Problem and Fixed Point Problem for Nonexpansive Semigroup without Prior Knowledge of Operator Norms

Authors: Haitao Che; Meixia Li
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.


---

## Body

## 1. Introduction

Recently, Moudafi [1] proposed the following split monotone variational inclusion problem (SMVIP): find a point x
∗
∈
H
1 such that (1)
0
∈
f
1
x
∗
+
B
1
x
∗
,
y
∗
=
A
x
∗
∈
H
2
solves
0
∈
f
2
y
∗
+
B
2
y
∗
,where H
1 and H
2 are two real Hilbert spaces with inner product 〈
·
,
·
〉 and induced norm · and B
1
:
H
1
→
2
H
1 and B
2
:
H
2
→
2
H
1 are multivalued maximal monotone mappings.Moudafi [1] shows that SMVIP (1) includes, as special cases, the split variational inequality problem, the split common fixed point problem, split zero problem, and split feasibility problem [1–7] which have already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning (see [5, 6]). This formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real-world problems, for instance, in sensor networks in computerized tomography and data compression [8, 9].Iff
1
≡
0 and f
2
≡
0, then SMVIP (1) can reduce to the following split variational inclusion problem (SVIP): find a point x
∗
∈
H
1 such that(2)
0
∈
B
1
x
∗
,
(3)
y
∗
=
A
x
∗
∈
H
2
s
o
l
v
e
s
0
∈
B
2
y
∗
.We know that (2) is the variational inclusion problem and denote its solution set by SOLVIP(B
1). SVIP ((2)-(3)) contains a pair of variational inclusion problems which need to be solved so that the image y
∗
=
A
x
∗ under a given bounded linear operator A, the solution x
∗ of SVIP (2) in H
1, is the solution of another SVIP (3) in another space H
2; we denote the solution set of SVIP (3) by SOLVIP(B
2). The solution set of SVIP ((2)-(3)) is denoted by U
=
{
x
∗
∈
H
1
:
x
∗
∈
S
O
L
V
I
P
(
B
1
) and A
x
∗
∈
S
O
L
V
I
P
(
B
2
)
}.Many works were devoted to the split variational inclusion problem ((2)-(3)). In 2012, Byrne et al. [4] proposed the weak and strong convergence of the following iterative method for SVIP ((2)-(3)): for x
0
∈
H
1, compute iterative sequence x
n generated by the following scheme:(4)
x
n
+
1
=
J
λ
B
1
x
n
+
γ
A
∗
J
λ
B
2
-
I
A
x
n
,where A
∗ is the adjoint of A, L is the spectral radius of the operator A
∗
A, and γ
∈
(
0
,
2
/
L
) and λ
>
0.In 2014, Kazmi and Rizvi [10] considered the strong convergence of the following iterative method: (5)
u
n
=
J
λ
B
1
x
n
+
γ
A
∗
J
λ
B
2
-
I
A
x
n
,
x
n
+
1
=
a
n
f
x
n
+
I
-
a
n
S
u
n
,where λ
>
0, A
∗ is the adjoint of A, L is the spectral radius of the operator A
∗
A, and γ
∈
(
0
,
1
/
L
). They proved the sequence {
x
n
} generated by (5) strongly converges to the fixed point of nonexpansive mapping S and the solution set Γ of SVIP ((2)-(3)).In 2015, Sitthithakerngkiet et al. [11] proposed the hybrid steepest descent method:(6)
u
n
=
J
λ
B
1
x
n
+
γ
A
∗
J
λ
B
2
-
I
A
x
n
,
x
n
+
1
=
a
n
ξ
f
x
n
+
I
-
a
n
D
S
n
u
n
,where S
n is a sequence of nonexpansive mappings, D is a strongly positive bounded linear operator, λ
>
0, A
∗ is the adjoint of A, L is the spectral radius of the operator A
∗
A, and γ
∈
(
0
,
1
/
L
) and λ
>
0. They revealed that the sequence {
x
n
} converges strongly to a point z, where z
=
P
Ω
(
I
-
D
+
ξ
f
)
(
z
) is a unique solution of the variational inequalities:(7)
D
-
ξ
f
z
,
z
-
x
≤
0
,
x
∈
Ω
.Note that, in algorithms (4), (5), and (6) mentioned above, the determination of the stepsize γ depends on the operator (matrix) norms A (or the largest eigenvalues of A
∗
A). This means that, in order to implement algorithms (4), (5), and (6), one has first to compute (or, at least, estimate) operator norms of A, which is not an easy work in practice.To overcome this difficulty, López et al. [12] and Zhao and Yang [13] presented useful method for choosing the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively.Motivated by the above results, we introduce a new choice of the stepsize sequence{
τ
n
} which depends on(8)
τ
n
=
σ
n
J
λ
B
2
-
I
A
x
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
,where σ
n
∈
[
a
,
b
]
⊂
(
0,1
). The advantage of our choice (8) of the stepsizes lies in the fact that no prior information about the operator norms of A is required, and still convergence is guaranteed.Following the work of Moudafi [1], Kazmi and Rizvi [10], and Byrne et al. [4], we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Numerical results are proposed to show that our algorithm is more suitable for SVIP ((2)-(3)) than the proposed algorithms (4) and (6).


