# Existence Theorems ofε-Cone Saddle Points for Vector-Valued Mappings

Authors: Tao Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

A new existence result ofε-vector equilibrium problem is first obtained. Then, by using the existence theorem of ε-vector equilibrium problem, a weakly ε-cone saddle point theorem is also obtained for vector-valued mappings.


---

## Body

## 1. Introduction

Saddle point problems are important in the areas of optimization theory and game theory. As for optimization theory, the main motivation of studying saddle point has been their connection with characterized solutions to minimax dual problems. Also, as for game theory, the main motivation has been the determination of two-person zero-sum games based on the minimax principle.In recent years, based on the development of vector optimization, a great deal of papers have been devoted to the study of cone saddle points problems for vector-valued mappings and set-valued mappings, such as [1–8]. Nieuwenhuis [5] introduced the notion of cone saddle points for vector-valued functions in finite-dimensional spaces and obtained a cone saddle point theorem for general vector-valued mappings. Gong [2] established a strong cone saddle point theorem of vector-valued functions. Li et al. [4] obtained an existence theorem of lexicographic saddle point for vector-valued mappings. Bigi et al. [1] obtained a cone saddle point theorem by using an existence theorem of a vector equilibrium problem. Zhang et al. [9] established a general cone loose saddle point for set-valued mappings. Zhang et al. [8] obtained a minimax theorem and an existence theorem of cone saddle points for set-valued mappings by using Fan-Browder fixed point theorem. Some other types of existence results can be found in [3, 10–18].On the other hand, in some situations, it may not be possible to find an exact solution for an optimization problem, or such an exact solution simply does not exist, for example, if the feasible set is not compact. Thus, it is meaningful to look for an approximate solution instead. There are also many papers to investigate the approximate solution problem, such as [19–21]. Kimura et al. [20] obtained several existence results for ε-vector equilibrium problem and the lower semicontinuity of the solution mapping of ε-vector equilibrium problem. Anh and Khanh [19] have considered two kinds of solution sets to parametric generalized ε-vector quasiequilibrium problems and established the sufficient conditions for the Hausdorff semicontinuity (or Berge semicontinuity) of these solution mappings. X. B. Li and S. J. Li [21] established some semicontinuity results on ε-vector equilibrium problem.The aim of this paper is to characterize theɛ-cone saddle point of vector-valued mappings. For this purpose, we first establish an existence theorem for ε-vector equilibrium problem. Then, by this existence result, we obtain an existence theorem for ε-cone saddle point of vector-valued mappings.


## 2. Preliminaries

LetX be a real Hausdorff topological vector space and let V be a real local convex Hausdorff topological vector space. Assume that S is a pointed closed convex cone in V with nonempty interior int
⁡
S
≠
∅. Let V
* be the topological dual space of V. Denote the dual cone of S by S
*:
(1)
S
*
=
{
s
*
∈
V
*
:
s
*
(
s
)
≥
0
,
∀
s
∈
S
}
.
Note that from Lemma 3.21 in [22] we have
(2)
z
∈
S
⟺
{
〈
z
*
,
z
〉
≥
0
,
∀
z
*
∈
S
*
}
,
z
∈
int
⁡
S
⟺
{
〈
z
*
,
z
〉
>
0
,
∀
z
*
∈
S
*
∖
{
0
}
}
.Definition 1 (see[7, 23]).
Letf
:
X
→
V be a vector-valued mapping. f is said to be S-upper semicontinuous on X if and only if, for each x
∈
X and any s
∈
int
⁡
S, there exists an open neighborhood U
x of x such that
(3)
f
(
u
)
∈
f
(
x
)
+
s
-
int
⁡
S
,
∀
u
∈
U
x
.
fis said to be S-lower semicontinuous on X if and only if -
f is S-upper semicontinuous on X.Lemma 2 (see [17]).
Letf
:
X
×
X
→
V be a vector-valued mapping and s
*
∈
S
*
∖
{
0
}. If f is S-lower semicontinuous, then s
*
∘
f is lower semicontinuous.Definition 3 (see[24]).
LetA and B be nonempty subsets of X and f
:
A
×
B
→
V be a vector-valued mapping.(i)
fis said to be S-concavelike in its first variable on A if and only if, for all x
1
,
x
2
∈
A and l
∈
[
0,1
], there exists x
¯
∈
A such that
(4)
f
(
x
¯
,
y
)
∈
l
f
(
x
1
,
y
)
+
(
1
-
l
)
f
(
x
2
,
y
)
+
S
,
∀
y
∈
B
.
(ii)
f is said to be S-convexlike in its second variable on B if and only if, for all y
1
,
y
2
∈
B and l
∈
[
0,1
], there exists y
¯
∈
B such that
(5)
f
(
x
,
y
¯
)
∈
l
f
(
x
,
y
1
)
+
(
1
-
l
)
f
(
x
,
y
2
)
-
S
,
∀
x
∈
A
.
(iii)
f is said to be S-concavelike-convexlike on A
×
B if and only if f is S-concavelike in its first variable and S-convexlike in its second variable.Definition 4.
LetA
⊂
V be a nonempty subset and ε
∈
int
⁡
S.(i)
A pointz
∈
A is said to be a weak  ε-minimal point of A if and only if A
∩
(
z
-
ε
-
int
⁡
S
)
=
∅ and Min
⁡


