# Iterative Algorithms for New General Systems of Set-Valued Variational Inclusions Involving(

Authors: Ting-jian Xiong; Heng-you Lan
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We introduce and study a class of new general systems of set-valued variational inclusions involving(
A
,
η
)-maximal relaxed monotone operators in Hilbert spaces. By using the general resolvent operator technique associated with (
A
,
η
)-maximal relaxed monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general system of set-valued variational inclusion problem and prove the convergence of this algorithm. Our results improve and extend some known results.


---

## Body

## 1. Introduction

It is well known that variational inequalities and variational inclusions, which have been extended and generalized in different directions by using novel and innovative techniques and ideas, provide mathematical models to some problems arising in economics, mechanics, engineering science, and other pure and applied sciences. Among these methods, the resolvent operator technique is very important. See, for example, [1–17] and the references therein.Recently, Huang and Fang [18] introduced a system of order complementarity problems and established some existence results for the system using fixed-point theory. Verma [19] introduced and studied some systems of the system variational inequalities and developed some iterative algorithms for approximating the solutions of the systems of variational inequalities. Cho et al. [20] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. The authors also proved some existence and uniqueness theorems of solutions for the system and also constructed an iterative algorithm for approximating the solution of the system of nonlinear variational inequalities. Further, Fang et al. [1], Yan et al. [2], Fang and Huang [3], and Cao [4] considered some new systems of variational inclusions involving H-monotone operators and (
H
,
η
)-monotone operators in Hilbert space, respectively. Using the corresponding resolvent operator associated with H-monotone operators and (
H
,
η
)-monotone operators, the authors proved the existence of solutions for these new systems of variational inclusions and constructed a new algorithm for approximating the solution of this system and discussed the convergence of the sequence of iterations generated by the algorithm.Very recently, Lan et al. [5] and Peng and Zhao [7] introduced and studied a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces, respectively. By using the concept and properties of A-monotone operators and the resolvent operator technique associated with A-monotone operators due to Verma [8], the author constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with A-monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. For more details, see, for example, [1–5, 7, 8, 10–19, 21–25].On the other hand, Lan [6] first introduced a new concept of (
A
,
η
)-monotone (the so-called (
A
,
η
)-maximal relaxed monotone [9]) operators, which generalizes the (
H
,
η
)-monotonicity, A-monotonicity, and other existing monotone operators as special cases, and studied some properties of (
A
,
η
)-monotone operators and defined resolvent operators associated with (
A
,
η
)-monotone operators.Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative(
A
,
η
)-maximal monotone operators in Hilbert spaces: find (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
∈
U
i
j
(
x
j
*
) such that
(1)
0
∈
F
i
(
u
i
1
,
…
,
u
i
i
-
1
,
x
i
*
,
u
i
i
+
1
,
…
,
u
i
m
)
+
M
i
(
x
i
*
)
,
where m is a given positive integer, F
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i, A
i
:
H
i
→
H
i, and η
i
:
H
i
×
H
i
→
H
i are single-valued operators, U
i
j
:
H
j
→
2
H
j is a set-valued operator, M
i
:
H
i
→
2
H
i is (
A
i
,
η
i
)-maximal relaxed monotone, and i
,
j
=
1,2
,
…
,
m and i
≠
j.Some special cases of the problem (1) had been studied by many authors. Here, we mention some of them as follows.Case 1.
Ifm
=
2, then the problem (1) reduces to the problem of finding (
x
1
*
,
x
2
*
)
∈
H
1
×
H
2 and u
1
∈
U
1
(
x
1
*
) and u
2
∈
U
2
(
x
2
*
) such that
(2)
0
∈
F
1
(
x
1
*
,
u
2
)
+
M
1
(
x
1
*
)
,
0
∈
F
2
(
u
1
,
x
2
*
)
+
M
2
(
x
2
*
)
.
The problem (2) is called a nonlinear set-valued variational inclusion system problem, which was considered and studied by Agarwal and Verma [9].Case 2.
Whenm
=
2 and M
i
(
x
i
)
=
∂
φ
i
(
x
i
), for all x
i
∈
H
i, i
=
1,2, where φ
i
:
H
i
→
R
∪
{
+
∞
} is proper, convex, and lower semicontinuous functional and ∂
φ
i denotes the subdifferential operator of φ
i, then problem (1) becomes the following system of set-valued mixed variational inequalities: find (
x
1
*
,
x
2
*
)
∈
H
1
×
H
2 and u
1
∈
U
1
(
x
1
*
) and u
2
∈
U
2
(
x
2
*
) such that
(3)
〈
F
1
(
x
1
*
,
u
2
)
,
x
-
x
1
*
〉
+
φ
1
(
x
)
-
φ
1
(
x
1
*
)
≥
0
,
∀
x
∈
H
1
,
〈
F
2
(
u
1
,
x
2
*
)
,
y
-
x
2
*
〉
+
φ
2
(
y
)
-
φ
2
(
x
2
*
)
≥
0
,
∀
y
∈
H
2
.