## 2. Preliminaries

Throughout this paper, we denoteC to be a nonempty closed convex subset of H
1. Let T
:
H
1
→
H
1 be a mapping. A point x
∈
H
1 is said to be a fixed point of T provided x
=
T
x. We use F
(
T
) to denote the fixed point set of T. We write x
n
⇀
x to indicate that the sequence {
x
n
} converges weakly to x, and x
n
→
x implies that {
x
n
} converges strongly to x. We use ω
w
(
x
k
)
=
{
x
:
∃
x
k
j
⇀
x
} standing for the weak ω-limit set of {
x
k
}. For any x
∈
H, there exists a unique nearest point in C, denoted by P
C
x, such that(9)
x
-
P
C
x
≤
x
-
y
,
∀
y
∈
C
.Before proceeding further, we need to introduce a few concepts.A mappingT
:
H
1
→
H
1 is called contraction, if there exists a constant α
∈
(
0,1
) such that (10)
T
x
-
T
y
≤
α
x
-
y
,
∀
x
,
y
∈
H
1
.Ifα
=
1, then T is called nonexpansive.A mappingT
:
H
1
→
H
1 is said to be firmly nonexpansive, if (11)
T
x
-
T
y
,
x
-
y
≥
T
x
-
T
y
2
,
∀
x
,
y
∈
H
1
.One-parameter family mappingΓ
=
{
T
(
t
)
:
0
≤
t
<
∞
} from H
1 into itself is said to be a nonexpansive semigroup if it satisfies the following conditions:(i)
T
(
0
)
x
=
x, for all x
∈
H
1.
(ii)
T
(
s
+
t
)
=
T
(
s
)
T
(
t
), for all s
,
t
≥
0.
(iii)
For eachx
∈
H
1, the mapping T
(
t
)
x is continuous.
(iv)
T
(
t
)
x
-
T
(
t
)
y
≤
x
-
y, for all x
,
y
∈
H
1 and t
≥
0.We denote byF
(
Γ
) the set of all common fixed points of Γ; that is, F
(
Γ
)
:
=
{
x
∈
H
1
:
T
(
t
)
x
=
x
,
∀
t
>
0
}. It is well known that F
(
Γ
) is closed and convex [14].Now, we give an example of a nonexpansive semigroup.Example 1.