ε
A denotes the set of all weak ε-minimal points of A.
(ii)
A pointz
∈
A is said to be a weak  ε-maximal point of A if and only if A
∩
(
z
+
ɛ
+
int
⁡
S
)
=
∅ and Max
⁡


ε
A denotes the set of all weak  ε-maximal points of A.Definition 5.
Letf
:
A
×
B
→
V be a vector-valued mapping and ε
∈
int
⁡
S. A point (
a
,
b
)
∈
A
×
B is said to be a weak  ε-S-saddle point of f on A
×
B if
(6)
f
(
a
,
b
)
∈
Max
⁡


ε
f
(
A
,
b
)
⋂
‍
Min
⁡


ε
f
(
a
,
B
)
.


## 3. Existence ofε-Vector Equilibrium Problem

In this section, we deal with the followingɛ-vector equilibrium problem (for short VAEP). Find x
¯
∈
E such that
(7)
f
(
x
,
y
)
+
ε
∉

-
int
⁡
S
,
∀
y
∈
E
,
where f
:
X
×
X
→
V is a vector-valued mapping, E is a nonempty subset of X
, and ε
∈
int
⁡
S.Iff
(
x
,
y
)
=
g
(
y
)
-
g
(
x
), x
,
y
∈
E, and if  x
¯
∈
E is a solution of VAEP, then x
¯
∈
E is a solution of ε-vector optimization of g, where g is a vector-valued mapping.Denote theε-solution set of (VAEP) by
(8)
S
(
ɛ
)
∶
=
{
x
¯
∈
E
:
f
(
x
,
y
)
+
ε
∉
-
int
⁡
S
,
∀
y
∈
E
}
.Lemma 6 (see [20]).
LetE be a nonempty subset of X. Suppose that f
:
X
×
X
→
V is a vector-valued mapping and the following conditions are satisfied:(i)
cl
⁡
E is a compact set;
(ii)
{
x
∈
cl
⁡
E
:
f
(
x
,
y
)
∉
-
int
⁡
S
,
∀
y
∈
cl
⁡
E
}
≠
∅;
(iii)
f is S-lower semicontinuous on cl
⁡
E
×
cl
⁡
E.Then, for each ε
∈
int
⁡
S, S
(
ɛ
)
≠
∅.Next, we give a sufficient condition for the condition (ii) in Lemma6.Lemma 7.
LetE be a nonempty subset of X. Suppose that f
:
X
×
X
→
V is a vector-valued mapping with f
(
x
,
x
)
=
0 for all x
∈
X and the following conditions are satisfied:(i)
cl
⁡
E is a compact set;
(ii)
f is S-concavelike-convexlike on cl
⁡
E
×
cl
⁡
E;
(iii)
for eachx
∈
cl
⁡
E, f
(
x
,
·
) is S-lower semicontinuous on cl
⁡
E.Then, there exists x
¯
∈
cl
⁡
E such that
(9)
f
(
x
¯
,
y
)
∉
-
int
⁡
S
,
∀
y
∈
cl
⁡
E
.Proof.
For anyt
<
0 and s
*
∈
S
*
∖
{
0
}, we define a multifunction G
:
cl
⁡
E
→
2
cl
⁡
E by
(10)
G
(
x
)
=
{
y
∈
cl
⁡
E
:
s
*
(
f
(
x
,
y
)
)
≤
t
}
,
∀
x
∈
cl
⁡
E
.
First, by assumptions, we must have(11)
⋂
x
∈
cl
⁡
E
‍
G
(
x
)
=
∅
.
In fact, if there exists y
¯
∈
cl
⁡
E such that y
¯
∈
G
(
x
), for all x
∈
cl
⁡
E, then
(12)
s
*
(
f
(
x
,
y
¯
)
)
≤
t
,
∀
x
∈
cl
⁡
E
.
Particularly, taking x
=
y
¯, we have 0
=
s
*
(
f
(
y
¯
,
y
¯
)
)
≤
t, which contradicts the assumption about t.
Then, by Lemma2, G
(
x
) is a closed set, for each x
∈
cl
⁡
E. By (11), for any y
∈
cl
⁡
E, we have
(13)
y
∈
V
∖
⋂
x
∈
cl
⁡
E
‍
G
(
x
)
=
⋃
x
∈
cl
⁡
E
‍
V
∖
G
(
x
)
.
Since cl
⁡
E is compact, there exists a finite point set {
x
1
,
x
2
,
…
,
x
n
} in cl
⁡
