# Stability for a New Class of GNOVI with (γ

Authors: Hong Gang Li; Yongqin Yang; Mao Ming Jin; Qinghua Zhang
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

By using ordered fixed point theory, we set up a new class of GNOVI structures (general nonlinear ordered variational inclusions) with(
γ
G
,
λ
)-weak-GRD mappings, discuss an existence theorem of solution, consider a perturbed Ishikawa iterative algorithm and the convergence of iterative sequences generated by the algorithm, and show the stability of algorithm for GNOVI structures in positive Hilbert spaces. The results in the instrument are obtained.


---

## Body

## 1. Introduction

Stability for variational inequality or general nonlinear ordered variational inclusions problems are of course powerful tools to deal with the problems occurring in control, nonlinear programming, economics, engineering sciences and optimization, and so forth. In recent years, there are some achievements in terms of systems of inequalities [1], weak vector variational inequality [2], differential mixed variational inequalities [3], and so forth. Moreover, Jin [4] studied the stability for strong nonlinear quasi-variational inclusion involving H-accretive operators in 2006. After that the authors investigated some the stability problems of perturbed Ishikawa iterative algorithms for nonlinear variational inclusion problems involving (
A
,
η
)-accretive mappings [5, 6].On the other hand, in 1972, Amann [7] had the number of solutions of nonlinear equations in ordered Banach spaces. Focusing on the work done related to the fixed points of nonlinear increasing operators in ordered Banach spaces, it is worth mentioning that work done by Du [8] is quite interesting and applicable in pure and applied sciences. From 2008, the authors have some results with regard to the approximation algorithm, the approximation solution for a variety of generalized nonlinear ordered variational inequalities, ordered equations and inclusions, and sensitivity analysis for a class of parametric variational inclusions in ordered Banach spaces (see [7–22]). For related work, we refer the reader to [1–36] and the references therein.Taking into account the importance of above-mentioned research works, in this paper, a new class of generalized nonlinear ordered variational inclusion structures, GNOVI structures, are introduced in positive Hilbert spaces. By using the resolvent operator for(
γ
G
,
λ
)-weak-GRD set-valued mappings and fixed point theory, an existence theorem of solution for the GNOVI frameworks is established, a perturbed Ishikawa iterative algorithm is suggested, and the stability and the convergence of iterative sequences generated by the algorithm are discussed in positive Hilbert spaces. In this field, the results in the instrument are obtained.


## 2. Preliminaries and a New Class of GNOVI Structures

Let us recall the following results and concepts for research stability for a new class of GNOVI with(
γ
G
,
λ
)-weak-GRD mappings in positive Hilbert spaces.LetR be real set, let H be Hilbert space with an inner product ·
,
·, a norm ·, and a zero element θ, let nonempty closed convex subsets C
⊆
H be a cone, let N be a normal constant of C, and let relation ≤ defined by a normal cone C be a partial ordered relation in H; then H formats an ordered Hilbert space for the ordered relation ≤ and x and y are said to be compared to each other (denoted by x
∝
y) for x
,
y
∈
H if x
≤
y or y
≤
x holds in H. If supper
⁡
u
,
v express the least upper bound of a binary set {
u
,
v
} and inf
⁡
u
,
v express the greatest lower bound of a binary set {
u
,
v
} on the partial ordered relation ≤ for any u
,
v
∈
H
P, supper
⁡
u
,
v and inf
⁡
u
,
v exist, and some binary operators can be defined as follows:(i)
x
∧
y
=
inf
⁡
u
,
v.
(ii)
u
∨
v
=
supper
⁡
u
,
v.
(iii)
u
⊕
v
=
(
u
-
v
)
∨
(
v
-
u
).
∧, ∨, and ⊕ are called AND, OR, and XOR operations, respectively; then (
H
,
∨
,
∧
,
≤
) is an ordered lattice [35].Definition 1.