IfU
1
=
U
2
≡
I, the identity operator, then the problem (3) reduces to the following problem of finding (
x
1
*
,
x
2
*
)
∈
H
1
×
H
2 such that
(4)
〈
F
1
(
x
1
*
,
x
2
*
)
,
x
-
x
1
*
〉
+
φ
1
(
x
)
-
φ
1
(
x
1
*
)
≥
0
,
∀
x
∈
H
1
,
〈
F
2
(
x
1
*
,
x
2
*
)
,
y
-
x
2
*
〉
+
φ
2
(
y
)
-
φ
2
(
x
2
*
)
≥
0
,
∀
y
∈
H
2
,
which is called the system of nonlinear variational inequalities considered by Cho et al. [20]. Some specializations of problem (4) are dealt by Kim and Kim [21].Case 3.
Ifm
=
2 and U
1
=
U
2
≡
I, then the problem (1) reduces to finding (
x
1
*
,
x
2
*
)
∈
H
1
×
H
2 such that
(5)
0
∈
F
1
(
x
1
*
,
x
2
*
)
+
M
1
(
x
1
*
)
,
0
∈
F
2
(
x
1
*
,
x
2
*
)
+
M
2
(
x
2
*
)
,
which was considered by Fang et al. [1].In brief, the problem (1) is the most general and unifying system form, so long as, for appropriate and suitable choices of positive integer m and operators F
i, A
i, η
i, M
i, and U
i
j for i
,
j
=
1,2
,
…
,
m, one can know that the problem (1) includes a number of known general problems of variational character, including variational inequality (system) problems and variational inclusion (system) problems as special cases. For more details, see [1–5, 7–25] and the reference therein.Furthermore, in this paper, we will construct some new iterative algorithms to approximate the solution of the general system of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms in Hilbert spaces.


## 2. Preliminaries

Thereafter, letH, H
i
(
i
=
1,2
,
…
,
m
) be real Hilbert spaces endowed with the norm ∥
·
∥ and inner product 〈
·
,
·
〉. Let 2
H and C
(
H
) denote the family of all the nonempty subsets of H and the family of all closed subsets of H, respectively.In order to get the main results of the paper, we need the following concepts and lemmas.Definition 1.
LetA
:
H
→
H be a single-valued operator. Then the map A is said to be (i)
α-strongly monotone, if there exists a constant α
>
0 such that
(6)
〈
A
(
x
)
-
A
(
y
)
,
x
-
y
〉
≥
α
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
;
(ii)
β-Lipschitz continuous, if there exists a constant β
>
0 such that
(7)
∥
A
x
-
A
y
∥
≤
β
∥
x
-
y
∥
,
∀
x
,
y
∈
H
.Definition 2.
Letη
:
H
×
H
→
H and A
:
H
→
H be single-valued operators; let M
:
H
→
2
H be set-valued operator. Then (i)
η is said to be τ-Lipschitz continuous, if there exists a constant τ
>
0 such that
(8)
∥
η
(
x
,
y
)
∥
≤
τ
∥
x
-
y
∥
,
∀
x
,
y
∈
H
;
(ii)
A is said to be η-monotone, if
(9)
〈
A
(
x
)
-
A
(
y
)
,
η
(
x
,
y
)
〉
≥
0
,
∀
x
,
y
∈
H
;
(iii)
A is said to be strictly η-monotone, if A is η-monotone and
(10)
〈
A
(
x
)
-
A
(
y
)
,
η
(
x
,
y
)
〉
=
0
iff
x
=
y
;
(iv)
A is said to be (
r
,
η
)-strongly monotone, if there exists a constant r
>
0 such that
(11)
〈
A
(
x
)
-
A
(
y
)
,
η
(
x
,
y
)
〉
≥
r
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
;
(v)
M is said to be (
m
,
η
)-relaxed monotone, if there exists a constant m
>
0 such that
(12)
〈
u
-
v
,
η
(
x
,
y
)
〉
≥
(
-
m
)
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
,
f
f
f
f
f
f
f
f
f
f
k
f
f
f
i
u
∈
M
(
x
)
,
v
∈
M
(
y
)
;
(vi)
M is said to be (
A
,
η
)-maximal relaxed monotone, if M is (
m
,
η
)-relaxed monotone and
(13)
(
A
+
λ
M
)
(
H
)
=
H
,
∀
λ
>
0
.Definition 3.
