# Extendedf-Vector Equilibrium Problem

Authors: Khushbu; Zubair Khan
Journal: International Journal of Computational Mathematics
 (2014)

---

## Abstract

We introduce and study extendedf-vector equilibrium problem. By using KKM-Fan Theorem as basic tool, we prove existence theorem in the setting of Hausdorff topological vector space and reflexive Banach space. Some examples are also given.


---

## Body

## 1. Introduction

Equilibrium problems have been extensively studied in recent years; the origin of this can be traced back to Blum and Oettli [1] and Noor and Oettli [2]. The equilibrium problem is a generalization of classical variational inequalities and provides us with a systematic framework to study a wide class of problems arising in finance, economics, operations research, and so forth. General equilibrium problems have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. Vector equilibrium problems have attracted increasing interest of many researchers and provide a unified model for several classes of problems, for example, vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems; see [1–4] and references therein. Many existence results for vector equilibrium problems have been established by several eminent researchers; see, for example, [5–13].The generalized monotonicity plays an important role in the literature of equilibrium problems and variational inequalities. There are a substantial number of papers on existence results for solving equilibrium problems and variational inequalities based on different monotonicity notions such as monotonicity, pseudomonotonicity, and quasimonotonicity.LetX and Y be two Hausdorff topological vector spaces, let K be a nonempty, closed, and convex subset of X, and let C be a pointed, closed, convex cone in Y with int
⁡
C
≠
∅. Given a vector-valued mapping f
:
K
×
K
→
Y, the vector equilibrium problem consists of finding x
-
∈
K such that
(1)
f
x
-
,
y
∉
-
int
⁡
C
,
∀
y
∈
K
.Inspired by the concept of monotonicity, KKM-Fan Theorem, and the other work done in the direction of generalization of vector equilibrium problems (see [14–16]) we introduce and study extended f-vector equilibrium problem and prove some existence results in the setting of Hausdorff topological vector spaces and reflexive Banach spaces.