An ordered Hilbert spaceH with an inner product 〈
·
,
·
〉 is said to be a positive Hilbert space (denoted by H
P) with a partially ordered relation ≤, if (1)
x
≥
θ
,
y
≥
θ
for  any
x
,
y
∈
H
;then 〈
x
,
y
〉
≥
0 holds, or H with an inner product 〈
·
,
·
〉 is said to be a nonpositive Hilbert space (denoted by H
N) with a partially ordered relation ≤.As an example, letC
1
=
{
(
u
,
v
)
∣
0
≤
u
,
v
,
u
,
v
∈
R
} be closed convex subsets and let ≤ defined by a normal cone C
1 be a partial ordered relation in R
2 (denoted by H
N
C
1); it is clear that R
2 is a positive Hilbert space with the partially ordered relation ≤. However, when letting C
2
=
{
(
u
,
v
)
∣
0
≤
u
,
v
≤
2
u
,
u
∈
R
}, then C
2 is closed convex subsets. Obviously, R
2 is a nonpositive Hilbert space with ≤ because 〈
(
u
,
u
)
,
(
u
,
-
1.5
u
)
〉
=
-
0.5
u
2
<
0 for θ
≤
(
u
,
u
), (
u
,
-
1.5
u
)
∈
C
2 (denoted by H
N
C
2).The following results and structural relationships are achievements gained by some folks in ordered Banach spaces (see [8, 9, 16–28, 30–35]), and they are as same as right in positive Hilbert space H
P.Theorem 2 (see [9, 35]).
LetR be real set, let H be an ordered Hilbert space, and let (
H
,
∨
,
∧
,
≤
) be an ordered lattice; then the following relations hold: (1)
Ifu and v can be compared, then θ
≤
u
⊕
v
.
(2)
θ
≤
u
⊕
θ.
(3)
Ifλ
≥
0, then λ
(
u
∨
v
)
=
λ
u
∨
λ
v.
(4)
Letλ be real; then (
λ
u
)
⊕
(
λ
v
)
=
λ
(
u
⊕
v
).
(5)
(
u
+
w
)
∧
(
v
+
w
) exists and (
u
+
w
)
∧
(
v
+
w
)
=
(
u
∧
v
)
+
w.
(6)
(
u
+
w
)
∨
(
v
+
w
) exists and (
u
+
w
)
∨
(
v
+
w
)
=
(
u
∨
y
)
+
w.
(7)
Ifu
,
v, and w can be compared to each other, then (
u
⊕
v
)
≤
u
⊕
w
+
w
⊕
v.
(8)
Let(
u
+
v
)
∨
(
w
+
z
) exist, and if x
∝
u
,
v and y
∝
u
,
v, then (
u
+
v
)
⊕
(
w
+
z
)
≤
(
u
⊕
w
+
v
⊕
z
)
∧
(
u
⊕
z
+
v
⊕
w
).
(9)
Ifu
,
v
,
w, and z can be compared to each other, then (
u
∧
v
)
⊕
(
w
∧
z
)
≤
(
(
u
⊕
w
)
∨
(
v
⊕
z
)
)
∧
(
(
u
⊕
z
)
∨
(
v
⊕
w
)
).
(10)
(
α
u
)
⊕
(
β
u
)
=
α
-
β
u for α
,
β
∈
R. For arbitrariness, u
,
v
,
w, and z
∈
H.Theorem 3 (see [8, 9, 16]).
LetH be an ordered Hilbert space and let ≤ be a partial ordered relation in H; then the following conclusions hold:(i)
Ifx
∝
y, then (1) supper
⁡
x
,
y and inf
⁡
x
,
y exist, (2) x
-
y
∝
y
-
x, and (3) θ
≤
(
x
-
y
)
∨
(
y
-
x
).
(ii)
If, for any natural numbern, x
∝
y
n and y
n
→
y
∗
(
n
→
∞
), then x
∝
y
∗.Theorem 4 (see [22]).
IfH
P is a positive Hilbert space and ≤ is a partial ordered relation in H
P, then the inequalities, (1)
ifθ
≤
z
,
u
≤
v, then u
,
z
≤
v
,
z,
(2)
ifθ
≤
z, then u
∧
v
,
z
≤
u
,
z
∧
v
,
z
,
u
,
z
∨
v
,
z
≤
u
∨
v
,
z,
(3)
ifθ
≤
z, then u
,
z
+
v
,
z
-
u
,
z
∧
v
,
z
≤
u
∨
v
,
z,
(4)
ifθ
≤
z, then u
,
z
∨
v
,
z
+
u
∧
v
,
z
≤
u
+
v
,
z,
(5)
ifθ
≤
z, then 〈
u
,
z
〉
⊕
〈
v
,
z
〉
≤
〈
u
⊕
v
,
z
〉,hold for u
,
v
,
z
,
w
∈
H
P.It is worth noting that(
1
)
–
(
5
) metric inequalities in Theorem 4 are failure in nonpositive Hilbert space H
N, for example, H
N
C
2.Definition 5.