Fori
=
1,2
,
…
,
m, let H
i be a Hilbert space and let A
i
:
H
i
→
H
i be a single-valued operator; then nonlinear operator F
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i is said to be (i)
μ
i-strongly monotone with respect to A
i in the ith argument, if there exist constants μ
i
>
0 such that, for x
i
1
,
x
i
2
∈
H
i, i
=
1,2
,
…
,
m,
(14)
〈
F
i
(
…
,
x
i
1
,
…
)
-
F
i
(
…
,
x
i
2
,
…
)
,
A
i
(
x
i
1
)
-
A
i
(
x
i
2
)
〉
≥
μ
i
∥
x
i
1
-
x
i
2
∥
2
;
(ii)
(
c
i
,
μ
i
)-relaxed cocoercive with respect to A
i (or relative (
c
i
,
μ
i
)-relaxed cocoercive) in the ith argument, if there exist constants c
i
,
μ
i
>
0 such that, for x
i
1
,
x
i
2
∈
H
i
,
i
=
1,2
,
…
,
m,
(15)
〈
F
i
(
…
,
x
i
1
,
…
)
-
F
i
(
…
,
x
i
2
,
…
)
,
A
i
(
x
i
1
)
-
A
i
(
x
i
2
)
〉
≥
(
-
c
i
)
∥
F
i
(
…
,
x
i
1
,
…
)
-
F
i
(
…
,
x
i
2
,
…
)
∥
2
+
μ
i
∥
x
i
1
-
x
i
2
∥
2
;
(iii)
ζ
i
j-Lipschitz continuous in the jth argument, if there exists constant ζ
i
j
>
0 such that, for x
j
,
y
j
∈
H
j, j
=
1,2
,
…
,
m,
(16)
∥
F
i
(
x
1
,
…
,
x
j
-
1
,
x
j
,
x
j
+
1
,
…
,
x
m
)
h
-
F
i
(
x
1
,
…
,
x
j
-
1
,
y
j
,
x
j
+
1
,
…
,
x
m
)
∥
≤
ζ
i
j
∥
x
j
-
y
j
∥
.Remark 4.
Whenm
=
1, then Definition 3 reduces to the corresponding concept of the relative strong monotonicity, relative relaxed cocoercive, and Lipschitz continuity.Definition 5.
Letη
:
H
×
H
→
H be a single-valued operator, let A
:
H
→
H be a strictly η-monotone operator, and let M
:
H
→
2
H be an (
A
,
η
)-maximal relaxed monotone operator. Then general resolvent operator R
M
,
λ
A
,
η
:
H
→
H is defined by
(17)
R
M
,
λ
A
,
η
(
z
)
=
(
A
+
λ
M
)
-
1
(
z
)
,
∀
z
∈
H
,
where λ
>
0 is a constant.Lemma 6 (see [6]).
Letη
:
H
×
H
→
H be a τ-Lipschitz continuous operator, let A
:
H
→
H be an (
r
,
η
)-strongly monotone operator, and let M
:
H
→
2
H be an (
A
,
η
)-maximal relaxed monotone operator. Then general resolvent operator R
M
,
λ
A
,
η
:
H
→
H is τ
/
(
r
-
λ
m
)-Lipschitz continuous; that is,
(18)
∥
R
M
,
λ
A
,
η
(
x
)
-
R
M
,
λ
A
,
η
(
y
)
∥
≤
τ
r
-
λ
m
∥
x
-
y
∥
,
∀
x
,
y
∈
H
,
where r
-
λ
m
>
0.Next, we define the Hausdorff pseudometricD
:
C
(
H
)
×
C
(
H
)
→
R
∪
{
+
∞
} as follows:
(19)
D
(
U
,
V
)
=
max
⁡
{
sup
⁡
x
∈
U
inf
⁡
y
∈
V
∥
x
-
y
∥
,
sup
⁡
y
∈
V
inf
⁡
x
∈
U
∥
x
-
y
∥
}
,
yyyyyyyyyyyyyyyyyyaaayyyyyyyyyy
∀
U
,
V
∈
C
(
H
)
.
Note that if C
(
H
) is restricted to closed bounded subsets of the family C
B
(
H
), then the Hausdorff pseudometric D reduces to Hausdorff metric H
^
:
C
B
(
H
)
×
C
B
(
H
)
→
R defined by
(20)
H
^
(
U
,
V
)
=
max
⁡
{
sup
⁡
x
∈
U
inf
⁡
y
∈
V
∥
x
-
y
∥
,
sup
⁡
y
∈
V
inf
⁡
x
∈
U
∥
x
-
y
∥
}
,
rrrrrrrrkkrrrrrrroorrrrrrrrrrrrrrrrr
∀
U
,
V
∈
C
B
(
H
)
.Definition 7.