E such that
(14)
cl
⁡
E
⊂
⋃
1
≤
i
≤
n
‍
V
∖
G
(
x
i
)
.
Namely, for each y
∈
cl
⁡
E, there exists i
∈
{
1,2
,
…
,
n
} such that
(15)
s
*
(
f
(
x
i
,
y
)
)
>
t
.
Now, we consider the set
(16)
M
∶
=
{
(
z
1
,
z
2
,
…
,
z
n
,
r
)
∈
R
n
+
1
∣
∃
y
∈
cl
⁡
E
,
e
s
*
(
f
(
x
i
,
y
)
)
≤
r
+
z
i
,
∀
i
=
1,2
,
…
,
n
(
z
1
,
z
2
,
…
,
z
n
,
r
)
∈
R
n
+
1
∣
∃
y
∈
cl
⁡
E
,
}
.
Obviously, by the condition (ii), M is a convex set. By (15), we have the fact that (
0
R
n
,
t
)
∉
M.
By the separation theorem of convex sets, there exists(
λ
1
,
λ
2
,
…
,
λ
n
,
r
¯
)
≠
0
R
n such that
(17)
∑
i
=
1
n
‍
λ
i
z
i
+
r
¯
r
≥
r
¯
t
,
∀
(
z
1
,
z
2
,
…
,
z
n
,
r
)
∈
M
.
Since M
+
R
n
+
1
⊂
M, we can get λ
i
≥
0  and  r
¯
≥
0, for all i
=
1,2
,
…
,
n. By the definition of M, for each y
∈
cl
⁡
E,
(18)
(
0
R
n
,
1
+
max
⁡
1
≤
i
≤
n
s
*
(
f
(
x
i
,
y
)
)
)
∈
int
⁡
M
,
(19)
(
s
*
(
f
(
x
1
,
y
)
)
-
r
,
w
s
*
(
f
(
x
2
,
y
)
)
-
r
,
…
,
s
*
(
f
(
x
n
,
y
)
)
-
r
,
r
)
∈
M
.
By (18), r
¯
>
0. Then, by (17) and (19),
(20)
∑
i
=
1
n
‍
λ
i
r
¯
s
*
(
f
(
x
i
,
y
)
)
+
r
(
1
-
∑
i
=
1
n
‍
λ
i
r
¯
)
≥
t
.
By (20), ∑
i
=
1
n
‍
(
λ
i
/
r
¯
)
=
1. Thus, by the condition (ii), for each y
∈
cl
⁡
E, there exists x
¯
∈
cl
⁡
E such that
(21)
s
*
(
f
(
x
¯
,
y
)
)
≥
∑
i
=
1
n
‍
λ
i
r
¯
s
*
(
f
(
x
i
,
y
)
)
≥
t
.
By the assumption about t and s
*, there exists x
¯
∈
cl
⁡
E such that
(22)
f
(
x
¯
,
y
)
∉
-
int
⁡
S
,
∀
y
∈
cl
⁡
E
.
This completes the proof.By Lemmas6 and 7, we can get the following result.Theorem 8.
LetE be a nonempty subset of X. Suppose that f
:
X
×
X
→
V is a vector-valued mapping with f
(
x
,
x
)
=
0 for all x
∈
X and the following conditions are satisfied:(i)
cl
⁡
E is a compact set;
(ii)
f is S-concavelike-convexlike on cl
⁡
E
×
cl
⁡
E;
(iii)
f is S-lower semicontinuous on cl
⁡
E
×
cl
⁡
E.Then, for each ε
∈
int
⁡
S, S
(
ε
)
≠
∅.Remark 9.
Note that the condition (i) does not require the fact thatcl
⁡
E is a convex set. So Theorem 8 is different from Theorem 3.2 in [20]. The following example explains this case.Example 10.
LetX
=
R, V
=
R
2, and E
=
[
0
,
1
/
3
]
∪
[
2
/
3
,
1
],
(23)
f
(
x
,
y
)
=
{
(
x
y
,
x
z
)
∈
R
2
∣
z
=
1
-
y
2
}
,
x
,
y
∈
X
,
S
=
{
(
x
,
y
)
∈
R
2
∣
x
≥
0
,
y
≥
0
}
.
Obviously, cl
⁡
E is a compact set. However, cl
⁡
E is not a convex set. So, Theorem 3.2 in [20] is not applicable. By the definition of f, f is S-concavelike-convexlike on cl
⁡
E
×
cl
⁡
E and S-lower semicontinuous on cl
⁡
E
×
cl
⁡
E. Thus, all conditions of Theorem 8 hold. Indeed, for each ɛ
=
(
ε
1
,
ε
2
)
∈
int
⁡
S,
(24)
f
(
0
,
y
)
+
ε
=
(
ε
1
,
ε
2
)
∉
-
int
⁡
S
,
∀
y
∈
E
.
Namely, 0
∈
S
(
ε
).