LetH
1
=
R and Γ
=
{
T
(
t
)
:
0
≤
t
<
∞
}, where T
(
t
)
x
=
1
/
1
0
t
x. Then, Γ
=
{
T
(
t
)
:
0
≤
t
<
∞
} is said to be a nonexpansive semigroup. In fact,(i)
T
(
0
)
x
=
1
/
1
0
0
x
=
x, for all x
∈
H
1,
(ii)
T
(
s
+
t
)
=
1
/
1
0
s
+
t
=
1
/
1
0
s
1
/
1
0
t
=
T
(
s
)
T
(
t
), for all s
,
t
≥
0,
(iii)
for eachx
∈
H
1, the mapping T
(
t
)
x
=
1
/
1
0
t
x is continuous,
(iv)
T
(
t
)
x
-
T
(
t
)
y
=
1
/
1
0
t
x
-
1
/
1
0
t
y
=
1
/
1
0
t
x
-
y
≤
x
-
y, for all x
,
y
∈
H
1 and t
≥
0.A set-valued mappingQ
:
H
1
→
2
H
1 is called monotone if, for all x
,
y
∈
H
1, f
∈
Q
x and g
∈
Q
y imply 〈
x
-
y
,
f
-
g
〉
≥
0.A monotone mappingQ
:
H
1
→
2
H
1 is called maximal if the graph G
(
Q
) of Q is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping Q is maximal if and only if for (
x
,
f
)
∈
H
×
H, 〈
x
-
y
,
f
-
g
〉
≥
0 for every (
y
,
g
)
∈
G
(
Q
) implies f
∈
Q
x.LetM
:
H
1
→
2
H
1 be a multivalued maximal monotone mapping. Then, the resolvent mapping J
λ
M
:
H
1
→
H
1, associated with M, is defined by (12)
J
λ
M
x
=
I
+
λ
M
-
1
x
,
∀
x
∈
H
1
,for some λ
>
0, where I stands for identity operator on H
1. We note that, for all λ
>
0, the resolvent operator J
λ
M is single-valued, nonexpansive, and firmly nonexpansive.The following principles play an important role in our argument.A mappingT
:
H
1
→
H
1 is called demiclosed at the origin if for each sequence {
x
n
} which weakly converges to x, and the sequence {
T
x
n
} strongly converges to 0, then T
x
=
0.T is said to be semicompact, if, for any bounded sequence {
x
n
}
⊂
H
1, lim
n
→
∞
⁡
x
n
-
T
x
n
⁡
⁡
=
0; then, there exists a subsequence {
x
n
i
}
⊂
{
x
n
} such that {
x
n
i
} converges strongly to some point x
∗
∈
H
1
.To establish our results, we need the following technical lemmas.Lemma 2 (see [15]).
Ifx
,
y
,
z
∈
H, then(a)
x
+
y
2
≤
x
2
+
2
〈
y
,
x
+
y
〉;
(b)
for anyλ
∈
[
0,1
];(13)
λ
x
+
1
-
λ
y
2
=
λ
x
2
+
1
-
λ
y
2
-
λ
1
-
λ
x
-
y
2
.
(c)
fora
,
b
,
c
∈
[
0,1
] with a
+
b
+
c
=
1,(14)
a
x
+
b
y
+
c
z
2
=
a
x
2
+
b
y
2
+
c
z
2
-
a
b
x
-
y
2
-
a
c
x
-
z
2
-
b
c
y
-
z
2
.Lemma 3 (see [1]).
SVIP ((2)-(3)) is equivalent to finding  x
∗
∈
H
1 such that y
∗
=
A
x
∗
∈
H
2, (15)
x
∗
=
J
λ
B
1
x
∗
,
y
∗
=
J
λ
B
2
y
∗
,
for
some
λ
>
0
.Lemma 4 (see [16]).
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let Γ
=
{
T
(
s
)
:
s
≥
0
} from C be a nonexpansive semigroup on C; then, for all h
≥
0, (16)
l
i
m
t
→
∞
sup
x
∈
C
⁡
1
t
∫
0
t
T
s
x
d
s
-
T
h
1
t
∫
0
t
T
s
x
d
s
=
0
.Lemma 5 (see [16]).
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H, let {
x
n
} be a sequence, and let Γ
=
{
T
(
s
)
:
s
≥
0
} from C be a nonexpansive semigroup on C; if the following conditions are satisfied,(i)
x
n
⇀
z,
(ii)
limsup
s
→
∞
⁡
limsup
n
→
∞
⁡
T
(
s
)
x
n
-
x
n
=
0,then, z
∈
F
(
Γ
).Lemma 6 (see [17]).
LetH be a Hilbert space and let {
u
n
} be a sequence in H such that there exists a nonempty set W
⊂
H satisfying the following:(i)
For everyw
∈
W, lim
n
→
∞
⁡
u
n
-
w exists.
(ii)
Each weak cluster point of the sequence{
w
n
} is in W.Then, there exists w
∗
∈
W such that {
u
n
} weakly converges to w
∗.Lemma 7.
