# Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

Authors: Ting-jian Xiong; Heng-you Lan
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce and study a new general system of nonlinear variational inclusions involving generalizedm-accretive mappings in Banach space. By using the resolvent operator technique associated with generalized m-accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.


---

## Body

## 1. Introduction

Letm be a given positive integer, for any i
∈
{
1,2
,
…
,
m
}, X
i a real Banach space with dual space X
i
*. X
i, X
i
* all endowed with the norm ∥
·
∥, and 〈
·
,
·
〉 the dual pair between X
i and X
i
* (as matter of convenience). Let 2
X
i denote the family of all the nonempty subsets of X
i, η
i
:
X
i
×
X
i
→
X
i
*, N
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i single-valued mappings, and M
i
:
X
i
→
2
X
i generalized m-accretive mapping for i
=
1,2
,
…
,
m. In this paper, we consider the following new general system for nonlinear variational inclusion involving generalized m-accretive mappings. Find (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m such that
(1)
0
∈
N
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
+
M
i
(
x
i
*
)
for all i
=
1,2
,
…
,
m. Some special cases of the problem (1) had been studied by many authors. See, for example, [1–34] and the reference therein. Here, we mention some of them as follows.Case 1.
The problem (1) with X
i
=
H
i
(
i
=
1,2
,
…
,
m
), the Hilbert spaces, was introduced and studied as general system of monotone nonlinear variational inclusions problems by Peng and Zhao [29].
IfJ
q
-
1
η
i
(
x
i
1
,
x
i
2
)
=
x
i
1
-
x
i
2 and M
i
=
∂
φ
i, φ
i
:
X
i
→
(
-
∞
,
+
∞
] is proper, convex, and lower semi-continuous functional on X
i, and ∂
φ
i denote the subdifferential operators of the φ
i for i
=
1,2
,
…
,
m, then the problem (1) is equivalent to finding (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
…
×
X
m such that
(2)
〈
N
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
,
j
(
x
i
-
x
i
*
)
〉
≥
ρ
i
(
φ
i
(
x
i
*
)
-
φ
i
(
x
i
)
)
,
∀
x
i
∈
X
i
.
WhenX
i
=
X, 2-uniformly smooth Banach space with the smooth constant K, C is a nonempty closed convex subset of X, N
i
(
x
1
,
x
2
,
…
,
x
m
)
=
ρ
i
A
i
(
x
i
+
1
)
+
x
i
-
x
i
+
1, where A
i
:
C
→
X and ρ
i
>
0 and x
m
+
1
=
x
1 for i
=
1,2
,
…
,
m; the problem (2) reduces to the following system of finding (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
C
×
C
×
⋯
×
C such that
(3)
〈
ρ
i
A
i
(
x
i
+
1
*
)
+
x
i
*
-
x
i
+
1
*
,
j
(
x
-
x
i
*
)
〉
≥
ρ
i
(
φ
i
(
x
i
*
)
-
φ
i
(
x
)
)
,
∀
x
∈
X
.
Further, in the problem (3), when φ
i is the indicator function of a nonempty closed convex set C, in X defined by
(4)
φ
i
(
y
)
=
{
0
,
y
∈
C
,
+
∞
,
y
∉
C
,
then the system (3) reduces to finding (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
C
×
C
×
⋯
×
C such that
(5)
〈
ρ
1
A
1
x
2
*
+
x
1
*
-
x
2
*
,
j
(
x
-
x
1
*
)
〉
≥
0
,
∀
x
∈
C
,
〈
ρ
2
A
2
x
3
*
+
x
2
*
-
x
3
*
,
j
(
x
-
x
2
*
)
〉
≥
0
,
∀
x
∈
C
,
〈
ρ
3
A
3
x
4
*
+
x
3
*
-
x
4
*
,
j
(
x
-
x
3
*
)
〉
≥
0
,
∀
x
∈
C
,
⋯
〈
ρ
m
A
m
x
1
*
+
x
m
*
-
x
1
*
,
j
(
x
-
x
m
*
)
〉
≥
0
,
∀
x
∈
C
,
which was introduced and studied by Zhu et al. [34].Case 2.
Ifm
=
3, then the system (3) is equivalent to finding (
x
1
*
,
x
2
*
,
x
3
*
)
∈
C
×
C
×
C such that
(6)
〈
ρ
1
A
1
x
2
*
+
x
1
*
-
x
2
*
,
j
(
x
-
x
1
*
)
〉
≥
ρ
1
(
φ
1
(
x
1
*
)
-
φ
1
(
x
)
)
,
∀
x
∈
C
,
〈
ρ
2
A
2
x
3
*
+
x
2
*
-
x
3
*
,
j
(
x
-
x
2
*
)
〉
≥
ρ
2
(
φ
2
(
x
2
*
)
-
φ
2
(
x
)
)
,
∀
x
∈
C
,
〈
ρ
3
A
3
x
1
*
+
x
3
*
-
x
1
*
,
j
(
x
-
x
3
*
)
〉
≥
ρ
3
(
φ
3
(
x
3
*
)
-
φ
3
(
x
)
)
,
∀
x
∈
C
.
It is easy to see that the mathematical model studied by Saewan and Kumam [31] is a variant of (6).Case 3.
Ifm
=
2, then the problem (1) reduces to find (
x
*
,
y
*
)
∈
X
1
×
X
2 such that
(7)
0
∈
N
1
(
x
*
,
y
*
)
+
M
1
(
x
*
)
,
0
∈
N
2
(
x
*
,
y
*
)
+
M
2
(
y
*
)
.
Problem (7) is called a system of strongly nonlinear quasivariational inclusion involving generalized m-accretive mappings, it is considered and studied by Lan [19]. There are many special cases of the problems (7) that can be found in [3, 7, 12–14, 17, 20, 28, 30] and the references cited therein.Case 4.
Ifm
=
1 and X
1
=
H, then the problem (1) reduces to finding x
*
∈
H such that
(8)
0
∈
N
(
x
*
)
+
M
(
x
*
)
,
which was introduced and studied by Fang and Huang [8]. We remark that for appropriate and suitable choices of positive integer m, the mappings η
i, N
i, and M
i, and the spaces X
i for i
=
1,2
,
…
,
m, one can know that the problem (1) includes a number of general class of variational character known problems, including minimization or maximization (whether constraint or not) of functions and minimax problems et al. as special cases. For more details, see [1–34] and the reference therein.