## 2. Preliminaries

The following definitions and concepts are needed to prove the results of this paper.Definition 1.
The Hausdorff topological vector spaceY is said to be an ordered space denoted by Y
,
C if ordering relations are defined in Y by a pointed, closed, convex cone C of Y as follows:
(2)
∀
x
,
y
∈
Y
,
y
≦
x
⟺
x
-
y
∈
C
,
∀
x
,
y
∈
Y
,
y
≤
x
⟺
x
-
y
∈
C
∖
0
,
∀
x
,
y
∈
Y
,
y
≰
x
⟺
x
-
y
∉
C
∖
0
.
If the interior of C is int
⁡
C
≠
∅, then the weak ordering relations in Y are also defined as follows:
(3)
∀
x
,
y
∈
Y
,
y
<
x
⟺
x
-
y
∈
int
⁡
C
,
∀
x
,
y
∈
Y
,
y
≮
x
⟺
x
-
y
∉
int
⁡
C
.
Throughout this paper, unless otherwise specified, we assume that Y
,
C is an ordered Hausdorff topological vector space with int
⁡
C
≠
∅.Definition 2.
LetK be a nonempty convex subset of a topological vector space X. A set-valued mapping A
:
K
→
2
X is said to be KKM-mapping, if, for each finite subset x
1
,
x
2
,
.
.
.
,
x
n of K, Co
x
1
,
x
2
,
.
.
.
,
x
n
⊆
⋃
i
=
1
n
A
x
i, where Co
x
1
,
x
2
,
.
.
.
,
x
n denotes the convex hull of x
1
,
x
2
,
.
.
.
,
x
n.The following KKM-Fan Theorem is important for us to prove the existence results of this paper.Theorem 3 (KKM-Fan Theorem).
LetK be a nonempty convex subset of a Hausdorff topological vector space X and let A
:
K
→
2
X be a KKM-mapping such that A
x is closed for all x
∈
K and A
x is compact for at least one x
∈
K; then
(4)
⋂
x
∈
K
A
x
≠
∅
.Lemma 4.
LetY
,
C be an ordered topological vector space with a pointed, closed, and convex cone C. Then for all x
,
y
∈
Y, one has the following:(i)
y
-
x
∈
int
⁡
C and y
∉
int
⁡
C imply x
∉
int
⁡
C;
(ii)
y
-
x
∈
C and y
∉
int
⁡
C imply x
∉
int
⁡
C;
(iii)
y
-
x
∈
-
int
⁡
C and y
∉
-
int
⁡
C imply x
∉
-
int
⁡
C;
(iv)
y
-
x
∈
-
C and y
∉
-
int
⁡
C imply x
∉
-
int
⁡
C.
LetX and Y be the Hausdorff topological vector spaces and let K be a nonempty, closed, convex subset of X. Let C be a pointed, closed, convex cone in Y with int
⁡
C
≠
∅. Let g
:
K
×
X
→
Y be a vector-valued mapping and let f
:
K
→
X be another mapping. We introduce the following extended f-vector equilibrium problem.
Findx
0
∈
K such that for all z
∈
K and λ
∈
(
0,1
]
(5)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
∈
K
.
Ifλ
=
1 and f
y
=
y
∈
K, the problem (5) reduces to the vector equilibrium problem of finding x
0
∈
K such that
(6)
g
x
0
,
y
∉
-
int
⁡
C
,
∀
y
∈
K
.
Problem (6) was introduced and studied by Xuan and Nhat [13].
In addition, ifY
=
R and C
=
R
+, then problem (5) reduces to the equilibrium problem of finding x
0
∈
K such that
(7)
g
x
0
,
y
≥
0
,
∀
y
∈
K
.
Problem (7) was introduced and studied by Blum and Oettli [1].
The fact that problem (5) is much more general than many existing equilibria, vector equilibrium problems, and so forth motivated us to study extended f-vector equilibrium problem given by (5).Definition 5.
LetX and Y be the Hausdorff topological vector spaces and let K be a nonempty, closed, convex subset of X and let C be a pointed, closed, convex cone in Y with int
⁡
C
≠
∅. Let g
:
K
×
X
→
Y and f
:
K
→
X be mappings. Then, g is said to be(i)