LetΓ
=
{
T
(
s
)
:
s
≥
0
} from H
1 into itself be a nonexpansive semigroup; then, for p
∈
F
(
Γ
), x
∈
H
1, and t
>
0, (17)
1
t
∫
0
t
T
s
x
d
s
-
x
2
≤
2
x
-
p
,
x
-
1
t
∫
0
t
T
s
x
d
s
.Proof.
Forp
∈
F
(
Γ
), x
∈
H
1, and t
>
0, it follows from the definition of nonexpansive semigroup that (18)
1
t
∫
0
t
T
s
x
d
s
-
p
2
≤
x
-
p
2
,which implies that(19)
1
t
∫
0
t
T
s
x
d
s
-
p
,
1
t
∫
0
t
T
s
x
d
s
-
p
≤
x
-
p
,
x
-
p
=
x
-
p
,
x
-
1
t
∫
0
t
T
s
x
d
s
+
x
-
p
,
1
t
∫
0
t
T
s
x
d
s
-
p
.Furthermore, (20)
1
t
∫
0
t
T
s
x
d
s
-
p
,
1
t
∫
0
t
T
s
x
d
s
-
x
≤
x
-
p
,
x
-
1
t
∫
0
t
T
s
x
d
s
.Then, (21)
1
t
∫
0
t
T
s
x
d
s
-
x
,
1
t
∫
0
t
T
s
x
d
s
-
x
+
x
-
p
,
1
t
∫
0
t
T
s
x
d
s
-
x
≤
x
-
p
,
x
-
1
t
∫
0
t
T
s
x
d
s
.Thus, (22)
1
t
∫
0
t
T
s
x
d
s
-
x
,
1
t
∫
0
t
T
s
x
d
s
-
x
≤
2
x
-
p
,
x
-
1
t
∫
0
t
T
s
x
d
s
.This completes the proof of Lemma 7.


## 3. Main Results

In this section, we first describe our algorithm and then reveal the convergence analysis of the algorithm.Now, we propose our algorithm.Algorithm 8.
Letx
0
∈
H
1 be arbitrary. Assume that x
n has been constructed and A
∗
(
J
λ
B
2
-
I
)
A
x
n
≠
0; then, calculate x
n
+
1 via the rule (
k
+
1
)th iterate via the following formula: (23)
u
n
=
J
λ
B
1
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
,
x
n
+
1
=
1
-
a
n
u
n
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
,where the stepsize τ
n is chosen by (8). If A
∗
(
J
λ
B
2
-
I
)
A
x
n
=
0, then x
n
+
1
=
x
n is the solution of ((2)-(3)) and the iterative process stops. Otherwise, we set k
:
=
k
+
1 and go to (23).Remark 9.
Notice that in (8) the choice of the stepsize τ
n is independent of the norms A.Remark 10.
The stepsize{
τ
n
} is bounded. Indeed, it follows from the condition on {
τ
n
} that(24)
i
n
f
A
∗
J
λ
B
2
-
I
A
x
n
≠
0
J
λ
B
2
-
I
A
x
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
-
τ
n
>
0
.On the other hand, since(25)
A
∗
J
λ
B
2
-
I
A
x
n
≤
A
J
λ
B
2
-
I
A
x
n
,we obtain(26)
J
λ
B
2
-
I
A
x
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
≥
1
A
2
.Thus, s
u
p
A
∗
(
J
λ
B
2
-
I
)
A
x
n
≠
0
τ
n
<
∞ and {
τ
n
} is bounded.Next, we will discuss the convergence analysis of algorithm (23) for approximating a common solution of SVIP ((2)-(3)) and fixed point problem for nonexpansive semigroups.Theorem 11.
LetH
1 and H
2 be two real Hilbert spaces and let A
:
H
1
→
H
2 be a bounded linear operator. Assume that B
1
:
H
1
→
2
H
1 and B
2
:
H
2
→
2
H
2 are maximal monotone mappings and Γ
:
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a one-parameter nonexpansive semigroup on H
1 such that Ω
=
F
(
Γ
)
∩
U
≠
∅. Let the sequences {
x
n
} be generated by (23). If 0
<
l
i
m
i
n
f
n
→
∞
a
n
<
l
i
m
s
u
p
n
→
∞
a
n
<
1 and {
t
n
}
∈
(
0
,
∞
), then {
x
n
} converges strongly to q
∈
Ω.Proof.