On the other hand, many authors discussed stability of the iterative sequence generated by the algorithm for solving the problems that they studied. Lan [19] introduced the notion of S-stable or stable with respect to S. Moreover, Agarwal et al. [1, 2], Jin [16], Kazmi and Bhat [18], and Lan and Kim [21] constructed some stability under suitable conditions, respectively.
Motivated and inspired by the above works, the main purpose of this paper is to introduce and study the new general system of nonlinear variational inclusions (1) involving generalized m-accretive mapping in uniformly smooth Banach spaces. By using the resolvent operator technique for generalized m-accretive, we prove the existence theorem of the solution for this kind of system of variational inclusions in Banach spaces and discuss the convergence and stability of a new perturbed iterative algorithm for solving this general system of nonlinear variational inclusions in Banach spaces.


## 2. Preliminaries

In order to get the main results of the paper, we need the following concepts and lemmas. LetX be a real Banach space with dual space X
*, 〈
·
,
·
〉 the dual pair between X and X
*, and 2
X denote the family of all the nonempty subsets of X. The generalized duality mapping J
q
:
X
→
2
X
* is defined by
(9)
J
q
(
x
)
=
{
f
*
∈
X
*
:
〈
x
,
f
*
〉
=
∥
x
∥
q
,
∥
f
*
∥
=
∥
x
∥
q
-
1
}
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
∀
x
∈
X
,
where q
>
1 is a constant. In particular, J
2 is the usual normalized duality mapping. It is known that if X
* is strictly convex or X is a uniformly smooth Banach space, then J
q is single-valued (see [33]), and if X
=
H, the Hilbert space, then J
2 becomes the identity mapping on H. We will denote the single-valued duality mapping by j
q.In order to construct convergence and stability for researching the problem (1), we need to be using the following definition and lemma.Definition 1.
LetX
i be Banach spaces, and let N
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i be single mappings for (
i
=
1,2
,
…
,
m
). Then N
i is said to be (i)
σ
j-strongly accretive with respect to jth argument if for any (
x
1
,
…
,
x
j
-
1
,
x
j
1
,
x
j
+
1
,
…
,
x
m
), (
x
1
,
…
,
x
j
-
1
,
x
j
2
,
x
j
+
1
,
…
,
x
m
)
∈
X
1
×
X
2
×
⋯
×
X
m, there exists j
q
j
(
x
j
1
-
x
j
2
)
∈
J
q
j
(
x
j
1
-
x
j
2
), such that
(10)
〈
j
q
j
(
x
j
1
-
x
j
2
)
N
i
(
x
1
,
…
,
x
j
-
1
,
x
j
1
,
x
j
+
1
,
…
,
x
m
)
-
N
i
(
x
1
,
…
,
x
j
-
1
,
x
j
2
,
x
j
+
1
,
…
,
x
m
)
,
j
q
j
(
x
j
1
-
x
j
2
)
〉
≥
σ
j
∥
x
j
1
-
x
j
2
∥
q
j
,
whereq
j
>
1 is a constant;
(ii)
(
ζ
i
1
,
…
,
ζ
i
j
,
…
,
ζ
i
m
)-Lipschitz continuous if there exists constants ζ
i
1
>
0
,
…
,
ζ
i
j
>
0
,
…, ζ
i
m
>
0, such that
(11)
∥
N
i
(
x
1
,
…
,
x
j
,
…
,
x
m
)
-
N
i
(
y
1
,
…
,
y
j
,
…
,
y
m
)
∥
≤
∑
j
=
1
m
‍
ζ
i
j
∥
x
j
-
y
j
∥
,
for allx
j
,
y
j
∈
X
j and j
=
1,2
…
,
m.Remark 2.
WhenX
i
=
H
i
(
i
=
1,2
,
…
,
m
), H
i is different or the same as Hilbert spaces, (i) and (ii) in Definition 1 reduce to strongly monotonicity with respect to jth argument of N
i and (
ζ
i
1
,
…
,
ζ
i
j
,
…
,
ζ
i
m
)-Lipschitz continuity of N
i, respectively (see [29]).Definition 3.
Letη
:
X
×
X
→
X
* be single-valued mapping. Then set-valued mapping M
:
X
→
2
X is said to be (i)
accretive if(12)
〈
u
-
v
,
J
q
(
x
-
y
)
〉
≥
0
,
∀
x
,
y
∈
X
,
u
∈
M
(
x
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
v
∈
M
(
y
)
;
(ii)
η-accretive if
(13)
〈
u
-
v
,
η
(
x
,
y
)
〉
≥
0
,
∀
x
,
y
∈
X
,
u
∈
M
(
x
)
,
v
∈
M
(
y
)
;
(iii)
m-accretive if M is accretive and (
I
+
ρ
M
)
(
X
)
=
X for all ρ
>
0, where I denotes the identity operator on X;
(iv)
generalizedm-accretive if M is η-accretive and (
I
+
ρ
M
)
(
X
)
=
X for all ρ
>
0.Remark 4.
WhenX
=
X
*
=
H, (i)–(iv) of Definition 3 reduce to the definitions of monotone operators, η-monotone operators, classical maximal monotone operators, and maximal η-monotone operators; if η
(
x
,
y
)
=
J
2
(
x
-
y
), then (ii) and (iv) of Definition 3 reduce to the definitions of accretive and m-accretive of uniformly smooth Banach spaces (see [10, 11]).Definition 5.
The mappingη
:
X
×
X
→
X
* is said to be (i)
δ-strongly monotone if there exists a constant δ
>
0 such that
(14)
〈
x
1
-
x
2
,
η
(
x
1
,
x
2
)
〉
≥
δ
∥
x
1
-
x
2
∥
2
,
∀
x
1
,
x
2
∈
X
;
(ii)
τ-Lipschitz continuous if there exists a constant τ
>
0 such that
(15)
∥
η
(
x
1
,
x
2
)
∥
≤
τ
∥
x
1
-
x
2
∥
,
∀
x
1
,
x
2
∈
X
.In [10], Huang and Fang show that for any ρ
i
>
0, inverse mapping (
I
+
ρ
i
M
i
)
-
1 is single-valued, if η
i
:
X
i
×
X
i
→
X
i
* is strict monotone and M
i
:
X
i
→
2
X
i is generalized m-accretive mapping, where I is the identity mapping. Based on this fact, Huang and Fang [10] gave the following definition.Definition 6.