A set-valued operatorU
:
H
→
2
H is said to be D-γ-Lipschitz continuous if there exists a constant γ
>
0 such that
(21)
D
(
U
(
x
)
,
U
(
y
)
)
≤
γ
∥
x
-
y
∥
,
∀
x
,
y
∈
H
.Lemma 8.
Letθ be a constant and 0
<
ω
<
1; then function f
(
ω
)
=
1
-
ω
+
ω
θ, for ω
∈
[
0,1
], is nonnegative and strictly decreases and f
(
ω
)
∈
[
0,1
]. Further, if ω
≠
0, then f
(
ω
)
∈
(
0,1
).Proof.
Sincef
(
ω
) is linear function, the conclusions immediately hold.


## 3. Iterative Algorithm and Convergence

In this section, we first prove the equivalence between the problem (1) and the problem of finding the fixed points of the general resolvent operator R
M
,
λ
A
,
η associated with (
A
,
η
)-maximal relaxed monotone operators. This equivalence is quite general and very important from a numerical point of view. Then, by using the equivalence, some new iterative algorithms for finding the approximation solutions of the problem (1) are analyzed. Further, the convergence criteria for the algorithms are also discussed.Lemma 9.
Let(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
∈
U
i
j
(
x
j
*
)
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
); then (
x
1
*
,
x
2
*
,
…
,
x
m
*
,
u
12
,
…
,
u
1
m
,
…
,
u
m
1
,
…
,
u
m
m
-
1
) (denoted by (
*
)) is a solution of the problem (1) if and only if (
*
) satisfy
(22)
x
i
*
=
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
*
)
-
ρ
i
F
i
(
u
i
1
,
…
,
u
i
i
-
1
,
x
i
*
,
u
i
i
+
1
,
…
,
u
i
m
)
]
,
where R
M
i
,
ρ
i
A
i
,
η
i
=
(
A
i
+
ρ
i
M
i
)
-
1 and ρ
i
>
0 is a constant, for i
=
1,2
,
⋯
,
m.Proof.
Let(
*
) satisfy the relation (22). By Definition 5 of general resolvent operator, the equality (22) holds if and only if
(23)
A
i
(
x
i
*
)
-
ρ
i
F
i
(
u
i
1
,
…
,
u
i
i
-
1
,
x
i
*
,
u
i
i
+
1
,
…
,
u
i
m
)
∈
(
A
i
+
ρ
i
M
i
)
(
x
i
*
)
;
that is,
(24)
0
∈
F
i
(
u
i
1
,
…
,
u
i
i
-
1
,
x
i
*
,
u
i
i
+
1
,
…
,
u
i
m
)
+
M
i
(
x
i
*
)
,
where i
=
1,2
,
…
,
m. Hence (
*
) are the solution of the problem (1). This completes the proof.By using formula (22) and Nadler [26], we can develop the following new iterative algorithms.Algorithm 10.
Consider the following.
Step  1. Choose (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
0
∈
U
i
j
(
x
j
0
) for i
,
j
=
1,2
,
…
,
m
,
j
≠
i.
Step  2. Let
(25)
x
i
n
+
1
=
(
1
-
λ
n
-
δ
n
)
x
i
n
+
λ
n
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
n
)
aaaaaaaaaaaaa
-
ρ
i
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
]
,
for all i
=
1,2
,
…
,
m and n
=
0,1
,
2
,
…, where λ
n and δ
n are nonnegative constants such that 0
<
λ
n
+
δ
n
≤
1 and lim
⁡
inf
⁡
n
≥
0
λ
n
>
0.
Step  3. Choose u
i
j
n
+
1
∈
U
i
j
(
x
j
n
+
1
)
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
) such that
(26)
∥
u
i
j
n
+
1
-
u
i
j
n
∥
≤
(
1
+
1
n
+
1
)
D
j
(
U
i
j
(
x
j
n
+
1
)
,
U
i
j
(
x
j
n
)
)
,
where D
j
(
·
,
·
) is the Hausdorff pseudometric on C
(
H
j
).
Step  4. If x
i
n
+
1 and u
i
j
n
+
1
(
i
,
j
=
1,2
,
…
,
m
) satisfy (25) to sufficient accuracy, stop. Otherwise, set n
:
=
n
+
1 and return to Step 2.Algorithm 11.
Consider the following.
Step  1. Choose (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
0
∈
U
i
j
(
x
j
0
), for i
,
j
=
1,2
,
…
,
m
,
j
≠
i.