LetH
P be a real positive Hilbert space, and let Q
:
H
P
×
H
P
→
H
P be a mapping. The mapping Q
:
H
×
H
→
H is said to be ordered Lipschitz continuous mapping with constants (
μ
,
ν
); if u
∝
v and w
∝
z, then Q
(
w
,
u
)
∝
Q
(
z
,
v
) and there exist constants μ
,
ν
>
0 such that (2)
Q
w
,
u
⊕
Q
z
,
v
≤
μ
w
⊕
z
+
ν
u
⊕
v
.Definition 6.
LetH
P be a real positive Hilbert space, let M
:
H
P
→
C
B
(
H
P
) be a set-valued mapping, and let G
:
H
P
→
H
P be a strong comparison and β-ordered compressed mapping.(1)
M is said to be a weak comparison mapping with respect to G; if, for any x
,
y
∈
X, x
∝
y, then there exist v
x
∈
M
(
G
(
x
)
) and v
y
∈
M
(
G
(
y
)
) such that x
∝
v
x, y
∝
v
y, and v
x
∝
v
y, where v
x and v
y are said to be weak-comparison elements, respectively.
(2)
M with respect to G is said to be a λ-weak ordered different comparison mapping with respect to G; if there exists a constant λ
>
0 such that, for any x
,
y
∈
H
P, there exist v
x
∈
M
(
G
(
x
)
), v
y
∈
M
(
G
(
y
)
), λ
(
v
x
-
v
y
)
∝
x
-
y holds, where v
x and v
y are said to be λ-elements, respectively.
(3)
M is said to be an ordered rectangular mapping, if, for each x
,
y
∈
H
P, and any v
x
∈
M
(
x
) and any v
y
∈
M
(
y
) such that 〈
v
x
⊙
v
y
,
-
(
x
⊕
y
)
〉
=
0 holds.
(4)
M is said to be a γ
G-ordered rectangular mapping with respect to G; if there exists a constant γ
G
≥
0, for any x
,
y
∈
H
P, there exist v
x
∈
M
(
G
(
x
)
) and v
y
∈
M
(
G
(
y
)
) such that (3)
v
x
⊙
v
y
,
-
G
x
⊕
y
≥
γ
G
G
x
⊕
G
y
2
holds, wherev
x and v
y are said to be γ
G-elements, respectively.
(5)
A weak comparison mappingM with respect to B is said to be a (
γ
G
,
λ
)-weak-G
R
D mapping with respect to B, if M is a γ
G-ordered rectangular and λ-weak ordered different comparison mapping with respect to B and (
G
+
λ
M
)
(
H
P
)
=
H
P for λ
>
0, and there exist v
x
∈
M
(
G
(
x
)
) and v
y
∈
M
(
G
(
y
)
) such that v
x and v
y are (
γ
G
,
λ
)-elements, respectively.Remark 7 (see [9]).
LetH
P be a real positive Hilbert space, let G
:
H
P
→
H
P be a single-valued mapping, and let M
:
H
P
→
C
B
(
H
P
) be a set-valued mapping; then one has the following:(i)
IfG
=
I (identical mapping), then a γ
I-ordered rectangular mapping must be ordered rectangularly in [15].
(ii)
An ordered RME mapping must beλ-weak-GRD in [15].
(iii)
Aλ-ordered monotone mapping must be λ-weak ordered different comparison [22].Theorem 8 (see [22]).
LetH
P be a real positive Hilbert space with normal constant N, and let G be a strong comparison and β-ordered compressed mapping. Let M
:
H
P
→
C
B
(
H
P
) be an α
I-weak ordered rectangular set-valued mapping and I is an identical mapping. Let mapping R
G
M
,
λ
=
(
G
+
λ
M
)
-
1
:
H
P
→
2
H
P be an inverse mapping of (
G
+
λ
M
).