Letη
i
:
X
i
×
X
i
→
X
i
* be strictly monotone mapping, and let M
i
:
X
i
→
2
X
i be generalized m-accretive mapping. Then the resolvent J
M
i
ρ
i for M
i is defined as follows:
(16)
J
M
i
ρ
i
(
x
i
)
=
(
I
+
ρ
i
M
i
)
-
1
(
x
i
)
,
∀
x
i
∈
X
i
,
where ρ
i
>
0 is a constant and I denotes the identity mapping on X
i for i
=
1,2
,
…
,
m.Lemma 7 (see [10, 11]).
Letη
i
:
X
i
×
X
i
→
X
i
* be τ
i-Lipschitz continuous and δ
i-strongly monotone, and let M
i
:
X
i
→
2
i
X be generalized m-accretive mapping. Then for any ρ
i
>
0, the resolvent operator J
M
i
ρ
i for M
i is τ
i
/
δ
i-Lipschitz continuous; that is,
(17)
∥
J
M
i
ρ
i
(
x
i
)
-
J
M
i
ρ
i
(
y
i
)
∥
≤
τ
i
δ
i
∥
x
i
-
y
i
∥
,
∀
x
i
,
y
i
∈
X
i
,
i
=
1,2
,
…
,
m
.The modules of smoothness is a measure, it is depicted geometric structure of the underlying Banach space. The modules of smoothness of Banach spaceX are the function ρ
X
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) defined by
(18)
ρ
X
(
t
)
=
sup
⁡
{
1
2
(
∥
x
+
y
∥
+
∥
x
-
y
∥
)
-
1
:
∥
x
∥
≤
1
,
∥
y
∥
≤
t
}
.
A Banach space X is called uniformly smooth if lim
⁡
t
→
0
(
ρ
X
(
t
)
/
t
)
=
0. X is called q-uniformly smooth if there exists a constant c
>
0 such that ρ
X
(
t
)
≤
c
t
q, where q
>
1 is a real number.Remark thatJ
q is single-valued if X is uniformly smooth, and Hilbert space and L
p (or l
p) (
2
≤
p
<
+
∞
) spaces are 2-uniformly smooth Banach spaces. In what follows, we will denote the single-valued generalized duality mapping by j
q.In the study of characteristic inequalities inq-uniformly smooth Banach spaces, Xu [35] proved the following result.Lemma 8.
Letq
>
1 be a given real number and let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant c
q
>
0 such that for all x
,
y
∈
X, j
q
(
x
)
∈
J
q
(
x
), there holds the following inequality:
(19)
∥
x
+
y
∥
q
≤
∥
x
∥
q
+
q
〈
y
,
j
q
(
x
)
〉
+
c
q
∥
y
∥
q
.Definition 9.
LetS be a self-map of X, x
0
∈
X, and let x
n
+
1
=
h
(
S
,
x
n
) define an iteration procedure which yields a sequence of points {
x
n
}
n
=
0
∞ in X. Suppose that {
x
∈
X
:
S
x
=
x
}
≠
∅ and {
x
n
}
n
=
0
∞ converges to a fixed point x
* of S. Let {
u
n
}
⊂
X and let ϵ
n
=
∥
u
n
+
1
-
h
(
S
,
u
n
)
∥. If lim
⁡
n
→
∞
ϵ
n
=
0 implies that u
n
→
x
*, then the iteration procedure defined by x
n
+
1
=
h
(
S
,
x
n
) is said to be S-stable or stable with respect to S.Lemma 10 (see [36]).
Let{
a
n
}, {
b
n
}, and {
c
n
} be three nonnegative real sequences satisfying the following condition: there exists a natural number n
0 such that
(20)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
b
n
t
n
+
c
n
,
∀
n
≥
n
0
,
where t
n
∈
[
0,1
], ∑
n
=
0
∞
‍
t
n
=
+
∞, lim
⁡
n
→
∞
b
n
=
0, and ∑
n
=
0
∞
‍
c
n
<
+
∞. Then a
n converges to 0 as n
→
∞.


## 3. Existence Theorem

In this section, we will give the existence theorem of the problem (1). The solvability of the problem (1) depends on the equivalence between (1) and the problem of finding the fixed point of the associated generalized resolvent operator. It follows from the definition of generalized resolvent operator J
M
i
ρ
i
(
i
=
1,2
,
…
,
m
) that we can obtain the following conclusion.Lemma 11.
Letη
i
:
X
i
×
X
i
→
X
i
*, N
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i single-valued mappings, and M
i
:
X
i
→
2
X
i generalized m-accretive mapping for (
i
=
1,2
,
…
,
m
). Then the following statements are mutually equivalent. (i)
An element(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m is a solution to the problem (1).
(ii)
There is an(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m such that
(21)
x
i
*
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
]
,
whereJ
M
i
ρ
i
=
(
I
+
ρ
i
M
i
)
-
1, and ρ
i
>
0 is constants for all i
=
1,2
,
…
,
m.
(iii)
For any given constantsρ
i
>
0, the map F
:
X
1
×
X
2
×
⋯
×
X
m
→
X
1
×
X
2
×
⋯
×
X
m is defined by
(22)
F
(
u
1
,
u
2
,
…
,
u
m
)
=
(
P
ρ
1
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
m
(
u
1
,
u
2
,
…
,
u
m
)
)
for allu
i
∈
X
i and i
=
1,2
,
…
,
m, has a fixed point (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m, where maps P
ρ
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i are defined by
(23)
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
=
J
M
i
ρ
i
[
u
i
-
ρ
i
N
i
(
u
1
,
…
,
u
i
-
1
,
u
i
,
u
i
+
1
,
…
,
u
m
)
]
foru
i
∈
X
i and i
=
1,2
,
…
,
m.Proof.