Step  2. Let
(27)
x
i
n
+
1
=
(
1
-
λ
-
δ
)
x
i
n
+
λ
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
n
)
a
a
a
a
a
a
a
a
a
a
a
-
ρ
i
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
]
,
for all i
=
1,2
,
…
,
m and n
=
0,1
,
2
,
…, where λ and δ are nonnegative constants such that 0
<
λ
+
δ
≤
1.
Step  3. Choose u
i
j
n
+
1
∈
U
i
j
(
x
j
n
+
1
)
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
) such that
(28)
∥
u
i
j
n
+
1
-
u
i
j
n
∥
≤
(
1
+
1
n
+
1
)
D
j
(
U
i
j
(
x
j
n
+
1
)
,
U
i
j
(
x
j
n
)
)
,
where D
j
(
·
,
·
) is the Hausdorff pseudometric on C
(
H
j
).
Step  4. If x
i
n
+
1 and u
i
j
n
+
1
(
i
,
j
=
1,2
,
…
,
m
) satisfy (27) to sufficient accuracy, stop. Otherwise, set n
:
=
n
+
1 and return to Step 2.Algorithm 12.
Consider the following.
Step  1. Choose (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
0
∈
U
i
j
(
x
j
0
), for i
,
j
=
1,2
,
…
,
m
,
j
≠
i.
Step  2. Let
(29)
x
i
n
+
1
=
(
1
-
λ
)
x
i
n
+
λ
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
n
)
aaaaaaaaaaaa
-
ρ
i
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
]
,
for all i
=
1,2
,
…
,
m and n
=
0,1
,
2
,
…, where λ is a nonnegative constant such that 0
<
λ
≤
1.
Step  3. Choose u
i
j
n
+
1
∈
U
i
j
(
x
j
n
+
1
)
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
) such that
(30)
∥
u
i
j
n
+
1
-
u
i
j
n
∥
≤
(
1
+
1
n
+
1
)
D
j
(
U
i
j
(
x
j
n
+
1
)
,
U
i
j
(
x
j
n
)
)
,
where D
j
(
·
,
·
) is the Hausdorff pseudometric on C
(
H
j
).
Step  4. If x
i
n
+
1 and u
i
j
n
+
1
(
i
,
j
=
1,2
,
…
,
m
) satisfy (29) to sufficient accuracy, stop. Otherwise, set n
:
=
n
+
1 and return to Step 2.Remark 13.
Letm
=
2; then Algorithms 10–12 reduce to Algorithms  4.1–4.3 of Agarwal and Verma [9], respectively.Now, we provide the main results concerning problem (1) with respect to Algorithms 10–12.Theorem 14.
Fori
=
1,2
,
…
,
m, let η
i
:
H
i
×
H
i
→
H
i be τ
i-Lipschitz continuous operator, let A
i
:
H
i
→
H
i be β
i-Lipschitz continuous and (
r
i
,
η
i
)-strongly monotone operator, and let M
i
:
H
i
→
2
H
i be (
A
i
,
η
i
)-maximal relaxed monotone operator. Suppose that U
i
j
:
H
j
→
C
(
H
j
) is D
j-γ
i
j-Lipschitz continuous for j
=
1,2
,
…
,
m and j
≠
i and nonlinear operator F
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i is (
c
i
,
μ
i
)-relaxed cocoercive with respect to A
i in the ith argument and ζ
i
j-Lipschitz continuous in the jth argument for j
=
1,2
,
…
,
m. If there exists constant ρ
i
>
0, for i
=
1,2
,
…
,
m, such that
(31)
τ
j
r
j
-
ρ
j
m
j
·
β
j
2
-
2
ρ
j
μ
j
+
2
ρ
j
c
j
ζ
j
j
2
+
ρ
j
2
ζ
j
j
2
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
<
1
,
where r
j
-
ρ
j
m
j
>
0 for j
=
1,2
,
…
,
m, then the problem (1) admits a solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
,
u
12
,
…
,
u
1
m
,
…
,
u
m
1
,
…
,
u
m
m
-
1
) (in short, (
*
)), where x
i
*
∈
H
i and u
i
j
∈
U
i
j
(
x
j
*
), for any i
,
j
=
1,2
,
…
,
m
,
j
≠
i, and sequences {
x
j
n
} and {
u
i
j
n
} generated by Algorithm 10 strongly converge to x
j
* and u
i
j
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
), respectively.Proof.