Ifα
I
λ
>
β
>
0, λ
(
α
I
∧
γ
G
)
>
β
>
0, and M
:
H
P
→
C
B
(
H
P
) is a (
γ
G
,
λ
)-weak-GRD set-valued mapping with respect to R
G
M
,
λ, then the resolvent operator R
G
M
,
λ of M is a single-valued comparison, and(4)
R
G
M
,
λ
u
⊕
R
G
M
,
λ
v
≤
1
γ
G
λ
-
β
u
⊕
v
,for z
u
∈
M
(
R
G
M
,
λ
(
u
)
) and z
v
∈
M
(
R
G
M
,
λ
(
v
)
), which are α
I, γ
G, and λ-elements, respectively.LetR be real set, and let H
P be a real positive Hilbert space with normal constant N, a norm ·, an inner product ·
,
·, and zero θ. Let M
:
H
P
→
CB
(
H
P
) and ρ
M
(
H
P
)
=
{
ρ
v
∣
v
∈
M
(
H
P
)
} be two set-valued mappings, and let g
:
H
P
→
H
P and F
:
H
P
×
H
P
→
H
P be two single-valued nonlinear ordered compression mappings. We consider the following structures.Forρ
>
0 and any ξ
∈
R, find x
∈
H
P such that(5)
θ
∈
ρ
M
x
-
ξ
F
x
,
g
x
,which is called a new class of general nonlinear ordered variational inclusion structures (GNOVI structures) in positive Hilbert spaces.Remark 9.
(i) Ifρ
=
1, ξ
=
0, and M
(
x
)
=
A
(
g
(
x
)
), then problem (5) becomes the ordered variational inequality A
(
g
(
x
)
)
≥
θ, which was studied by Li [9].
(ii) Ifρ
=
1 and ξ
=
0, then problem (5) becomes the ordered variational inequality θ
∈
M
(
x
), which was studied by Li [10].
(iii) Ifρ
=
ω, ξ
=
-
1, and F
(
x
,
g
(
x
)
)
=
f
(
x
)
-
w
(
w
∈
X
), then problem (5) becomes the ordered variational inequality w
∈
f
(
x
)
+
ω
M
(
x
), which was studied by Li et al. [20].


## 3. Existence Theorem of the Solution for GNOVI Structures

In this section, by using Definition1 and Theorems 2–4 and 8, we study a new class of general nonlinear ordered variational inclusion structures in positive Hilbert spaces.Lemma 10.
LetH
P be a real positive Hilbert space with normal constant N, let G be a strong comparison and β-ordered compressed mapping, and let M
:
H
P
→
C
B
(
H
P
) be a (
γ
G
,
λ
)-weak ordered GRD set-valued mapping with respect to R
G
M
,
λ. Let g
:
H
P
→
H
P and F
:
H
P
×
H
P
→
H
P be two single-valued nonlinear mappings. Then inclusion problem (5) has a solution x
∗ if and only if x
∗
=
R
G
M
,
λ
(
G
(
x
∗
)
+
λ
ξ
/
ρ
F
(
x
∗
,
g
(
x
∗
)
)
) in H
P.Proof.
Forρ
>
0, take notice of the fact that θ
∈
ρ
M
(
x
)
-
ξ
F
(
x
,
g
(
x
)
) if and only if G
(
x
)
+
λ
ξ
/
ρ
F
(
x
,
g
(
x
)
)
∈
G
(
x
)
+
λ
M
(
x
); this directly follows from the definition of R
G
ρ
M
,
λ and problem (5).Theorem 11.