We first prove that (i)⇔ (ii). Let (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m satisfy the relation in (ii). Then, the definition of resolvent operator J
M
i
ρ
i implies that this equality holds if and only if
(24)
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
∈
(
I
+
ρ
i
M
i
)
(
x
i
*
)
for i
=
1,2
,
…
,
m; that is
(25)
0
∈
N
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
+
M
i
(
x
i
*
)
,
where i
=
1,2
,
…
,
m. Thus (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m is the solution of the problem (1).
Next, we show (ii)⇔ (iii). If (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m satisfy following relation:
(26)
x
i
*
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
]
,
then, for any i
=
1,2
,
…
,
m, it follows from
(27)
P
ρ
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
]
that
(28)
P
ρ
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
=
x
i
*
.
Hence, (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m is a fixed point of the mapping
(29)
F
(
u
1
,
u
2
,
…
,
u
m
)
=
(
P
ρ
1
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
m
(
u
1
,
u
2
,
…
,
u
m
)
)
.
Conversely, if (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m is a fixed point of the mapping F
:
X
1
×
X
2
×
⋯
×
X
m
→
X
1
×
X
2
×
⋯
×
X
m, then
(30)
P
ρ
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
=
x
i
*
for i
=
1,2
,
…
,
m. Hence, from
(31)
P
ρ
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
]
,
we have
(32)
x
i
*
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
…
,
x
i
-
1
*
,
x
i
*
,
x
i
+
1
*
,
…
,
x
m
*
)
]
for i
=
1,2
,
…
,
m. Therefore (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m satisfy the relation of (ii).Theorem 12.
LetX
i be a real q
i-uniformly smooth Banach space with q
i
>
1 and let η
i
:
X
i
×
X
i
→
X
i
* be τ
i-Lipschitz continuous and δ
i-strongly monotone for any i
=
1,2
,
…
,
m. Suppose that M
i
:
X
i
→
2
X
i is generalized m-accretive mapping, and N
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i is σ
i-strongly accretive in the ith argument and (
ζ
i
1
,
…
,
ζ
i
i
,
…
,
ζ
i
m
)-Lipschitz continuous for i
=
1,2
,
…
,
m. If
(33)
τ
j
δ
j
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
+
∑
i
=
1
,
i
≠
j
m
‍
ζ
i
j
ρ
i
τ
i
δ
i
<
1
,
where c
q
j is the constants as in Lemma 8 for j
=
1,2
,
…
,
m, then problem (1) has a unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m.Proof.
For any givenρ
i
>
0 and i
=
1,2
,
…
,
m, we first define P
ρ
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i as follows:
(34)
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
=
J
M
i
ρ
i
[
u
i
-
ρ
i
N
i
(
u
1
,
u
2
,
…
,
u
m
)
]
,
for all u
i
∈
X
i. Now define ∥
·
∥
* on X
1
×
X
2
×
⋯
×
X
m by
(35)
∥
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
=
∑
i
=
1
m
‍
∥
u
i
∥
,
∀
(
u
1
,
u
2
,
…
,
u
m
)
∈
X
1
×
X
2
×
⋯
×
X
m
.
It is easy to see that (
X
1
×
X
2
×
⋯
×
X
m
,
∥
·
∥
*
) is a Banach space. In fact (i)
∥
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
=
∑
i
=
1
m
‍
∥
u
i
∥
≥
0, the negative being satisfied;
(ii)
for all real numberα,
(36)
∥
α
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
=
∥
(
α
u
1
,
α
u
2
,
…
,
α
u
m
)
∥
*
=
∑
i
=
1
m
‍
∥
α
u
i
∥
=
∑
i
=
1
m
‍
|
α
|
∥
u
i
∥
=
|
α
|
∑
i
=
1
m
‍
∥
u
i
∥
=
|
α
|
∥
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
homogeneity being satisfied;
(iii)
for all(
u
1
,
u
2
,
…
,
u
m
)
,
(
v
1
,
v
2
,
…
,
v
m
)
∈
X
1
×
X
2
×
⋯
×
X
m,
(37)
∥
(
u
1
,
u
2
,
…
,
u
m
)
+
(
v
1
,
v
2
,
…
,
v
m
)
∥
*
=
∥
(
u
1
+
v
1
,
u
2
+
v
2
,
…
,
u
m
+
v
m
)
∥
*
=
∑
i
=
1
m
‍
∥
u
i
+
v
i
∥
≤
∑
i
=
1
m
‍
(
∥
u
i
∥
+
∥
v
i
∥
)
=
∑
i
=
1
m
‍
∥
u
i
∥
+
∑
i
=
1
m
‍
∥
v
i
∥
=
∥
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
+
∥
(
v
1
,
v
2
,
…
,
v
m
)
∥
*
,
the triangle inequality being satisfied;
(iv)
let∥
(
u
1
,
u
2
,
…
,
u
m
)
∥
*
=
0; that is, ∑
i
=
1
m
‍
∥
u
i
∥
=
0; this implies that ∥
u
i
∥
=
0
(
i
=
1,2
,
…
,
m
); thus u
i
=
0
(
i
=
1,2
,
…
,
m
); we get ∥
·
∥
* is a norm on the X
1
×
X
2
×
⋯
×
X
m;
(v)
let(
u
1
n
,
u
2
n
,
…
,
u
m
n
)
∈
X
1
×
X
2
×
⋯
×
X
m is Cauchy sequence; that is, for ∀
ϵ
>
0, there exists a positive integer N; let n
>
N; we have
(38)
∥
(
u
1
n
+
p
,
u
2
n
+
p
,
…
,
u
m
n
+
p
)
-
(
u
1
n
,
u
2
n
,
…
,
u
m
n
)
∥
*
=
∑
i
=
1
m
‍
∥
u
i
n
+
p
-
u
i
n
∥
<
ϵ
.
Thus, for alli
∈
{
1,2
,
3
,
…
,
m
}, we have ∥
u
i
n
+
p
-
u
i
n
∥
<
ϵ
(
n
>
N
,
p
=
1,2
,
3
,
…
,
); that is, {
u
i
n
}
⊂
X
i is also Cauchy sequence; thus lim
⁡
n
→
∞
u
i
n
=
u
i
∈
X
i for i
=
1,2
,
…
,
m; we get (
u
1
,
u
2
,
…
,
u
m
)
∈
X
1
×
X
2
×
⋯
×
X
m and (
u
1
,
u
2
,
…
,
u
m
) is a cluster point on the (
X
1
×
X
2
×
⋯
×
X
m
,
∥
·
∥
*
); we claim (
X
1
×
X
2
×
⋯
×
X
m
,
∥
·
∥
*
) is a Banach space.
