Stable- Proximal Iterative Algorithms for General Mixed Variatio...
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Stable- Proximal Iterative Algorithms for General
Mixed Variational Inequalities
Xing cui
*
Department of MathematicsLiaoning Technical University, Liaoning Fuxin, China (123000)
E-mail：xingsnail@yahoo.com.cn
Abstract
This paper is mainly considered the algorithms of generalized mixed quasi-variational inequality,where
the algorithms is based on the projection methods. We presents a stable- proximal iterative algorithm
for general mixed variational inequalities which has a set-valued mapping for finding the common
element, by finding fixed point of nonexpansion mappings and the set of solutions of generalized
mixed variational inequalities. Moreover, we also discuss the convergence criteria.
Keywords: set-valued mappings; stable- proximal iterative algorithms; general mixed variational
inequalities.
1 Introduction
Variational inequality theory provides us unified, general and elegant models for studying
many problems arising in various research problems, such as mathematics, physics, economics,
engineering science, control and optimization theory.
At the present, it is very active for the research of variational inequalities on both theory and
algorithms, especially on algorithms. And now they have a lot of effective algorithms to solve a
number of models. However, variational inequality with the set value mapping which is close to
economic problem has more theoretical research than algorithms,and research on algorithms is
quite few. This paper based on the algorithm for variational inequalities with Single-valued
mapping researches variational inequalities with a set-valued mapping ,and presents Stable-
Proximal Iterative Algorithms for General Mixed Variational Inequalities.
2 Stability Results
Let
H
be a real Hilbert space. Let ,TAand
g
be mapping from
H
into itself. Let V be a
set-valued mapping from
H
into 2
H
.Given : H
ϕ
→ {}R
+
∞∪ a proper convex and lower
semi-continuous function, considering the Problem of finding
,()
x
Hy Vx
∈
∈ ， such that
,()(())0,Tx Ay z x z g x z H
ϕ
ϕ
〈+ −〉+ − ≥ ∈ （1）
which is called a general mixed variational inequality problem.
Defnition 1
[1]
If A is a maximal monotone operator, the resolvent operator is defined as
HxxAIxJ
A
∈∀+=
−
),()()(
1
ρ
,
and
0>
ρ
is a Constant,
I
is an identity operator.
Defnition 2
[2]
:VH→ 2
H
is
H
- Lipschitz continuous with constant 0
η
> if for
every
(,)
H
Tu Tv u v
η
≤
−
·
Defnition 3
[1]
:
g
HH→ is Lipschitz continuous with constant 0u > if for every
,uv H∈
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