LetR be real set, and let H
P be a real positive Hilbert space with an inner product 〈
·
,
·
〉 and a normal constant N. Let G be a strong comparison and β-ordered compressed mapping, let M
:
H
P
→
C
B
(
H
P
) be an α
I-ordered rectangular and (
γ
G
,
λ
)-weak-GRD set-valued mapping with respect to R
G
M
,
λ, and let v
x
∈
M
(
R
G
M
,
λ
(
x
)
) and v
y
∈
M
(
R
G
M
,
λ
(
y
)
) be α
I, λ, and γ
G-elements, respectively. Let F
:
H
P
×
H
P
→
H
P be an ordered Lipschitz continuous mapping with constants (
μ
,
ν
), and let g
:
H
P
→
H
P be single-valued nonlinear φ-ordered compression mapping. If M
,
G
,
g
,
λ
ξ
/
ρ
F
(
x
,
g
(
x
)
) and G
(
x
)
+
λ
ξ
/
ρ
F
(
x
,
g
(
x
)
) are compared to each other and β and λ satisfy(6)
0
<
β
<
λ
γ
G
2
∧
α
I
∧
1
,
(7)
ρ
β
N
+
1
+
λ
μ
+
ν
φ
ξ
N
<
ρ
γ
G
λ
,then there exists a solution x
∗ of GNOVI structures (5), which is a fixed point of R
G
M
,
λ
(
G
(
x
)
+
λ
ξ
/
ρ
F
(
x
,
g
(
x
)
)
).Proof.
LetH
P be a positive Hilbert space with an inner product 〈
·
,
·
〉 and a normal constant N, let G be a strong comparison and β-ordered compression mapping, and let M
(
x
)
=
{
v
∣
v
∈
M
(
x
)
}
:
H
P
→
CB
(
H
P
)
(
ρ
>
0
) be a (
γ
G
,
λ
)-weak-GRD set-valued mapping with respect to R
G
M
,
λ.Sinceα
I
,
β
,
γ
G
,
λ
>
0 and by condition (6), we have (8)
λ
α
I
∧
γ
G
≥
λ
γ
G
2
∧
α
=
λ
γ
G
2
∧
λ
α
I
>
β
>
0
,
1
>
β
γ
G
λ
-
β
>
0
. By Theorems 4 and 8
(
4
) in [9] and the conditions, if x
1
∝
x
2, then R
G
M
,
λ
(
G
+
λ
ξ
/
ρ
F
(
·
,
g
(
·
)
)
)
(
x
1
)
∝
R
G
M
,
λ
(
G
(
·
)
+
λ
ξ
/
ρ
F
(
·
,
g
(
·
)
)
)
(
x
2
) for x
1
,
x
2
∈
H
P, and(9)
R
G
ρ
M
,
λ
G
+
λ
ξ
ρ
F
·
,
g
·
x
1
⊕
R
G
ρ
M
,
λ
G
+
λ
ξ
ρ
F
·
,
g
·
x
2
≤
1
γ
G
λ
-
β
G
+
λ
ξ
ρ
F
·
,
g
·
x
1
⊕
G
+
λ
ξ
ρ
F
·
,
g
·
x
2
≤
N
γ
G
λ
-
β
G
x
1
⊕
G
x
2
+
λ
∣
ξ
∣
ρ
F
x
1
,
g
x
1
⊕
F
x
2
,
g
x
2
≤
N
γ
G
λ
-
β
β
x
1
⊕
x
2
+
λ
ξ
ρ
μ
x
1
⊕
x
2
+
ν
g
x
1
⊕
g
x
2
≤
N
γ
G
λ
-
β
β
x
1
⊕
x
2
+
λ
ξ
ρ
μ
+
ν
φ
x
1
⊕
x
2
≤
N
ρ
β
+
λ
μ
+
ν
φ
ξ
ρ
γ
G
λ
-
β
x
1
⊕
x
2
.It follows that R
G
M
,
λ
(
G
+
λ
/
ρ
w
) has a fixed point x
∗, which is a solution x
∗ for GNOVI (5), from Lemma 10 and N
ρ
β
+
λ
(
μ
+
ν
φ
ξ
)
/
ρ
(
γ
G
λ
-
β
)
<
1 for (7).


## 4. Stability of Algorithm for GNOVI Structures

Definition 12.