Takingp
∈
Ω, we have p
=
J
λ
B
1
p, A
p
=
J
λ
B
2
(
A
p
), and T
(
s
)
p
=
p. From (23) and Lemma 3, one has (27)
u
n
-
p
2
=
J
λ
B
1
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
=
J
λ
B
1
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
J
λ
B
1
p
2
≤
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
=
x
n
-
p
2
+
τ
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
+
2
τ
n
x
n
-
p
,
A
∗
J
λ
B
2
-
I
A
x
n
.Notice that (28)
2
τ
n
x
n
-
p
,
A
∗
J
λ
B
2
-
I
A
x
n
=
2
τ
n
A
x
n
-
p
,
J
λ
B
2
-
I
A
x
n
=
2
τ
n
A
x
n
-
A
p
+
J
λ
B
2
-
I
A
x
n
-
J
λ
B
2
-
I
A
x
n
,
J
λ
B
2
-
I
A
x
n
=
2
τ
n
J
λ
B
2
A
x
n
-
p
,
J
λ
B
2
-
I
A
x
n
-
2
τ
n
J
λ
B
2
-
I
A
x
n
2
=
τ
n
J
λ
B
2
A
x
n
-
A
p
2
+
J
λ
B
2
-
I
A
x
n
2
-
A
x
n
-
A
p
2
-
2
τ
n
J
λ
B
2
-
I
A
x
n
2
≤
τ
n
J
λ
B
2
-
I
A
x
n
2
-
2
τ
n
J
λ
B
2
-
I
A
x
n
2
=
-
τ
n
J
λ
B
2
-
I
A
x
n
2
.It follows from (28), (27), and (8) that (29)
u
n
-
p
2
≤
x
n
-
p
2
+
τ
n
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
2
-
J
λ
B
2
-
I
A
x
n
2
≤
x
n
-
p
2
.Furthermore, (23), (29), and Lemma 2 lead to(30)
x
n
+
1
-
p
2
=
1
-
a
n
u
n
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
p
2
=
1
-
a
n
u
n
-
p
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
p
2
=
1
-
a
n
u
n
-
p
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
1
t
n
∫
0
t
n
T
s
p
d
s
2
=
1
-
a
n
u
n
-
p
2
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
1
t
n
∫
0
t
n
T
s
p
d
s
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
1
-
a
n
u
n
-
p
2
+
a
n
u
n
-
p
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
=
u
n
-
p
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
x
n
-
p
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
x
n
-
p
2
≤
⋯
≤
x
0
-
p
2
.Hence, {
x
n
} is bounded and we also obtain that {
u
n
} is bounded.
On the other hand, from Lemma7, we obtain (31)
x
n
+
1
-
p
2
=
1
-
a
n
u
n
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
p
2
=
u
n
-
p
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
=
u
n
-
p
2
+
a
n
2
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
+
2
a
n
u
n
-
p
,
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
≤
u
n
-
p
2
+
a
n
2
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
x
n
-
p
2
+
a
n
a
n
-
1
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
,which means that (32)
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
x
n
-
p
2
-
x
n
+
1
-
p
2
.From (30), we have that {
x
n
-
p
} is nonincreasing, and then {
x
n
-
p
} is convergent. Obviously,(33)
lim
n
→
∞
⁡
x
n
-
p
2
-
x
n
+
1
-
p
2
=
0
.This together with (32) and the condition on a
n deduce that (34)
lim
n
→
∞
⁡
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
=
0
.Observe that (35)
u
n
-
T
u
u
n
≤
u
n
-
1
t
n
∫
0
t
n
T
s
u
n
d
s
+
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
T
u
1
t
n
∫
0
t
n
T
s
u
n
d
s
+
T
u
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
T
u
u
n
≤
2
u
n
-
1
t
n
∫
0
t
n
T
s
u
n
d
s
+
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
T
u
1
t
n
∫
0
t
n
T
s
u
n
d
s
.It follows from (34) and Lemma 4 that (36)
lim
n
→
∞
⁡
u
n
-
T
u
u
n
=
0
.