f-monotone with respect to C, if and only if, for all x
,
y
,
z
∈
K, λ
∈
(
0,1
],
(8)
g
λ
x
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
∈
-
C
;
(ii)
f-hemicontinuous, if and only if, for all x
,
y
∈
K, t
∈
0,1, the mapping t
→
g
t
y
+
1
-
t
x
,
f
x is continuous at 0
+;
(iii)
f-pseudomonotone, if and only if, for all x
,
y
,
z
∈
K, λ
∈
(
0,1
],
(9)
g
λ
x
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
implies
g
λ
y
+
1
-
λ
z
,
f
x
∉
int
⁡
C
;
(iv)
f-generally convex, if and only if, for all x
,
y
,
z
,
∈
K, λ
∈
(
0,1
],
(10)
g
z
,
f
x
∉
-
int
⁡
C
,
g
z
,
f
y
∉
-
int
⁡
C
imply
g
z
,
f
λ
x
+
1
-
λ
y
∉
-
int
⁡
C
.
In support of Definition5, we have the following examples.Example 6.
LetX
=
R, K
=
R
+, Y
=
R
2, and C
=
x
,
y
:
x
≤
0
,
y
≤
0.
Letg
:
K
×
X
→
Y andf
:
K
→
X be mappings such that
(11)
g
x
,
y
=
y
,
x
2
,
∀
x
,
y
∈
K
,
f
x
=
x
2
,
∀
x
∈
K
.
Then,
(12)
g
λ
x
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
=
f
y
,
λ
x
+
1
-
λ
z
2
+
f
x
,
λ
y
+
1
-
λ
z
2
=
y
2
,
λ
x
+
1
-
λ
z
2
+
x
2
,
λ
y
+
1
-
λ
z
2
∈
-
C
;
that is, g
λ
x
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
∈
-
C.
Hence,g is f-monotone with respect to C.Example 7.
LetX
=
R, K
=
R
+, Y
=
R
2, and C
=
x
,
y
:
x
≤
0
,
y
≤
0. Let F
:
0,1
→
Y be a mapping such that
(13)
F
t
=
g
t
y
+
1
-
t
x
,
f
x
,
∀
t
∈
0,1
.
Let g
:
K
×
X
→
Y and f
:
K
→
X be mappings such that
(14)
g
x
,
y
=
y
,
x
2
,
∀
x
,
y
∈
K
,
f
x
=
x
2
,
∀
x
∈
K
.
Then,
(15)
F
t
=
g
t
y
+
1
-
t
x
,
f
x
=
f
x
,
t
y
+
1
-
t
x
2
=
x
2
,
t
y
+
1
-
t
x
2
which implies that t
→
g
t
y
+
1
-
t
x
,
f
x is continuous at 0
+. Hence, g is f-hemicontinuous.Example 8.
LetX
=
R, K
=
R
+, Y
=
R
2, and C
=
x
,
y
:
x
≥
0
,
y
≥
0.
Letg
:
K
×
X
→
Y and f
:
K
→
X be mappings such that
(16)
g
x
,
y
=
y
,
-
x
2
,
∀
x
,
y
∈
K
,
f
x
=
m
x
,
∀
x
∈
K
,
m
is
constant
.
Then,
(17)
g
λ
x
+
1
-
λ
z
,
f
y
=
f
y
,
-
λ
x
+
1
-
λ
z
2
=
m
y
,
-
λ
x
+
1
-
λ
z
2
∉
-
int
⁡
C
implies m
>
0, so it follows that
(18)
g
λ
y
+
1
-
λ
z
,
f
x
=
f
x
,
-
λ
y
+
1
-
λ
z
2
=
m
x
,
-
λ
x
+
1
-
λ
z
2
∉
int
⁡
C
.
Hence, g is f-pseudomonotone with respect to C.Example 9.
LetX
=
R, K
=
R
+, Y
=
R
2, and C
=
x
,
y
:
x
≥
0
,
y
≤
0.
Letg
:
K
×
X
→
Y and f
:
K
→
X be mappings such that
(19)
g
x
,
y
=
y
,
x
2
,
∀
x
,
y
∈
K
,
f
x
=
m
x
,
∀
x
∈
K
,
m
is
constant
.
Then,
(20)
g
z
,
f
x
=
f
x
,
z
2
=
m
x
,
z
2
∉
-
int
⁡
C
implies m
>
0, and
(21)
g
z
,
f
y
=
f
y
,
z
2
=
m
y
,
z
2
∉
-
int
⁡
C
again implies m
>
0, so it follows that
(22)
g
z
,
f
λ
x
+
1
-
λ
y
=
f
λ
x
+
1
-
λ
y
,
z
2
=
m
λ
x
+
1
-
λ
y
,
z
2
=
λ
m
x
+
1
-
λ
m
y
,
z
2
∉
-
int
⁡
C
.
Hence, g is f-generally convex.Definition 10.
A mappingη
:
K
×
K
→
X is said to be affine in the first argument if and only if, for all x
,
y
,
z
∈
K and t
∈
0,1,
(23)
η
t
x
+
1
-
t
y
,
z
=
t
η
x
,
z
+
1
-
t
η
y
,
z
.
Similarly, one can define the affine property of η with respect to the second argument.Definition 11.
LetY
,
C be an ordered topological vector space. A mapping T
:
X
→
Y is called C-convex if and only if, for each pair x
,
y
∈
K and λ
∈
(
0,1
],
(24)
T
λ
x
+
1
-
λ
y
≤
C
λ
T
x
+
1
-
λ
T
y
.