Now, by (34), for any given ρ
i
>
0, define mapping F
:
X
1
×
X
2
×
⋯
×
X
m
→
X
1
×
X
2
×
⋯
×
X
m by
(39)
F
(
u
1
,
u
2
,
…
,
u
m
)
=
(
P
ρ
1
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
,
…
,
P
ρ
m
(
u
1
,
u
2
,
…
,
u
m
)
)
,
where u
i
∈
X
i for i
=
1,2
,
…
,
m.
In the sequel, we prove thatF is a contractive mapping on the (
X
1
×
X
2
×
⋯
×
X
m
,
∥
·
∥
*
). In fact, for any u
i
,
v
i
∈
X
i and i
=
1,2
,
…
,
m, it follows from (34) and Lemma 7 that
(40)
∥
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
-
P
ρ
i
(
v
1
,
v
2
,
…
,
v
m
)
∥
=
∥
J
M
i
ρ
i
[
u
i
-
ρ
i
N
i
(
u
1
,
u
2
,
…
,
u
m
)
]
-
J
M
i
ρ
i
[
v
i
-
ρ
i
N
i
(
v
1
,
v
2
,
…
,
v
m
)
]
∥
≤
τ
i
δ
i
∥
u
i
-
v
i
-
ρ
i
(
N
i
(
u
1
,
u
2
,
…
,
u
m
)
-
N
i
(
v
1
,
v
2
,
…
,
v
m
)
)
∥
≤
τ
i
δ
i
∥
u
i
-
v
i
h
h
h
h
h
h
-
ρ
i
(
N
i
(
u
1
,
…
,
u
i
-
1
,
u
i
,
u
i
+
1
,
…
,
u
m
)
h
h
h
h
h
h
h
h
h
h
h
-
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
)
∥
+
ρ
i
τ
i
δ
i
∥
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
h
h
h
h
h
h
h
h
-
N
i
(
v
1
,
…
,
v
i
-
1
,
v
i
,
v
i
+
1
,
…
,
v
m
)
∥
.
By assumptions and Lemma 8, we have
(41)
∥
u
i
-
v
i
-
ρ
i
(
N
i
(
u
1
,
…
,
u
i
-
1
,
u
i
,
u
i
+
1
,
…
,
u
m
)
h
h
h
h
h
h
h
h
h
h
h
h
-
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
)
∥
q
i
≤
∥
u
i
-
v
i
∥
q
i
+
c
q
i
ρ
i
q
i
∥
N
i
(
u
1
,
…
,
u
i
-
1
,
u
i
,
u
i
+
1
,
…
,
u
m
)
h
h
h
h
h
h
h
h
h
h
-
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
∥
q
i
-
q
i
ρ
i
〈
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
,
j
q
i
(
u
i
-
v
i
)
N
i
(
u
1
,
…
,
u
i
-
1
,
u
i
,
u
i
+
1
,
…
,
u
m
)
h
h
h
h
h
h
h
h
-
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
,
j
q
i
(
u
i
-
v
i
)
〉
≤
(
1
-
q
i
ρ
i
σ
i
+
c
q
i
ρ
i
q
i
ζ
i
i
q
i
)
∥
u
i
-
v
i
∥
q
i
,
∥
N
i
(
u
1
,
…
,
u
i
-
1
,
v
i
,
u
i
+
1
,
…
,
u
m
)
-
N
i
(
v
1
,
…
,
v
i
-
1
,
v
i
,
v
i
+
1
,
…
,
v
m
)
∥
≤
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
u
j
-
v
j
∥
.
From (40)-(41), we obtain
(42)
∥
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
-
P
ρ
i
(
v
1
,
v
2
,
…
,
v
m
)
∥
≤
ρ
i
τ
i
δ
i
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
u
j
-
v
j
∥
+
τ
i
δ
i
1
-
q
i
ρ
i
σ
i
+
c
q
i
ρ
i
q
i
ζ
i
i
q
i
q
i
∥
u
i
-
v
i
∥
for i
=
1,2
,
…
,
m. Equation (42) implies that
(43)
∑
j
=
1
m
‍
∥
P
ρ
j
(
u
1
,
u
2
,
…
,
u
m
)
-
P
ρ
j
(
v
1
,
v
2
,
…
,
v
m
)
∥
=
∑
i
=
1
m
‍
∥
P
ρ
i
(
u
1
,
u
2
,
…
,
u
m
)
-
P
ρ
i
(
v
1
,
v
2
,
…
,
v
m
)
∥
≤
∑
i
=
1
m
(
∑
j
=
1
,
j
≠
i
m
‍
τ
i
δ
i
1
-
q
i
ρ
i
σ
i
+
c
q
i
ρ
i
q
i
ζ
i
i
q
i
q
i
∥
u
i
-
v
i
∥
‍
h
h
h
h
h
h
h
h
h
+
ρ
i
τ
i
δ
i
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
u
j
-
v
j
∥
)
≤
∑
i
=
1
m
‍
τ
i
δ
i
1
-
q
i
ρ
i
σ
i
+
c
q
i
ρ
i
q
i
ζ
i
i
q
i
q
i
∥
u
i
-
v
i
∥
+
∑
i
=
1
m
‍
ρ
i
τ
i
δ
i
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
u
j
-
v
j
∥
=
∑
j
=
1
m
‍
τ
j
δ
j
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
∥
u
j
-
v
j
∥
+
∑
j
=
1
m
‍
(
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
δ
i
)
∥
u
j
-
v
j
∥
=
∑
j
=
1
m
‍
[
τ
j
δ
j
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
δ
i
]
×
∥
u
j
-
v
j
∥
≤
k
∑
j
=
1
m
‍
∥
u
j
-
v
j
∥
,
where k
=
max
⁡
1
≤
j
≤
m
{
(
τ
j
/
δ
j
)
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
+
∑
i
=
1
,
i
≠
j
m
‍
(
ζ
i
j
ρ
i
τ
i
/
δ
i
)
}. By (33), we know that 0
≤
k
<
1. It follows from (43) that
(44)
∥
F
(
u
1
,
u
2
,
…
,
u
m
)
-
F
(
v
1
,
v
2
,
…
,
v
m
)
∥
*
≤
k
∥
(
u
1
,
u
2
,
…
,
u
m
)
-
(
v
1
,
v
2
,
…
,
v
m
)
∥
*
.