From (29) and (30), we deduce (37)
x
n
+
1
-
p
2
=
1
-
a
n
u
n
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
p
2
=
1
-
a
n
u
n
-
p
2
+
a
n
u
n
-
p
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
=
u
n
-
p
2
-
a
n
1
-
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
2
≤
u
n
-
p
2
≤
x
n
-
p
2
+
τ
n
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
2
-
J
λ
B
2
-
I
A
x
n
2
=
x
n
-
p
2
+
σ
n
σ
n
-
1
J
λ
B
2
-
I
A
x
n
4
A
∗
J
λ
B
2
-
I
A
x
n
2
.It yields that (38)
σ
n
1
-
σ
n
J
λ
B
2
-
I
A
x
n
4
A
∗
J
λ
B
2
-
I
A
x
n
2
≤
x
n
-
p
2
-
x
n
+
1
-
p
2
.Thus, (39)
lim
n
→
∞
⁡
J
λ
B
2
-
I
A
x
n
2
A
∗
J
λ
B
2
-
I
A
x
n
=
0
.Since(40)
A
∗
J
λ
B
2
-
I
A
x
n
≤
A
J
λ
B
2
-
I
A
x
n
,it is easy to show that (41)
J
λ
B
2
-
I
A
x
n
≤
A
J
λ
B
2
-
I
A
x
n
2
A
∗
J
λ
B
2
-
I
A
x
n
.Consequently, we obtain(42)
lim
n
→
∞
⁡
J
λ
B
2
-
I
A
x
n
=
0
.Furthermore,(43)
lim
n
→
∞
⁡
A
∗
J
λ
B
2
-
I
A
x
n
=
0
.
From (23), we have (44)
u
n
-
p
2
=
J
λ
B
1
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
=
J
λ
B
1
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
J
λ
B
1
p
2
≤
u
n
-
p
,
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
=
1
2
u
n
-
p
2
+
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
-
u
n
-
p
-
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
=
1
2
u
n
-
p
2
+
x
n
+
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
-
p
2
-
u
n
-
x
n
-
τ
n
A
∗
J
λ
B
2
-
I
A
x
n
2
=
1
2
u
n
-
p
2
+
x
n
-
p
2
+
τ
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
+
2
τ
n
x
n
-
p
,
A
∗
J
λ
B
2
-
I
A
x
n
-
u
n
-
x
n
2
+
2
τ
n
u
n
-
x
n
,
A
∗
J
λ
B
2
-
I
A
x
n
-
τ
n
2
A
∗
J
λ
B
2
-
I
A
x
n
2
.Hence, (45)
u
n
-
p
2
≤
x
n
-
p
2
+
2
τ
n
x
n
-
p
,
A
∗
J
λ
B
2
-
I
A
x
n
-
u
n
-
x
n
2
+
2
τ
n
u
n
-
x
n
,
A
∗
J
λ
B
2
-
I
A
x
n
,which implies that (46)
u
n
-
x
n
2
≤
x
n
-
p
2
-
u
n
-
p
2
+
2
τ
n
x
n
-
p
A
∗
J
λ
B
2
-
I
A
x
n
+
2
τ
n
u
n
-
x
n
A
∗
J
λ
B
2
-
I
A
x
n
.Moreover, from (30), we have (47)
u
n
-
x
n
2
≤
x
n
-
p
2
-
x
n
+
1
-
p
2
+
2
τ
n
x
n
-
p
A
∗
J
λ
B
2
-
I
A
x
n
+
2
τ
n
u
n
-
x
n
A
∗
J
λ
B
2
-
I
A
x
n
.Therefore, (43) yields that (48)
lim
n
→
∞
⁡
u
n
-
x
n
=
0
.We compute (49)
x
n
+
1
-
u
n
=
1
-
a
n
u
n
+
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
=
a
n
1
t
n
∫
0
t
n
T
s
u
n
d
s
-
u
n
. Equation (34) implies that (50)
lim
n
→
∞
⁡
x
n
+
1
-
u
n
=
0
.Consequently,(51)
lim
n
→
∞
⁡
x
n
+
1
-
x
n
≤
lim
n
→
∞
⁡
x
n
+
1
-
u
n
+
u
n
-
x
n
=
0
.Furthermore, (52)
lim
n
→
∞
⁡
u
n
+
1
-
u
n
≤
lim
n
→
∞
⁡
u
n
+
1
-
x
n
+
1
+
x
n
+
1
-
u
n
=
0
.