LetT
:
H
P
→
H
P be a self-mapping, u
0
∈
H
P, and let u
n
+
1
=
h
(
T
,
u
n
) define an iteration procedure which yields a sequence of points {
u
n
}
u
=
0
∞ in H
P. Suppose that {
u
∈
H
P
:
T
u
=
u
}
≠
∅ and {
u
n
}
n
=
0
∞ converge to a fixed point u
∗ of T. Let {
w
n
}
⊂
H
P and let ε
n
=
w
n
+
1
-
h
(
T
,
w
n
). If lim
n
→
∞
⁡
ε
n
=
0 implies that u
n
→
u
∗, then the iteration procedure defined by u
n
+
1
=
h
(
T
,
u
n
) is said to be T-stable or stable with respect to T.Lemma 13 (see [36]).
Let{
ξ
n
}
n
=
0
∞ be a nonnegative real sequence and let {
ζ
n
}
n
=
0
∞ be a real sequence in [
0,1
] such that ∑
n
=
0
∞
ζ
n
=
∞. If there exists a positive integer n
1 such that (10)
ξ
n
+
1
≤
1
-
ζ
n
ξ
n
+
ζ
n
η
n
,
∀
n
≥
n
1
,where η
n
≥
0 for all n
≥
0 and η
n
→
0
(
n
→
∞
), then lim
n
→
∞
⁡
ξ
n
=
0.Based on Theorem11, we can develop a new Ishikawa iterative sequence for solving problem (5) as follows.Algorithm 14.
LetR be real set, and let H
P be a real positive Hilbert space with normal constant N. Let {
ω
n
}
n
=
0
∞ and {
σ
n
}
n
=
0
∞ be two sequences such that ω
n
,
σ
n
∈
[
0,1
] and ∑
n
=
0
∞
ω
n
=
∞. Let {
a
n
}
n
=
0
∞ and {
b
n
}
n
=
0
∞ be two sequences in H
P introduced to take into account possible inexact computation, where a
n
⊕
θ
=
a
n and b
n
⊕
θ
=
b
n
(
n
=
0,1
,
2
,
⋯
). For any given x
0
∈
H
P, the perturbed Ishikawa iterative sequence {
x
n
}
n
=
0
∞ is defined by(11)
x
n
+
1
=
1
-
ω
n
x
n
+
ω
n
R
G
M
,
λ
G
y
n
+
λ
ξ
ρ
F
y
n
,
g
y
n
+
ω
n
a
n
,
y
n
=
1
-
σ
n
x
n
+
σ
n
R
G
M
,
λ
G
x
n
+
λ
ξ
ρ
F
x
n
,
g
x
n
+
σ
n
b
n
. Let {
z
n
}
n
=
0
∞ be any sequence in X and define {
ε
n
}
n
=
0
∞ by(12)
ε
n
=
z
n
+
1
-
R
G
M
,
λ
G
t
n
+
λ
ξ
ρ
F
t
n
,
g
t
n
+
ω
n
a
n
,
t
n
=
1
-
σ
n
z
n
+
σ
n
R
G
M
,
λ
G
z
n
+
λ
ξ
ρ
F
z
n
,
g
z
n
+
ω
n
b
n
,where ρ
,
λ
>
0, ξ
∈
R and n
=
0
,
1,2
,
…
.Remark 15.
For a suitable choice of the mappingsA, g, f, B, σ
n, ω
n, ρ, and ξ and space H
P, then Algorithm 14 can be degenerated to known the algorithms in [9].Theorem 16.
LetR
,
H
P
,
M
,
g
,
F be the same as in Theorem 11, and let {
ω
n
}
n
=
0
∞ and {
σ
n
}
n
=
0
∞ be two sequences such that ω
n
,
σ
n
∈
[
0,1
] and ∑
n
=
0
∞
ω
n
=
∞. Let {
a
n
}
n
=
0
∞ and {
b
n
}
n
=
0
∞ be two sequences in H
P introduced to take into account possible inexact computation, where a
n
⊕
θ
=
a
n and b
n
⊕
θ
=
b
n
(
n
=
0,1
,
2
,
⋯
). If condition(13)
ρ
β
+
λ
μ
+
ν
φ
ξ
<
ρ
λ
γ
G
-
β
min
⁡
1
N
,
1
2holds, then one has the following:(i)
Iflim
n
→
∞
⁡
a
n
∨
-
a
n
=
lim
n
→
∞
⁡
b
n
∨
-
b
n
=
0, then sequence {
x
n
} generated by (11) converges strongly to x
∗
∈
H
P, and x
∗ is a unique solution of problem (5).