## 3. Existence Results

We prove the following equivalence lemma which we need for the proof of our main results.Lemma 12.
LetX be a Hausdorff topological vector space, let K be a closed, convex subset of X, and let Y
,
C be an ordered Hausdorff topological vector space with int
⁡
C
≠
∅. Let g
:
K
×
X
→
Y be a vector-valued mapping which is C-convex in the second argument, f-monotone with respect to C, positive homogeneous in the second argument, andf-hemicontinuous and let f
:
K
→
X be a continuous, affine mapping such that, for z
∈
K, λ
∈
(
0,1
], g
λ
x
+
1
-
λ
z
,
f
x
=
0 for all x
∈
K. Then for all z
∈
K and λ
∈
(
0,1
], the following statements are equivalent. Find x
0
∈
K such that (i)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C, ∀
y
∈
K;
(ii)
g
λ
y
+
1
-
λ
z
,
f
x
0
∉
int
⁡
C, ∀
y
∈
K.Proof.
(i)⇒(ii). For all z
∈
K and λ
∈
(
0,1
], let x
0 be a solution of (i); then we have
(25)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
.
Since g is f-monotone with respect to C, we have
(26)
g
λ
x
0
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
0
∈
-
C
⟹
g
λ
x
0
+
1
-
λ
z
,
f
y
∈
-
C
-
g
λ
y
+
1
-
λ
z
,
f
x
0
.
Suppose to the contrary that (ii) is false. Then, there exists y
∈
K such that
(27)
g
λ
y
+
1
-
λ
z
,
f
x
0
∈
int
⁡
C
.
By (26), we obtain
(28)
g
λ
x
0
+
1
-
λ
z
,
f
y
∈
-
C
-
int
⁡
C
⊂
-
int
⁡
C
which contradicts (i). Thus, (i) ⇒(ii).
(ii)⇒(i). Conversely, suppose that (ii) holds. Let y
∈
K be arbitrary and taking y
α
=
α
y
+
1
-
α
x
0, α
∈
(
0,1
) so y
α
∈
K as K is convex.
Therefore,(29)
g
λ
y
α
+
1
-
λ
z
,
f
x
0
∉
int
⁡
C
⟹
1
-
α
g
λ
y
α
+
1
-
λ
z
,
f
x
0
∉
int
⁡
C
.
As g
λ
x
+
1
-
λ
z
,
f
x
=
0, we have
(30)
g
λ
y
α
+
1
-
λ
z
,
f
y
α
=
0
⟹
g
λ
y
α
+
1
-
λ
z
,
f
α
y
+
1
-
α
x
0
=
0
.
Since f is affine, we have
(31)
g
λ
y
α
+
1
-
λ
z
,
α
f
y
+
1
-
α
f
x
0
=
0
.
Since g is C-convex in the second argument, therefore we have
(32)
0
=
g
λ
y
α
+
1
-
λ
z
,
α
f
y
+
1
-
α
f
x
0
≤
C
α
g
λ
y
α
+
1
-
λ
z
,
f
y
+
1
-
α
g
λ
y
α
+
1
-
λ
z
,
f
x
0
⟹
1
-
α
g
λ
y
α
+
1
-
λ
z
,
f
x
0
+
α
g
λ
y
α
+
1
-
λ
z
,
f
y
∈
C
.
By (29), as 1
-
α
g
λ
y
α
+
1
-
λ
z
,
f
x
0
∉
int
⁡
C and using Lemma 4, we have from (32)(33)
-
α
g
λ
y
α
+
1
-
λ
z
,
f
y
∉
int
⁡
C
⟹
g
λ
y
α
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
.
Since g is f-hemicontinuous, therefore, for α
→
0
+, we have
(34)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
∈
K
.
Thus, (ii)⇒(i). This completes the proof.Theorem 13.
LetX be a Hausdorff topological vector space and let K be a compact and convex subset of X and let Y
,
C be an ordered Hausdorff topological vector space with int
⁡
C
≠
∅. Let g
:
K
×
X
→
Y be a vector-valued, affine mapping which is f-monotone with respect to C, positive homogeneous in the second argument, and f-hemicontinuous and let f
:
K
→