This proves that F
:
X
1
×
X
2
×
⋯
×
X
m
→
X
1
×
X
2
×
⋯
×
X
m is a contraction mapping. Hence, there exists a unique (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m such that
(45)
F
(
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
;
that is, P
ρ
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
=
x
i
* for i
=
1,2
,
…
,
m; that is,
(46)
x
i
*
=
J
M
i
ρ
i
[
x
i
*
-
ρ
i
N
i
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
]
.
By Lemma 11, (
x
1
*
,
x
2
*
,
…
,
x
m
*
) is the unique solution of problem (1). This completes the proof.Remark 13.
Ifm
=
2, then Theorem 12 reduces to Theorem 3.2 of Lan [19].Corollary 14.
LetH
i be real Hilbert space and η
i
:
H
i
×
H
i
→
H
i be τ
i-Lipschitz continuous and δ
i-strongly monotone for any i
=
1,2
,
…
,
m. Suppose that M
i
:
H
i
→
2
H
i is maximal η
i-monotone mapping, N
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i is σ
i-strongly monotone in the ith argument, and (
ζ
i
1
,
…
,
ζ
i
i
,
…
,
ζ
i
m
)-Lipschitz continuous for i
=
1,2
,
…
,
m. If
(47)
τ
j
δ
j
1
-
2
ρ
j
σ
j
+
ρ
j
2
ζ
j
j
2
2
+
∑
i
=
1
,
i
≠
j
m
‍
ζ
i
j
ρ
i
τ
i
δ
i
<
1
,
then problem (1) has a unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m.Corollary 15.
LetH
i be real Hilbert space for any i
=
1,2
,
…
,
m. Suppose that φ
i
:
H
i
→
(
-
∞
,
+
∞
] is proper, convex, and lower semicontinuous functional on H
i and N
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i is σ
i-strongly monotone in the ith argument and (
ζ
i
1
,
…
,
ζ
i
i
,
…
,
ζ
i
m
)-Lipschitz continuous for i
=
1,2
,
…
,
m. If
(48)
1
-
2
ρ
j
σ
j
+
ρ
j
2
ζ
j
j
2
2
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
ζ
i
j
<
1
,
then problem (2) has a unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m.


## 4. Perturbed Iterative Algorithms

In this section, by using Definition9 and Lemma 10, we construct a new perturbed iterative algorithm with mixed errors for solving problem (1) and prove the convergence and stability of the iterative sequence generated by the algorithm.Algorithm 16.
Letη
i
:
X
i
×
X
i
→
X
i
* and N
i
:
X
1
×
X
2
×
⋯
×
X
m
→
X
i be single-valued mappings and let M
i
:
X
i
→
2
X
i be generalized m-accretive mapping for i
=
1,2
,
…
,
m. For any given initial point (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
X
1
×
X
2
×
⋯
×
X
m, the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} for problem (1) is defined by
(49)
x
i
n
+
1
=
(
1
-
α
n
)
x
i
n
+
α
n
J
M
i
ρ
i
[
x
i
n
-
ρ
i
N
i
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
]
+
α
n
u
i
n
+
w
i
n
,
where n
≥
0, i
=
1,2
,
…
,
m, {
α
n
} is a sequence in [
0,1
], and {
u
i
n
}
,
{
w
i
n
}
⊂
X
i are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: (i)
u
i
n
=
u
i
′
n
+
u
i
′′
n;
(ii)
lim
⁡
n
→
∞
∥
u
i
′
n
∥
=
0;
(iii)
∑
n
=
0
∞
‍
∥
u
i
′′
n
∥
<
+
∞, ∑
n
=
0
∞
‍
∥
w
i
n
∥
<
+
∞ for i
=
1,2
,
…
,
m.
Let{
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
} be any sequence in X
1
×
X
2
×
⋯
×
X
m and define {
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
} by
(50)
ϵ
i
n
=
∥
z
i
n
+
1
-
{
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
h
h
h
h
h
h
h
h
h
h
h
+
α
n
J
M
i
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
h
h
h
h
h
h
h
h
h
h
h
+
α
n
u
i
n
+
w
i
n
-
{
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
}
∥
for i
=
1,2
,
…
,
m.Algorithm 17.
Letη
i
:
H
i
×
H
i
→
H
i and N
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i be single-valued mappings and let M
i
:
H
i
→
2
H
i be maximal η
i-monotone mapping for i
=
1,2
,
…
,
m. For any given initial point (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
H
1
×
H
2
×
⋯
×
H
m, the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} for problem (1) is defined by
(51)
x
i
n
+
1
=
(
1
-
α
n
)
x
i
n
+
α
n
J
M
i
ρ
i
[
x
i
n
-
ρ
i
N
i
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
]
+
α
n
u
i
n
+
w
i
n
,
where n
≥
0, i
=
1,2
,
…
,
m, {
α
n
} is a sequence in [
0,1
], and {
u
i
n
}
,
{
w
i
n
}
⊂
X
i are errors to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: (i)
lim
⁡
n
→
∞
∥
u
i
n
∥
=
0;
(ii)
∑
n
=
0
∞
‍
∥
w
i
n
∥
<
+
∞ for i
=
1,2
,
…
,
m.
Let{
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
} be any sequence in H
1
×
H
2
×
⋯
×
H
m and define {
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
} by
(52)
ϵ
i
n
=
∥
z
i
n
+
1
-
{
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
h
h
h
h
h
h
h
h
h
h
h
+
α
n
J
M
i
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
h
h
h
h
h
h
h
h
h
h
h
+
α
n
u
i
n
+
w
i
n
{
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
}
∥
for i
=
1,2
,
…
,
m.Algorithm 18.