Fori
=
1,2
,
…
,
m, applying Algorithm 10 and Lemma 6, we have
(32)
∥
x
i
n
+
1
-
x
i
n
∥
≤
(
1
-
λ
n
-
δ
n
)
∥
x
i
n
-
x
i
n
-
1
∥
+
λ
n
∥
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
n
)
g
g
g
g
g
g
h
k
i
g
g
-
ρ
i
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
]
fffffffffif

-
R
M
i
,
ρ
i
A
i
,
η
i
[
A
i
(
x
i
n
-
1
)
g
g
g
g
g
h
k
f
f
h
f
g
-
ρ
i
F
i
(
u
i
1
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
-
1
)
]

R
M
i
,
ρ
i
A
i
,
η
i
∥
≤
(
1
-
λ
n
)
∥
x
i
n
-
x
i
n
-
1
∥
+
λ
n
τ
i
r
i
-
ρ
i
m
i
∥
A
i
(
x
i
n
)
-
A
i
(
x
i
n
-
1
)
hhhhhhhhhh
-
ρ
i
[
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
hhhhhhhhhi
-
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
∥
+
λ
n
τ
i
ρ
i
r
i
-
ρ
i
m
i
∥
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
ggggggggggig
-
F
i
(
u
i
1
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
-
1
)
∥
.
SinceA
i is β
i-Lipschitz continuous, F
i is (
c
i
,
μ
i
)-relaxed cocoercive with respect to A
i in the ith argument, and F
i is ζ
i
j-Lipschitz continuous in the j-th argument for j
=
1,2
,
…
,
m, then we have
(33)
∥
A
i
(
x
i
n
)
-
A
i
(
x
i
n
-
1
)
y
-
ρ
i
[
x
i
n
-
1
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
y
-
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
]
∥
2
=
∥
A
i
(
x
i
n
)
-
A
i
(
x
i
n
-
1
)
∥
2
-
2
ρ
i
〈
x
i
n
-
1
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
hhhhhhh
-
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
,
hhhhhhh
A
i
(
x
i
n
)
-
A
i
(
x
i
n
-
1
)
〉
+
ρ
i
2
∥
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
jjjjjjjjhjjjj
-
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
∥
2
≤
β
i
2
∥
x
i
n
-
x
i
n
-
1
∥
2
-
2
ρ
i
[
∥
x
i
n
-
x
i
n
-
1
∥
2
(
-
c
i
)
∥
x
i
n
-
1
,
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
hhhhhhllllilhhi
-
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
∥
2
jjjjjjjjjjjijj
+
μ
i
∥
x
i
n
-
x
i
n
-
1
∥
2
]
+
ρ
i
2
ζ
i
i
2
∥
x
i
n
-
x
i
n
-
1
∥
2
≤
(
β
i
2
-
2
ρ
i
μ
i
+
2
ρ
i
c
i
ζ
i
i
2
+
ρ
i
2
ζ
i
i
2
)
∥
x
i
n
-
x
i
n
-
1
∥
2
.
By D
j-γ
i
j-Lipschitz continuity of U
i
j and (26), we get
(34)
∥
F
i
(
u
i
1
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
-
F
i
(
u
i
1
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
-
1
)
∥
≤
∥
F
i
(
u
i
1
n
,
u
i
2
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
s
s
f
f
-
F
i
(
u
i
1
n
-
1
,
u
i
2
n
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
∥
+
⋯
+
∥
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
h
h
h
h
h
h
h
h
-
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
∥
+
∥
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
,
…
,
u
i
m
n
)
f
f
f
f
-
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
)
∥
+
⋯
+
∥
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
)
ddddddd
-
F
i
(
u
i
1
n
-
1
,
u
i
2
n
-
1
,
…
,
u
i
i
-
1
n
-
1
,
x
i
n
-
1
,
u
i
i
+
1
n
-
1
,
…
,
u
i
m
n
-
1
)
∥
≤
ζ
i
1
∥
u
i
1
n
-
u
i
1
n
-
1
∥
+
⋯
+
ζ
i
i
-
1
∥
u
i
i
-
1
n
-
u
i
i
-
1
n
-
1
∥
+
ζ
i
i
+
1
∥
u
i
i
+
1
n
-
u
i
i
+
1
n
-
1
∥
+
⋯
+
ζ
i
m
∥
u
i
m
n
-
u
i
m
n
-
1
∥
=
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
u
i
j
n
-
u
i
j
n
-
1
∥
≤
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
(
1
+
1
n
)
D
j
(
U
i
j
(
x
j
n
)
,
U
i
j
(
x
j
n
-
1
)
)
≤
(
1
+
1
n
)
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
γ
i
j
∥
x
j
n
-
x
j
n
-
1
∥
.