X be a continuous, affine mapping such that, for z
∈
K, λ
∈
(
0,1
], g
λ
x
+
1
-
λ
z
,
f
x
=
0 for all x
∈
K and let the mapping x
→
g
λ
y
+
1
-
λ
x
,
f
x be continuous. Then problem (5) admits a solution; that is, for all z
∈
K and λ
∈
(
0,1
], there exists x
0
∈
K such that
(35)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
∈
K
.Proof.
Fory
∈
K, we define
(36)
M
y
=
x
∈
K
:
g
λ
x
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
S
y
=
x
∈
K
:
g
λ
y
+
1
-
λ
z
,
f
x
∉
int
⁡
C
.
Clearly M
y
≠
∅, as y
∈
M
y. We divide the proof into three steps.
Step
1. We claim that M
:
K
→
2
K is a KKM-mapping. If M is not a KKM-mapping, then there exists u
∈
Co
y
1
,
y
2
,
…
,
y
n such that, for all t
i
∈
0,1,  i
=
1,2
,
…
,
n with ∑
i
=
1
n
t
i
=
1, we have
(37)
u
=
∑
i
=
1
n
t
i
y
i
∉
⋃
i
=
1
n
M
y
i
.
Thus, we have
(38)
g
λ
u
+
1
-
λ
z
,
f
y
i
∈
-
int
⁡
C
.
Since g is affine in the second argument and g
λ
u
+
1
-
λ
z
,
f
u
=
0, we have
(39)
∑
i
=
1
n
t
i
g
λ
u
+
1
-
λ
z
,
f
y
i
∈
-
int
⁡
C
⟹
g
λ
u
+
1
-
λ
z
,
f
∑
i
=
1
n
t
i
y
i
∈
-
int
⁡
C
⟹
g
λ
u
+
1
-
λ
z
,
f
u
∈
-
int
⁡
C
.
It follows that 0
∈
-
int
⁡
C which contradicts to the pointedness of C
x and hence M is a KKM-mapping.
Step
2. One has ⋂
y
∈
K
M
y
=
⋂
y
∈
K
S
y and S is also a KKM-mapping.
Ifx
∈
M
y, then g
λ
x
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C. By the f-monotonicity of g with respect to C, we have
(40)
g
λ
x
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
∈
-
C
⟹
g
λ
x
+
1
-
λ
z
,
f
y
∈
-
C
-
g
λ
y
+
1
-
λ
z
,
f
x
.
Suppose that x
∉
S
y. Then, we have
(41)
g
λ
y
+
1
-
λ
z
,
f
x
∈
int
⁡
C
.
It follows from (40) that
(42)
g
λ
x
+
1
-
λ
z
,
f
y
∈
-
C
-
int
⁡
C
⊂
-
int
⁡
C
which contradicts the fact that x
∈
M
y. Therefore, x
∈
S
y; that is, M
y
⊂
S
y. Then,
(43)
⋂
y
∈
k
M
y
⊂
⋂
y
∈
K
S
y
.
On the other hand, suppose that x
∈
⋂
y
∈
K
S
y. We have
(44)
g
λ
y
+
1
-
λ
z
,
f
x
∉
int
⁡
C
,
∀
y
∈
K
.
By Lemma 12, we have
(45)
g
λ
x
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
∈
K
.
That is, x
∈
⋂
y
∈
K
M
y. Hence,
(46)
⋂
y
∈
k
M
y
⊃
⋂
y
∈
K
S
y
.
So,
(47)
⋂
y
∈
k
M
y
=
⋂
y
∈
K
S
y
.
Also⋂
y
∈
K
S
y
≠
∅, since y
∈
S
y. From above, we know that M
y
⊂
S
y and, by Step 1, we know that M is a KKM-mapping. Thus, S is also a KKM-mapping.
Step
3. For all y
∈
K, S
y is closed.
Letx
n be a sequence in S
y such that x
n converges to x
∈
K. Then,
(48)
g
λ
y
+
1
-
λ
z
,
f
x
n
∉
int
⁡
C
∀
n
.
Since the mapping x
→
g
λ
y
+
1
-
λ
z
,
f
x is continuous, we have
(49)
g
λ
y
+
1
-
λ
z
,
f
x
n
⟶
g
λ
y
+
1
-
λ
z
,
f
x
∉
int
⁡
C
.
We conclude that x
∈
S
y; that is, S
y is a closed subset of a compact set K and hence compact.
By KKM-Theorem3, ⋂
y
∈
K
S
y
≠
∅ and also ⋂
y
∈
K
M
y
≠
∅. Hence, there exists x
0
∈
⋂
y
∈
k
M
y
=
⋂
y
∈
K
S
y; that is, there exists x
0
∈
K such that
(50)
g
λ
x
0
+
1
-
λ
z
,
f
y