LetN
i
:
H
1
×
H
2
×
⋯
×
H
m
→
H
i be single-valued mappings and φ
i
:
H
i
→
(
-
∞
,
+
∞
] is proper, convex, and lower semi-continuous functional on H
i for i
=
1,2
,
…
,
m. For any given initial point (
x
1
0
,
x
2
0
,
…
,
x
m
0
)
∈
H
1
×
H
2
×
⋯
×
H
m, the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} for problem (2) is defined by
(53)
x
i
n
+
1
=
(
1
-
α
n
)
x
i
n
+
α
n
J
φ
i
[
x
i
n
-
ρ
i
N
i
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
]
+
w
i
n
,
where n
≥
0, i
=
1,2
,
…
,
m, {
α
n
} is a sequence in [
0,1
], {
w
i
n
}
⊂
X
i are errors to take into account a possible inexact computation of the resolvent operator point satisfying the condition ∑
n
=
0
∞
‍
∥
w
i
n
∥
<
+
∞ for i
=
1,2
,
…
,
m. Let {
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
} be any sequence in H
1
×
H
2
×
⋯
×
H
m and define {
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
} by
(54)
ϵ
i
n
=
∥
z
i
n
+
1
-
{
(
1
-
α
n
)
z
i
n
+
α
n
J
φ
i
h
h
h
h
h
h
h
h
h
h
+
α
n
J
φ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
h
h
h
h
h
h
h
h
h
h
+
w
i
n
{
(
1
-
α
n
)
z
i
n
+
α
n
J
φ
i
}
∥
for i
=
1,2
,
…
,
m.Remark 19.
Ifm
=
2, then Algorithm 16 reduces to Algorithm 4.3 of Lan [19].
Next we will show the convergence and stability of Algorithm16.Theorem 20.
Suppose thatX
i, η
i, N
i, and M
i
(
i
=
1,2
,
…
,
m
) are the same as in Theorem 12. If ∑
n
=
0
∞
‍
α
n
=
+
∞ and condition (33) holds, then the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} defined by Algorithm 16 converges strongly to the unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m of the problem (1). Moreover, if there exists a
∈
(
0
,
α
n
] for all n
≥
0, then
(55)
lim
⁡
n
→
∞
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
if and only if
(56)
lim
⁡
n
→
∞
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
=
(
0,0
,
…
,
0
)
︸
m
,
where (
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
) is defined by (50).Proof.
From Theorem12, we know that problem (1) has a unique solution
(57)
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
X
1
×
X
2
×
⋯
×
X
m
.
It follows from (49) and the proof of (42) in Theorem 12 that, for i
=
1,2
,
…
,
m,
(58)
∥
x
i
n
+
1
-
x
i
*
∥
≤
(
1
-
α
n
)
∥
x
i
n
-
x
i
*
∥
+
α
n
{
∑
j
=
1
,
j
≠
i
m
‍
τ
i
δ
i
1
-
q
i
ρ
i
σ
i
+
c
q
i
ρ
i
q
i
ζ
i
i
q
i
q
i
∥
x
i
n
-
x
i
*
∥
h
h
h
h
h
h
h
+
ρ
i
τ
i
δ
i
∑
j
=
1
,
j
≠
i
m
‍
ζ
i
j
∥
x
j
n
-
x
j
*
∥
}
+
α
n
∥
u
i
′
n
∥
+
(
∥
u
i
′′
n
∥
+
∥
w
i
n
∥
)
.
It follows from (58), we have
(59)
∑
i
=
1
m
‍
∥
x
i
n
+
1
-
x
i
*
∥
≤
(
1
-
α
n
)
∑
i
=
1
m
‍
∥
x
i
n
-
x
i
*
∥
+
α
n
∑
j
=
1
m
‍
[
τ
j
δ
j
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
δ
i
]
×
∥
x
j
n
-
x
j
*
∥
+
α
n
∑
i
=
1
m
‍
∥
u
i
′
n
∥
+
∑
i
=
1
m
‍
∥
u
i
′′
n
∥
+
∑
i
=
1
m
‍
∥
w
i
n
∥
=
(
1
-
α
n
)
∑
j
=
1
m
‍
∥
x
j
n
-
x
j
*
∥
+
α
n
∑
j
=
1
m
‍
[
τ
j
δ
j
1
-
q
j
ρ
j
σ
j
+
c
q
j
ρ
j
q
j
ζ
j
j
q
j
q
j
+
∑
i
=
1
,
i
≠
j
m
‍
ρ
i
τ
i
ζ
i
j
δ
i
]
×
∥
x
j
n
-
x
j
*
∥
+
α
n
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
≤
[
1
-
α
n
(
1
-
k
)
]
∑
j
=
1
m
‍
∥
x
j
n
-
x
j
*
∥
+
α
n
(
1
-
k
)
·
1
1
-
k
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
(
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
)
,
where k is the same as in (43). Letting t
n
=
α
n
(
1
-
k
)
∈
[
0,1
], b
n
=
(
1
/
(
1
-
k
)
)
∑
j
=
1
m
‍
∥
u
j
′
n
∥, and c
n
=
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
(
n
≥
0
), then it follows from ∑
n
=
0
∞
‍
α
n
=
+
∞ and (i)–(iii) of Algorithm 16 that
(60)
∑
n
=
0
∞
‍
t
n
=
+
∞
,
lim
⁡
n
→
∞
b
n
=
1
1
-
k
∑
j
=
1
m
‍
(
lim
⁡
n
→
∞
∥
u
j
′
n
∥
)
=
0
,
∑
n
=
0
∞
‍
c
n
=
∑
j
=
1
m
‍
∑
n
=
0
∞
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∑
n
=
0
∞
‍
∥
w
j
n
∥
<
+
∞
.
Settinga
n
=
∑
j
=
1
m
‍
∥
x
j
n
-
x
j
*
∥, then (59) can be rewritten as
(61)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
b
n
t
n
+
c
n
,
n
=
0,1
,
2
,
…
.
It follows from Lemma 10 that lim
⁡
n
→
∞
a
n
=
0; that is,
(62)
lim
⁡
n
→
∞
∑
j
=
1
m
‍
∥
x
j
n
-
x
j
*
∥
=
0
;
thus
(63)
x
j
n
⟶
x
j
*
(
n
⟶
∞
)
,
(
j
=
1,2
,
…
,
m
)
.