(ii)
Moreover, if0
<
φ
≤
ω
n, then lim
n
→
∞
⁡
z
n
=
x
∗ if and only if lim
n
→
∞
⁡
ε
n
=
0, where ε
n is defined by (12); that is, sequence {
x
n
} generated by (11) is T-stable.Proof.
LetR
,
H
P
,
M
,
g
,
F be the same as in Theorem 11. If (13) holds then (7) is true.
In the first place, we show that (i) is right.
Letx
∗ be a unique solution of problem (5); then we have(14)
x
∗
=
1
-
ω
n
x
∗
+
ω
n
R
G
M
,
λ
G
x
∗
+
λ
ξ
ρ
F
x
∗
,
g
x
∗
=
1
-
σ
n
x
∗
+
σ
n
R
G
M
,
λ
G
x
∗
+
λ
ξ
ρ
F
x
∗
,
g
x
∗
.From (11), (14), and (9) and Theorems 2, 4, and 11, it follows that(15)
θ
≤
x
n
+
1
⊕
x
∗
≤
1
-
ω
n
x
n
⊕
x
∗
+
ω
n
a
n
⊕
θ
+
ω
n
R
G
M
,
λ
G
y
n
+
λ
ξ
ρ
F
y
n
,
g
y
n
⊕
R
G
M
,
λ
G
x
∗
+
λ
ξ
ρ
F
x
∗
,
g
x
∗
≤
1
-
ω
n
x
n
⊕
x
∗
+
h
ω
n
y
n
⊕
x
∗
+
ω
n
a
n
⊕
θ
,where(16)
h
=
ρ
β
+
λ
μ
+
ν
φ
ξ
ρ
γ
G
λ
-
β
.Similarly, we can prove that(17)
θ
≤
y
n
⊕
x
∗
≤
1
-
σ
n
x
n
⊕
x
∗
+
σ
n
b
n
⊕
θ
+
σ
n
R
G
M
,
λ
G
y
n
+
λ
ξ
ρ
F
y
n
,
g
y
n
⊕
R
G
M
,
λ
G
x
∗
+
λ
ξ
ρ
F
x
∗
,
g
x
∗
≤
1
-
σ
n
x
n
⊕
x
∗
+
σ
n
h
x
n
⊕
x
∗
+
σ
n
b
n
⊕
θ
.It follows that(18)
θ
≤
x
n
+
1
⊕
x
∗
≤
1
-
ω
n
x
n
⊕
x
∗
+
h
ω
n
1
-
σ
n
x
n
⊕
x
∗
+
σ
n
h
x
n
⊕
x
∗
+
σ
n
b
n
+
ω
n
a
n
⊕
θ
≤
1
-
ω
n
x
n
⊕
x
∗
+
h
1
-
σ
n
x
n
⊕
x
∗
+
σ
n
h
x
n
⊕
x
∗
+
σ
n
b
n
⊕
θ
+
ω
n
a
n
⊕
θ
≤
1
-
ω
n
1
-
2
h
x
n
⊕
x
∗
+
ω
n
h
σ
n
b
n
⊕
θ
+
a
n
⊕
θ
,from (15), (16), and (17) and Theorems 2 and 4.
By assumption (13), we have 0
<
1
-
2
h
<
1 and deduce(19)
x
n
+
1
-
x
∗
≤
1
-
ω
n
1
-
2
h
N
x
n
-
x
∗
≤
+
ω
n
1
-
2
h
N
h
b
n
∨
-
b
n
+
a
n
∨
-
a
n
1
-
2
h
,for (18) and Theorem 11, and a
n
⊕
θ
=
a
n
∨
-
a
n and b
n
⊕
θ
=
b
n
∨
-
b
n.
Let(20)
η
n
=
x
n
-
x
∗
,
ζ
n
=
ω
n
1
-
2
h
N
,
χ
n
=
h
b
n
∨
-
b
n
+
a
n
∨
-
a
n
1
-
2
h
;then (20) can be written as(21)
η
n
+
1
≤
1
-
ζ
n
ξ
n
+
ζ
n
χ
n
.It follows from Lemma 13 and lim
n
→
∞
⁡
a
n
∨
-
a
n
=
lim
n
→
∞
⁡
b
n
∨
-
b
n
=
0 that ξ
n
→
0
(
n
→
∞
), and so {
x
n
} converges strongly to unique solution x
∗ of problem (5).