∉
-
int
⁡
C
∀
y
,
z
∈
K
,
λ

∈

0,1
.
Thus, x
0 is a solution of problem (5).In support of Theorem13, we give the following example.Example 14.
Let=
Y
=
R, K
=
R
+, and C
=
x
:
x
≥
0.
Letg
:
K
×
X
→
Y and f
:
K
→
X be mappings such that
(51)
g
x
,
y
=
x
y
,
∀
x
,
y
∈
K
,
f
x
=
-
m
x
∀
x
∈
K
,
m
>
0
.
Then, (i)
for anyx
,
y
,
z
∈
K,
(52)
g
λ
x
+
1
-
λ
z
,
f
y
+
g
λ
y
+
1
-
λ
z
,
f
x
=
λ
x
+
1
-
λ
z
-
m
y
+
λ
y
+
1
-
λ
z
-
m
x
=
-
2
λ
m
x
y
-
1
-
λ
m
y
z
+
m
z
x
=
-
m
2
λ
x
y
+
1
-
λ
y
z
+
z
x
∈
-
C
for
m
>
0
;
that is,g is f-monotone with respect to C;
(ii)
for anyr
>
0,
(53)
g
x
,
r
y
=
x
r
y
=
r
x
y
=
r
g
x
,
y
;
that is,g is positive homogeneous in the second argument;
(iii)
letF
:
0,1
→
Y be a mapping such that F
t
=
g
t
y
+
1
-
t
x
,
f
x, ∀
t
∈
0,1; then,
(54)
F
t
=
g
t
y
+
1
-
t
x
,
f
x
=
t
y
+
1
-
t
x
-
m
x
which is a continuous mapping; that is,t
→
g
t
y
+
1
-
t
x
,
f
x is continuous at 0
+. Hence, g is f-hemicontinuous.
(iv)
LetG
:
K
→
Y be a mapping such that
(55)
G
x
=
g
λ
y
+
1
-
λ
x
,
f
x
,
∀
x
∈
K
;
then,(56)
G
x
=
g
λ
y
+
1
-
λ
x
,
f
x
=
λ
y
+
1
-
λ
x
-
m
x
which implies thatx
→
g
λ
y
+
1
-
λ
x
,
f
x is continuous mapping.
Hence, all the conditions of Theorem13 are satisfied.
In addition,(57)
g
λ
x
+
1
-
λ
z
,
f
y
=
λ
x
+
1
-
λ
z
-
m
y
=
-
m
λ
x
+
1
-
λ
z
y
∉
int
⁡
C
for
m
>
0
.
Thus, it follows that x is a solution of problem (5) for all z
∈
K and λ
∈
(
0,1
].Corollary 15.
LetK be a compact and convex subset of X and let Y
,
C be an ordered topological vector space with int
⁡
C
≠
∅. Let g
:
K
×
X
→
Y be a vector-valued mapping which is f-pseudomonotone with respect to C and let f
:
K
→
X be a continuous, affine mapping such that, for z
∈
K, λ
∈
(
0,1
], g
λ
x
+
1
-
λ
z
,
f
x
=
0 for all x
∈
K. Let the mapping x
→
g
λ
x
+
1
-
λ
z
,
f
y be continuous. Then, problem (5) is solvable.Proof.
By Step 1 of Theorem13, it follows that M is a KKM-mapping. Also it follows from f-pseudomonotonicity of g that M
y
⊂
S
y; thus S is also a KKM-mapping. By Step 3 of Theorem 13, the conclusion follows.Theorem 16.
LetX be a reflexive Banach space; let Y
,