Hence, we know that the sequence{
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} converges strongly to the unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
) of the problem (1).
Now we prove the second conclusion. By (50), now we know
(64)
∥
z
i
n
+
1
-
x
i
*
∥
≤
∥
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
h
h
h
h
h
h
h
h
+
α
n
u
i
n
+
w
i
n
-
x
i
*
(
1
-
α
n
)
z
i
n
+
α
n
J
M
i
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
∥
+
ϵ
i
n
,
where i
=
1,2
,
…
,
m. As the proof of inequality (59), we have
(65)
∑
j
=
1
m
‍
∥
(
1
-
α
n
)
z
j
n
+
α
n
J
M
j
ρ
j
[
z
j
n
-
ρ
j
N
j
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
+
α
n
u
j
n
+
w
j
n
-
x
j
*
(
1
-
α
n
)
z
j
n
+
α
n
J
M
j
ρ
j
[
z
j
n
-
ρ
j
N
j
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
∥
≤
[
1
-
α
n
(
1
-
k
)
]
∑
j
=
1
m
‍
∥
z
j
n
-
x
j
*
∥
+
α
n
(
1
-
k
)
·
1
1
-
k
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
(
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
)
.
Since 0
<
a
≤
α
n
(
n
=
0,1
,
2
,
…
,
), it follows from (64) and (65) that
(66)
∑
j
=
1
m
‍
∥
z
j
n
+
1
-
x
j
*
∥
≤
[
1
-
α
n
(
1
-
k
)
]
∑
j
=
1
m
‍
∥
z
j
n
-
x
j
*
∥
+
α
n
(
1
-
k
)
·
1
1
-
k
(
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
1
a
∑
j
=
1
m
‍
ϵ
j
n
)
+
(
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
)
.
Suppose that lim
⁡
n
→
∞
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
=
(
0,0
,
…
,
0
)
︸
m. Letting b
n
′
=
(
1
/
(
1
-
k
)
)
(
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
(
1
/
a
)
∑
j
=
1
m
‍
ϵ
j
n
) and a
n
′
=
∑
j
=
1
m
‍
∥
z
j
n
-
x
j
*
∥, then (66) implies that
(67)
a
n
+
1
′
≤
(
1
-
t
n
)
a
n
′
+
b
n
′
t
n
+
c
n
,
n
=
0,1
,
2
,
…
,
where t
n, c
n are the same as previously. Since lim
⁡
n
→
∞
∥
u
j
′
n
∥
=
0 and lim
⁡
n
→
∞
ϵ
j
n
=
0
(
j
=
1,2
,
…
,
m
),
(68)
lim
⁡
n
→
∞
b
n
′
=
1
1
-
k
[
∑
j
=
1
m
‍
(
lim
⁡
n
→
∞
∥
u
j
′
n
∥
)
+
1
a
∑
j
=
1
m
‍
(
lim
⁡
n
→
∞
ϵ
j
n
)
]
=
0
.
It again follows from Lemma 10, we have lim
⁡
n
→
∞
a
n
′
=
0 and so
(69)
lim
⁡
n
→
∞
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
)
.
Conversely, if lim
⁡
n
→
∞
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
), it follows from (50), then we get
(70)
ϵ
i
n
≤
∥
z
i
n
+
1
-
x
i
*
∥
+
∥
(
1
-
α
n
)
z
i
n
+
α
n
J
M
j
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
+
α
n
u
i
n
+
w
i
n
-
x
i
*
J
M
j
ρ
i
[
z
i
n
-
ρ
i
N
i
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
]
∥
,
∀
i
=
1,2
,
…
,
m
.
Combining (65) with (70), we have
(71)
∑
i
=
1
m
‍
ϵ
i
n
≤
∑
i
=
1
m
‍
∥
z
i
n
+
1
-
x
i
*
∥
+
[
1
-
α
n
(
1
-
k
)
]
∑
j
=
1
m
‍
∥
z
j
n
-
x
j
*
∥
+
α
n
(
1
-
k
)
·
1
1
-
k
∑
j
=
1
m
‍
∥
u
j
′
n
∥
+
(
∑
j
=
1
m
‍
∥
u
j
′′
n
∥
+
∑
j
=
1
m
‍
∥
w
j
n
∥
)
⟶
0
(
n
⟶
∞
)
.
This completes the proof.Corollary 21.
Suppose thatH
i, η
i, N
i, and M
i
(
i
=
1,2
,
…
,
m
) are the same as in Corollary 14. If ∑
n
=
0
∞
‍
α
n
=
+
∞ and condition (47) holds, then the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} defined by Algorithm 17 converges strongly to the unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m of the problem (1). Moreover, if there exists a
∈
(
0
,
α
n
] for all n
≥
0, then lim
⁡
n
→
∞
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
) if and only if
(72)
lim
⁡
n
→
∞
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
=
(
0,0
,
…
,
0
)
︸
m
,
where (
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
) is defined by (50).Corollary 22.
Assume thatH
i, N
i, and φ
i
(
i
=
1,2
,
…
,
m
) are the same as in Corollary 15. If ∑
n
=
0
∞
‍
α
n
=
+
∞ and condition (48) holds, then the perturbed iterative sequence {
(
x
1
n
,
x
2
n
,
…
,
x
m
n
)
} defined by Algorithm 18 converges strongly to the unique solution (
x
1
*
,
x
2
*
,
…
,
x
m
*
)
∈
H
1
×
H
2
×
⋯
×
H
m of the problem (2). Moreover, if there exists a
∈
(
0
,
α
n
] for all n
≥
0, then lim
⁡
n
→
∞
(
z
1
n
,
z
2
n
,
…
,
z
m
n
)
=
(
x
1
*
,
x
2
*
,
…
,
x
m
*
) if and only if
(73)
lim
⁡
n
→
∞
(
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
)
=
(
0,0
,
…
,
0
)
︸
m
,
where (
ϵ
1
n
,
ϵ
2
n
,
…
,
ϵ
m
n
) is defined by (54).Remark 23.
Ifm
=
2, then Theorem 20 reduces to Theorem 4.3 of Lan [19]. Further, one can easily see that our results presented in this paper may be viewed as an refinement and improvement of the previously known results.



---

*Source: 659870-2014-10-14.xml*