There is one more point; we prove (ii).
LetQ
(
x
)
=
R
G
M
,
λ
(
G
(
x
)
+
λ
ξ
/
ρ
F
(
x
,
g
(
x
)
)
) for x
∈
X
P. By (12) and Theorems 2 and 4 and (14), we obtain(22)
θ
≤
z
n
+
1
⊕
x
∗
≤
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
+
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
⊕
x
∗
≤
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
+
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
⊕
1
-
ω
n
x
∗
+
ω
n
Q
x
∗
≤
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
+
1
-
ω
n
z
n
⊕
x
∗
+
ω
n
Q
t
n
⊕
Q
x
∗
+
ω
n
a
n
⊕
θ
≤
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
⊕
θ
+
1
-
ω
n
1
-
2
h
z
n
⊕
x
∗
+
ω
n
h
σ
n
b
n
⊕
θ
+
a
n
⊕
θ
.As in the proof of inequality (19) and Theorem 11, we have(23)
z
n
+
1
-
x
∗
≤
N
z
n
+
1
-
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
+
N
1
-
ω
n
1
-
2
h
z
n
-
x
∗
+
N
ω
n
h
σ
n
b
n
∨
-
b
n
+
a
n
∨
-
a
n
≤
N
ε
n
+
N
1
-
ω
n
1
-
2
h
z
n
-
x
∗
+
N
ω
n
h
b
n
∨
-
b
n
+
a
n
∨
-
a
n
.Since 0
<
φ
≤
ω
n, by (23), we have(24)
z
n
+
1
-
x
∗
≤
1
-
ω
n
1
-
2
h
N
z
n
-
x
∗
+
1
-
2
h
ω
n
N
h
b
n
∨
-
b
n
+
a
n
∨
-
a
n
1
-
2
h
+
ε
n
φ
1
-
2
h
.Suppose that lim
n
→
∞
⁡
ε
n
=
0; we have lim
n
→
∞
⁡
z
n
=
x
∗ for ∑
n
=
0
∞
ω
n
=
∞, Theorem 3, and lim
n
→
∞
⁡
a
n
∨
-
a
n
=
lim
n
→
∞
⁡
b
n
∨
-
b
n
=
0.
Conversely, iflim
n
→
∞
⁡
z
n
=
x
∗, then, by (14) and lim
n
→
∞
⁡
a
n
∨
-
a
n
=
lim
n
→
∞
⁡
b
n
∨
-
b
n
=
0, we get(25)
θ
≤
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
≤
z
n
+
1
⊕
x
∗
+
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
⊕
x
∗
≤
z
n
+
1
⊕
x
∗
+
1
-
ω
n
z
n
⊕
x
∗
+
ω
n
Q
t
n
⊕
Q
x
∗
+
ω
n
a
n
⊕
θ
≤
z
n
+
1
⊕
x
∗
+
1
-
ω
n
1
-
2
h
z
n
⊕
x
∗
≤
+
ω
n
h
σ
n
b
n
⊕
θ
+
a
n
⊕
θ
.From (12) and Theorem 11, it follows that (26)
ε
n
=
z
n
+
1
-
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
≤
N
z
n
+
1
⊕
1
-
ω
n
z
n
+
ω
n
Q
t
n
+
ω
n
a
n
⊕
θ
≤
N
z
n
+
1
⊕
x
∗
+
N
1
-
ω
n
1
-
2
h
z
n
-
x
∗
≤
+
N
ω
n
h
b
n
∨
-
b
n
+
a
n
∨
-
a
n
;we can have(27)
lim
n
→
∞
⁡
ε
n
=
0
.Sequence {
x
n
} generated by (11) is T-stable. This completes the proof.Remark 17.
For a suitable choice of the mappingsM
,
g
,
F
,
ρ
,
ξ, we can obtain known results [9, 22] as special cases of Theorem 11.



---

*Source: 9217091-2016-09-26.xml*