It follows from (32)–(34) that
(35)
∥
x
i
n
+
1
-
x
i
n
∥
≤
(
1
-
λ
n
)
∥
x
i
n
-
x
i
n
-
1
∥
+
λ
n
τ
i
r
i
-
ρ
i
m
i
[
∑
j
=
1
,
j
≠
i
m
‍
β
i
2
-
2
ρ
i
μ
i
+
2
ρ
i
c
i
ζ
i
i
2
+
ρ
i
2
ζ
i
i
2
∥
x
i
n
-
x
i
n
-
1
∥
h
h
h
h
h
h
h
h
h
h
i
+
(
1
+
1
n
)
ρ
i
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
γ
i
j
∥
x
j
n
-
x
j
n
-
1
∥
]
,
which implies that
(36)
∑
j
=
1
m
‍
∥
x
j
n
+
1
-
x
j
n
∥
=
∑
i
=
1
m
‍
∥
x
i
n
+
1
-
x
i
n
∥
≤
∑
i
=
1
m
‍
[
∑
j
=
1
,
j
≠
i
m
‍
(
1
-
λ
n
)
∥
x
i
n
-
x
i
n
-
1
∥
fffffffyfff
+
λ
n
τ
i
r
i
-
ρ
i
m
i
(
∑
j
=
1
,
j
≠
i
m
‍
β
i
2
-
2
ρ
i
μ
i
+
2
ρ
i
c
i
ζ
i
i
2
+
ρ
i
2
ζ
i
i
2
h
h
h
h
h
h
h
h
y
h
h
h
h
h
h
h
h
×
∥
x
i
n
-
x
i
n
-
1
∥
+
(
1
+
1
n
)
ρ
i
h
h
h
h
h
h
h
h
y
h
h
h
h
h
h
h
×
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
γ
i
j
∥
x
j
n
-
x
j
n
-
1
∥
)
]
=
∑
i
=
1
m
‍
[
λ
n
τ
i
r
i
-
ρ
i
m
i
(
1
-
λ
n
)
+
λ
n
τ
i
r
i
-
ρ
i
m
i
g
g
g
i
i
g
g
×
β
i
2
-
2
ρ
i
μ
i
+
2
ρ
i
c
i
ζ
i
i
2
+
ρ
i
2
ζ
i
i
2
λ
n
τ
i
r
i
-
ρ
i
m
i
]
∥
x
i
n
-
x
i
n
-
1
∥
+
(
1
+
1
n
)
λ
n
∑
i
=
1
m
‍
∑
j
=
1
,
j
≠
i
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
∥
x
j
n
-
x
j
n
-
1
∥
=
∑
j
=
1
m
‍
[
(
1
-
λ
n
)
+
λ
n
τ
j
r
j
-
ρ
j
m
j
d
d
d
i
d
i
d
×
β
j
2
-
2
ρ
j
μ
j
+
2
ρ
j
c
j
ζ
j
j
2
+
ρ
j
2
ζ
j
j
2
λ
n
τ
j
r
j
-
ρ
j
m
j
]
∥
x
j
n
-
x
j
n
-
1
∥
+
(
1
+
1
n
)
λ
n
∑
j
=
1
m
‍
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
∥
x
j
n
-
x
j
n
-
1
∥
=
∑
j
=
1
m
‍
[
∑
i
=
1
,
i
≠
j
m
‍
(
1
-
λ
n
)
+
λ
n
s
s
s
s
s
i
u
s
s
×
(
∑
i
=
1
,
i
≠
j
m
‍
τ
j
r
j
-
ρ
j
m
j
β
j
2
-
2
ρ
j
μ
j
+
2
ρ
j
c
j
ζ
j
j
2
+
ρ
j
2
ζ
j
j
2
ddddgggdd
+
(
1
+
1
n
)
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
)
]
∥
x
j
n
-
x
j
n
-
1
∥
=
∑
j
=
1
m
‍
[
(
1
-
λ
n
)
+
λ
n
θ
j
n
]
∥
x
j
n
-
x
j
n
-
1
∥
≤
∑
j
=
1
m
‍
[
(
1
-
ω
)
+
ω
θ
j
n
]
∥
x
j
n
-
x
j
n
-
1
∥
≤
f
n
(
ω
)
∑
j
=
1
m
‍
∥
x
j
n
-
x
j
n
-
1
∥
,
where
(37)
ω
=
lim
⁡
inf
⁡
n
≥
0
λ
n
>
0
,
f
n
(
ω
)
=
max
⁡
1
≤
j
≤
m
{
(
1
-
ω
)
+
ω
θ
j
n
}
,
θ
j
n
=
τ
j
r
j
-
ρ
j
m
j
β
j
2
-
2
ρ
j
μ
j
+
2
ρ
j
c
j
ζ
j
j
2
+
ρ
j
2
ζ
j
j
2
+
(
1
+
1
n
)
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
.
By condition (31), we know sequence {
θ
j
n
} is monotonely decreasing and θ
j
n
→
θ
j as n
→
∞. Therefore,
(38)
f
(
ω
)
=
lim
⁡
n
→
∞
f
n
(
ω
)
=
max
⁡
1
≤
j
≤
m
{
(
1
-
ω
)
+
ω
θ
j
}
.