## 4. Existence ofε-Cone Saddle Points

Lemma 11.
LetE be a nonempty subset of X and E
=
A
×
B. Let ε
∈
int
⁡
S and let f
:
X
×
X
→
V be a vector-valued mapping with f
(
x
,
y
)
=
g
(
a
,
v
)
-
g
(
u
,
b
), where x
=
(
a
,
b
), y
=
(
u
,
v
), a
,
u
∈
A, and v
,
b
∈
B. If there exists x
¯
=
(
a
¯
,
b
¯
)
∈
E such that
(25)
f
(
x
¯
,
y
)
+
ε
∉
-
int
⁡
S
,
∀
y
∈
E
,
then (
a
¯
,
b
¯
)
∈
A
×
B is a weak  ε-S-saddle point of g on A
×
B.Proof.
By assumptions, we have(26)
f
(
x
¯
,
y
)
+
ɛ
∉
-
int
⁡
S
,
∀
y
∈
E
.
Then,
(27)
g
(
a
¯
,
v
)
-
g
(
u
,
b
¯
)
+
ε
∉
-
int
⁡
S
,
∀
(
u
,
v
)
∈
A
×
B
.
By (27), taking u
=
a
¯,
(28)
g
(
a
¯
,
v
)
-
g
(
a
¯
,
b
¯
)
+
ε
∉
-
int
⁡
S
,
∀
v
∈
B
,
which implies g
(
a
¯
,
b
¯
)
∈
Min
⁡
ε
g
(
a
¯
,
B
). Then, by (27), taking v
=
b
¯,
(29)
g
(
a
¯
,
b
¯
)
-
g
(
u
,
b
¯
)
+
ε
∉
-
int
⁡
S
,
∀
u
∈
A
,
which implies g
(
a
¯
,
b
¯
)
∈
Max
⁡
ε
g
(
A
,
b
¯
). Thus, (
a
¯
,
b
¯
)
∈
A
×
B is a weak  ε-S-saddle point of g on A
×
B. This completes the proof.Theorem 12.
LetA and B be nonempty sets and ε
∈
int
⁡
S. Suppose that g is a vector-valued mapping and the following conditions are satisfied:(i)
cl
⁡
A and cl
⁡
B are compact sets;
(ii)
g is S-concavelike-convexlike on cl
⁡
A
×
cl
⁡
B;
(iii)
g is S-upper semicontinuous on cl
⁡
A
×
cl
⁡
B;
(iv)
g is S-lower semicontinuous on cl
⁡
A
×
cl
⁡
B.Then, g has a weak  ε-S-saddle point on A
×
B.Proof.
LetA
×
B
=
E and f
:
cl
⁡
E
×
cl
⁡
E
→
V be a vector-valued mappings by
(30)
f
(
x
,
y
)
=
g
(
a
,
v
)
-
g
(
u
,
b
)
,
∀
x
=
(
a
,
b
)
∈
cl
⁡
E
,
y
=
(
u
,
v
)
∈
cl
⁡
E
.
Next, we show that all assumptions of Theorem 8 are satisfied by g.
Clearly, by the condition (i),cl
⁡
E is compact. Then, by the condition (ii), we have the fact that, for each a
1
,
a
2
∈
cl
⁡
A and l
∈
[
0,1
], there exists a
3
∈
cl
⁡
A such that
(31)
g
(
a
3
,
b
)
∈
l
g
(
a
1
,
b
)
+
(
1
-
l
)
g
(
a
2
,
b
)
+
S
,
∀
b
∈
cl
⁡
B
and, for each b