C be an ordered topological vector space with int
⁡
C
≠
∅. Let K be a nonempty, bounded, and convex subset of X. Let g
:
K
×
X
→
Y be a vector-valued mapping which is f-monotone with respect to C, positive homogeneous in the second argument, f-hemicontinuous, and f-generally convex on K. Let f
:
K
→
X be a continuous, affine mapping such that, for z
∈
K, λ
∈
(
0,1
], g
λ
x
+
1
-
λ
z
,
f
x
=
0 for all x
∈
K. Then, problem (5) admits a solution, that is, for all z
∈
K and λ
∈
(
0,1
], there exists x
0
∈
K such that
(58)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
∈
K
.Proof.
For eachy
∈
K, let
(59)
M
y
=
x
∈
K
:
g
λ
x
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
S
y
=
x
∈
K
:
g
λ
y
+
1
-
λ
z
,
f
x
∉
int
⁡
C
for all z
∈
K and λ
∈
(
0,1
].
From the proof of Theorem13, we know that S
y is closed and S is a KKM-mapping. We also know that
(60)
⋂
y
∈
k
M
y
=
⋂
y
∈
K
S
y
.
Since K is a bounded, closed, and convex subset of a reflexive Banach space X, therefore, K is weakly compact.
Now, we show thatS
y is convex. Suppose that y
1
,
y
2
∈
S
y and t
1
,
t
2
≥
0 with t
1
+
t
2
=
1.
Then,(61)
g
λ
y
+
1
-
λ
z
,
f
y
i
∉
int
⁡
C
,
i
=
1,2
.
Since g is f-generally convex, we have
(62)
g
λ
y
+
1
-
λ
z
,
f
t
1
y
1
+
t
2
y
2
∉
int
⁡
C
;
that is, t
1
y
1
+
t
2
y
2
∈
S
y, which implies that S
y is convex. Since S
y is closed and convex, S
y is weakly closed.
AsS is a KKM-mapping, S
y is weakly closed subset of K; therefore S
y is weakly compact. By KKM-Theorem 3, there exists x
0
∈
K such that x
0
∈
⋂
y
∈
k
M
y
=
⋂
y
∈
K
S
y
≠
∅. That is, there exists x
0
∈
K such that
(63)
g
λ
x
0
+
1
-
λ
z
,
f
y
∉
-
int
⁡
C
,
∀
y
,
z
∈
K
,
λ
∈
0,1
.
Hence, problem (5) is solvable.


## 4. Conclusion

In this paper, some existence results for extendedf-vector equilibrium problem are proved in the setting of Hausdorff topological vector spaces and reflexive Banach spaces. The concept of monotonicity plays an important role in obtaining existence results. The results of this paper can be viewed as generalizations of some known equilibrium problems as explained by (6) and (7).



---

*Source: 643230-2014-12-03.xml*