Since{
x
n
} and {
u
n
} are bounded, we consider a weak cluster point q of {
x
n
}. Without loss of generality, we may assume that subsequence {
x
n
j
} of {
x
n
} converges weakly to q. From (48), we have {
u
n
j
} of {
u
n
}, which converges weakly to q. Furthermore, u
n
j
=
J
λ
B
1
(
x
n
j
+
τ
n
j
A
∗
(
J
λ
B
2
-
I
)
A
x
n
j
) can be rewritten as(53)
x
n
j
-
u
n
j
+
τ
n
j
A
∗
J
λ
B
2
-
I
A
x
n
j
λ
∈
B
1
u
n
j
.Taking limit j
→
∞ in (53) and taking into account (43) and (48), together with the fact that the graph of a maximal monotone operator is weakly strongly closed, we have 0
∈
B
1
(
q
); that is, q
∈
S
O
L
V
I
P
(
B
1
). Furthermore, from the asymptotical behavior of {
x
n
} and {
u
n
}, we deduce that {
A
x
n
j
} weakly converges to A
q. Applying (43) and the fact that the resolvent J
λ
B
2 is nonexpansive, together with Lemma 3, we have A
q
∈
B
2
(
A
q
); that is, A
q
∈
S
O
L
V
I
P
(
B
2
). Thus, q
∈
U.
Next, we will prove thatq
∈
F
(
T
). For the sake of contradiction, suppose that T
(
u
)
q
≠
q. It follows from Opial condition and (36) that (54)
lim
j
→
∞
inf
⁡
u
n
j
-
q
<
l
i
m
j
→
∞
inf
⁡
u
n
j
-
T
u
q
≤
lim
j
→
∞
inf
⁡
u
n
j
-
T
u
u
n
j
+
T
u
u
n
j
-
T
u
q
≤
lim
j
→
∞
inf
⁡
u
n
j
-
T
u
u
n
j
+
u
n
j
-
q
≤
lim
j
→
∞
inf
⁡
u
n
j
-
q
.This is a contradiction, which shows that q
∈
F
(
Γ
). Thus, x
∈
Ω. Furthermore, from Lemma 6, we deduce that x
n
⇀
q, u
n
⇀
q, and q
∈
Ω. Due to (36), we obtain that T
(
u
) is semicompact, and then {
x
n
} and {
u
n
} converge strongly to q. This completes the proof of Theorem 11.Theorem 12.
LetH
1 and H
2 be two real Hilbert spaces and let A
:
H
1
→
H
2 be a bounded linear operator. Assume that B
1
:
H
1
→
2
H
1 and B
2
:
H
2
→
2
H
2 are maximal monotone mappings and let T
:
H
1
→
H
1 be a nonexpansive mapping such that Ω
=
F
(
T
)
∩
U
≠
∅. Let the sequences {
x
n
} be generated by (23). If 0
<
l
i
m
i
n
f
n
→
∞
a
n
<
l
i
m
s
u
p
n
→
∞
a
n
<
1, then, {
x
n
} converges strongly to q
∈
Ω.Proof.
Clearly, Theorem12 is valid for a nonexpansive mapping. Therefore, the desired conclusion follows immediately from Theorem 11. This completes the proof.


## 4. Numerical Examples

We now pay our attention to show a numerical example to demonstrate the performance and convergence of our result. In the experiment, the stopping criterion isA
∗
(
J
λ
B
2
-
I
)
A
x
n
≤
1
0
-
10. In [4, 11], the stopping criterion is x
n
+
1
-
x
n
≤
1
0
-
10. ST denotes the initial point, IT denotes the iterative number, and SOL denotes a solution of the test problem.Example 13 (see [11]).