1
,
b
2
∈
cl
⁡
B and l
∈
[
0,1
], there exists b
3
∈
cl
⁡
B
(32)
g
(
a
,
b
3
)
∈
l
g
(
a
,
b
1
)
+
(
1
-
l
)
g
(
a
,
b
2
)
-
S
,
∀
a
∈
cl
⁡
A
.
By (31) and (32), for each (
a
1
,
b
1
)
,
(
a
2
,
b
2
)
∈
cl
⁡
E and l
∈
[
0,1
], there exists (
a
3
,
b
3
)
∈
cl
⁡
E such that
(33)
g
(
a
3
,
b
)
-
g
(
a
,
b
3
)
∈
l
(
g
(
a
1
,
b
)
-
g
(
a
,
b
1
)
)
(
g
(
a
1
,
b
)
-
g
(
a
,
b
1
)
)
+
(
1
-
l
)
(
g
(
a
2
,
b
)
-
g
(
a
,
b
2
)
)
+
S
,
(
g
(
a
1
,
b
)
-
g
(
a
,
b
1
)
)
(
g
(
i
a
1
,
b
)
-
g
(
,
b
1
)
)
∀
(
a
,
b
)
∈
cl
⁡
E
,
g
(
a
,
b
3
)
-
g
(
a
3
,
b
)
∈
l
(
g
(
a
,
b
1
)
-
g
(
a
1
,
b
)
)
(
g
(
a
,
b
1
)
-
g
(
a
1
,
b
)
)
+
(
1
-
l
)
(
g
(
a
,
b
2
)
-
g
(
a
2
,
b
)
)
-
S
,
(
g
(
a
,
b
1
)
-
g
(
a
1
,
b
)
)
(
g
(
,
i
b
1
)
-
g
(
a
1
,
b
)
)
∀
(
a
,
b
)
∈
cl
⁡
E
.
Namely, f is S-concavelike-convexlike on cl
⁡
E
×
cl
⁡
E.
Now, we show thatf is S-lower semicontinuous on cl
⁡
E
×
cl
⁡
E. By the condition (iii), for each (
a
,
v
)
∈
cl
⁡
A
×
cl
⁡
B and s
∈
int
⁡
S, there exists an open neighborhood U
a of a and U
v of v such that
(34)
g
(
u
a
,
u
v
)
∈
g
(
a
,
v
)
-
s
2
+
int
⁡
S
,
∀
u
a
∈
U
a
,
u
v
∈
U
v
,
and, for each (
u
,
b
)
∈
cl
⁡
A
×
cl
⁡
B and s
∈
int
⁡
S, there exists an open neighborhood U
u of u and U
b of b such that
(35)
g
(
u
u
,
u
b
)
∈
g
(
u
,
b
)
+
s
2
-
int
⁡
S
,
∀
u
u
∈
U
u
,
u
b
∈
U
b
.
By (34) and (35), we have the fact that, for any (
(
a
,
b
)
,
(
u
,
v
)
)
∈
cl
⁡
E
×
cl
⁡
E,
(36)
g
(
u
a
,
u
v
)
-
g
(
u
u
,
u
b
)
∈
g
(
a
,
v
)
-
g
(
u
,
b
)
-
s
+
int
⁡
S
,
∀
(
(
u
a
,
u
b
)
,
(
u
u
,
u
v
)
)
∈
U
a
×
U
b
×
U
u
×
U
v
.
Namely, f is S-lower semicontinuous on cl
⁡
E
×
cl
⁡
E. Therefore, by Lemma 11, g has a weak  ε-S-saddle point on A
×
B. This completes the proof.Remark 13.
The conditions (iii) and (iv) of Theorem12 do not imply that g is continuous (see [23]).



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*Source: 478486-2014-02-25.xml*