Since 0
<
θ
j
<
1, for j
=
1,2
,
…
,
m, we get θ
=
max
⁡
1
≤
j
≤
m
{
θ
j
}
∈
(
0,1
). By Lemma 8, we have f
(
ω
)
=
1
-
ω
+
ω
θ
∈
(
0,1
). From (36), it follows that {
x
j
n
} are Cauchy sequences and there exists x
j
*
∈
H
j such that x
j
n
→
x
j
* as n
→
∞, for j
=
1,2
,
…
,
m.
Next, we show thatu
i
j
n
→
u
i
j
∈
U
i
j
(
x
j
*
) as n
→
∞ for i
,
j
=
1,2
,
…
,
m and j
≠
i.
It follows from (34) that {
u
i
j
n
} are also Cauchy sequences. Hence, there exists u
i
j
∈
H
j such that u
i
j
n
→
u
i
j as n
→
∞, for i
,
j
=
1,2
,
…
,
m
,
j
≠
i. Furthermore,
(39)
d
(
u
i
j
,
U
i
j
(
x
j
*
)
)
=
inf
⁡
{
∥
u
i
j
-
t
∥
:
t
∈
U
i
j
(
x
j
*
)
}
≤
∥
u
i
j
-
u
i
j
n
∥
+
d
(
u
i
j
n
,
U
i
j
(
x
j
*
)
)
≤
∥
u
i
j
-
u
i
j
n
∥
+
D
j
(
U
i
j
(
x
j
n
)
,
U
i
j
(
x
j
*
)
)
≤
∥
u
i
j
-
u
i
j
n
∥
+
γ
i
j
∥
x
j
n
-
x
j
*
∥
→
0
(
n
→
∞
)
.
Since U
i
j
(
x
j
*
) is closed, we have u
i
j
∈
U
i
j
(
x
j
*
), for j
=
1,2
,
…
,
m
,
j
≠
i.
Using continuity,(
*
) that is, (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m and u
i
j
∈
U
i
j
(
x
j
*
)
(
i
,
j
=
1,2
,
…
,
m
,
j
≠
i
) satisfy (22) and so, in light of Lemma 9, (
*
), is a solution to problem (1). This completes the proof.Remark 15.
Ifm
=
2, then Theorem 14 reduces to Theorem  4.2 in [9].From Theorem14, we have the following results.Corollary 16.
Fori
,
j
=
1,2
,
…
,
m and j
≠
i, assume that η
i, A
i, M
i, U
i
j, and H
i are the same as in Theorem 14. Let F
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i be μ
i-strongly monotone with respect to A
i in the ith argument and ζ
i
j-Lipschitz continuous in the jth argument for i
,
j
=
1,2
,
…
,
m and j
≠
i. If there exist constants ρ
i
>
0, for i
=
1,2
,
…
,
m, such that
(40)
τ
j
r
j
-
ρ
j
m
j
·
β
j
2
-
2
ρ
j
μ
j
+
ρ
j
2
ζ
j
j
2
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
γ
i
j
r
i
-
ρ
i
m
i
<
1
,
where r
j
-
ρ
j
m
j
>
0, for j
=
1,2
,
…
,
m, then problem (1) admits a solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
,
u
12
,
…
,
u
1
m
,
…
,
u
m
1
,
…
,
u
m
m
-
1
), where, for any i
,
j
=
1,2
,
…
,
m
,
j
≠
i, x
i
*
∈
H
i and u
i
j
∈
U
i
j
(
x
j
*
) and sequences {
x
j
n
}, {
u
i
j
n
} generated by Algorithm 11 strongly converge to x
j
* and u
i
j, respectively.Corollary 17.
Fori
,
j
=
1,2
,
…
,
m and j
≠
i, let η
i, A
i, M
i, F
i, and H
i be the same as in Corollary 16 and let U
i
j
:
H
j
→
C
B
(
H
j
) be H
^-γ
i
j-Lipschitz continuous. If condition (40) in Corollary 16 holds, then problem (1) admits a solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
,
u
12
,
…
,
u
1
m
,
…
,
u
m
1
,
…
,
u
m
m
-
1
), where x
i
*
∈
H
i and u
i
j
∈
U
i
j
(
x
j
*
), and sequences {
x
j
n
}, {
u
i
j
n
}
(
i
,
j
=
1,2
,
…
,
m, j
≠
i
) generated by Algorithm 12 strongly converge to x
j
* and u
i
j, for any i
,
j
=
1,2
,
…
,
m,  j
≠
i, respectively.



---

*Source: 698593-2014-06-05.xml*