LetH
1
=
H
2
=
R
2, and let two operators of matrix multiplication B
1
:
R
2
→
R
2 and B
2
:
R
2
→
R
2 be defined by B
1
(
x
)
=
T
1
(
x
) and B
2
(
x
)
=
T
2
(
x
), where T
1
=
8
0
0
2 and T
2
=
3
0
0
6
. Observe that T
1 and T
2 are positive linear operators; then, they are maximal monotone. So, we can define the resolvent mappings J
λ
B
1
=
(
I
+
λ
B
1
)
-
1 and J
λ
B
2
=
(
I
+
λ
B
2
)
-
1 on R
2 associated with B
1 and B
2, where λ
>
0. Let A
∈
R
2
×
2 be a nonsingular matrix operator in which elements are random and let A
∗ be an adjoint of A. Assume that λ
=
0.5, a
n
=
1
/
2, σ
n
=
0.8
+
1
/
10
n, and Γ
=
{
T
(
t
)
:
0
≤
t
<
∞
}, where T
(
t
)
=
1
0
-
t
x, for x
∈
R
2.Example 14.
Let two operators of matrix multiplicationB
1
:
R
I
→
R
I and B
2
:
R
I
→
R
J be defined by B
1
(
x
)
=
T
1
(
x
) and B
2
(
x
)
=
T
2
(
x
), which are generated by the codes as (55)
L
=
30
∗
rand
⁡
I
,
1
,
B
1
=
diag
⁡
L
,
H
=
20
∗
rand
⁡
J
,
1
,
B
2
=
d
i
a
g
H
.Observe that T
1 and T
2 are positive linear operators; then, they are maximal monotone. So, we can define the resolvent mappings J
λ
B
1
=
(
I
+
λ
B
1
)
-
1 on R
I and J
λ
B
2
=
(
I
+
λ
B
2
)
-
1 on R
J associated with B
1 and B
2, where λ
>
0. Let A
=
(
a
i
j
)
J
×
I be a matrix operator which is generated by the codes as(56)
A
=
10
∗
r
a
n
d
J
,
I
+
5
.We choose the initial point x
1
=
10
∗
rand
⁡
(
I
,
1
)
+
20.Figure1 reveals that the more the iteration steps are, the more slowly the sequence (
x
n
,
y
n
) converges to (
0,0
). From Table 1, we see that our method is as completive as the methods of [4, 11], and the sequence {
x
n
} is more closed to the same point (
0,0
). From Table 2, as the size increases, our method performs better than the algorithms of [4, 11]. Furthermore, it indicates that our method is promising for solving large scale problems. Thus, we could observe that our algorithm is more suitable for SVIP ((2)-(3)) than the proposed algorithms of [4, 11].Table 1
Numerical  comparison  between  our  algorithm  and  the  algorithms  of [4, 11] for Example 13.
Our algorithm
The algorithm of [11]
The algorithm of [4]
ST
IT
SOL
IT
SOL
IT
SOL
(
1,1
)
⊤
12
10
-
12
(
0.0019,0.9722
)
⊤
21
10
-
10
(
0.1953,0.2653
)
⊤
14
10
-
10
(
0.0003,0.5509
)
⊤
(
10,10
)
⊤
12
10
-
11
(
0.0019,0.9722
)
⊤
20
10
-
10
(
0.1554,0.1697
)
⊤
15
10
-
10
(
0.0003,0.8953
)
⊤
(
100,100
)
⊤
13
10
-
11
(
0.0009,0.7899
)
⊤
21
10
-
10
(
0.1787,0.1891
)
⊤
17
10
-
10
(
0.0000,0.2364
)
⊤
(
1000,1000
)
⊤
14
10
-
11
(
0.0004,0.6418
)
⊤
22
10
-
10
(
0.3249,0.3865
)
⊤
18
10
-
10
(
0.0000,0.3842
)
⊤Table 2
Numerical  comparison  between  our  algorithm  and  the  algorithms  of  [4, 11]  for  Example 14.
Our algorithm
The algorithm of [11]
The algorithm of [4]
I
J
IT
‖
x
n
‖
IT
‖
x
n
‖
IT
‖
x
n
‖
10
20
13
5.4634
e
-
017
435
1.0608
e
-
011
104
NaN
30
30
45
2.0376
e
-
015
1449
1.5019
e
-
011
84
NaN
50
100
33
4.5147
e
-
016
195
NaN
71
NaN
100
200
31
6.2001
e
-
017
134
NaN
64
NaNFigure 1
Behavior ofx
n at the initial point (
100,100
)
T with λ
=
0.5 for Example 13.



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*Source: 408165-2015-11-04.xml*