# System of Operator Quasi Equilibrium Problems

Authors: Suhel Ahmad Khan
Journal: International Journal of Analysis
 (2014)

---

## Abstract

We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involvingΦ-condensing mappings.


---

## Body

## 1. Introduction

In 2002, Domokos and Kolumbán [1] gave an interesting interpretation of variational inequality and vector variational inequalities (for short, VVI) in Banach space settings in terms of variational inequalities with operator solutions (for short, OVVI). The notion and viewpoint of OVVI due to Domokos and Kolumbán [1] look new and interesting even though it has a limitation in application to VVI. Recently, Kazmi and Raouf [2] introduced the operator equilibrium problem which generalizes the notion of OVVI to operator vector equilibrium problems (for short, OVEP) using the operator solution. They derived some existence theorems of solution of OVEP with pseudomonotonicity, without pseudomonotonicity, and with B-pseudomonotonicity. However, they dealt with only the single-valued case of the bioperator. It is very natural and useful to extend a single-valued case to a corresponding set-valued one from both theoretical and practical points of view.The system of vector equilibrium problems and the system of vector quasi equilibrium problems were introduced and studied by Ansari et al. [3, 4]. Inspired by above cited work, in this paper, we consider a system of operator quasi equilibrium problems (for short, SOQEP) in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings according to [5] as basic tool, we derive some existence theorems for solutions to SOQEP with and without involving Φ-condensing mappings.Further, we consider a system of generalized quasi operator equilibrium problems (for short, SGQOEP) in topological vector spaces and give some of its special cases and derive some existence theorems for solutions to SOQEP with and without involvingΦ-condensing mappings by using well-known maximal element theorem [5] for a family of set-valued mappings, and, consequently, we also get some existence theorems for solutions to a system of operator equilibrium problems.


## 2. Preliminaries

LetI be an index set, for each i
∈
I, and let X
i be a Hausdorff topological vector space. We denote L
(
X
i
,
Y
i
), the space of all continuous linear operators from X
i into Y
i, where Y
i is topological vector space for each i
∈
I. Consider a family of nonempty convex subsets {
K
i
}
i
∈
I with K
i in L
(
X
i
,
Y
i
).Let(1)
X
=
∏
i
∈
I
‍
X
i
,
K
=
∏
i
∈
I
‍
K
i
.
Let C
i
:
K
→
2
Y
i be a set-valued mapping such that, for each f
∈
K, C
i
(
f
) is solid, open, and convex cone such that 0
∉
C
i
(
f
) and P
i
=
⋂
f
∈
K
C
i
(
f
).For eachi
∈
I, let F
i
:
K
×
K
i
→
Y
i be a bifunction and let A
i
:
K
→
2
K
i be a set-valued mapping with nonempty values. We consider the followingsystem of operator quasi equilibrium problems (for short, SOQEP). Find f
-
∈
K such that, for each i
∈
I,
(2)
f
-
i
∈
A
i
(
f
-
)
,
F
i
(
f
-
,
g
i
)
∉
-
C
i
(
f
-
)
,
∀
g
i
∈
A
i
(
f
-
)
.We remarked that, for the suitable choices ofI
,
F
i
,
K
i
,
X
i
,
Y
i
,
C
i, and A
i, SOQEP (2) reduces to the problems considered and studied by [3–6] and the references therein.Now, we will give the following concepts and results which are used in the sequel.Definition 1.
LetM be a nonempty and convex subset of a topological vector space, and let Z be a topological vector space with a closed and convex cone P with apex at the origin. A vector-valued function ϕ
:
M
→
Z is said to be as follows: (i)
P-function if and only if ∀
x
,
y
∈
M and λ
∈
[
0,1
]:
(3)
ϕ
(
λ
x
+
(
1
-
λ
)
y
)
∈
λ
ϕ
(
x
)
+
(
1
-
λ
)
ϕ
(
y
)
-
P
;
(ii)
natural P-quasifunction if and only if ∀
x
,
y
∈
M and λ
∈
[
0,1
]:
(4)
ϕ
(
λ
x
+
(
1
-
λ
)
y
)
∈
C
o
{
ϕ
(
x
)
,
ϕ
(
y
)
}
-
P
,
where C
o
B denotes the convex hull of B;
(iii)
P-quasifunction if and only if ∀
α
∈
Z and the set {
x
∈
M
:
ϕ
(
x
)
-
α
∈
-
P
} is convex.Definition 2 (see [7]).
LetX be a topological vector space and let L be a lattice with a minimal element, denoted by 0. A mapping ϕ
:
2
X
→
L is called a measure of noncompactness provided that the following conditions hold for any M
,
N
∈
2
X: (i)
ϕ
(
C
-
o
M
)
=
ϕ
(
M
), where C
-
o
M denotes the closed convex hull of M;
(ii)
ϕ
(
M
)
=
o if and only if M is precompact;
(iii)
ϕ
(
M
∪
N
)
=
max
⁡
{
ϕ
(
M
)
,
ϕ
(
N
)
}.Definition 3 (see [7]).
LetX be a topological vector space, D
⊂
X, and let ϕ be a measure of noncompactness on X. A set-valued mapping T
:
D
→
2
X is called ϕ-condensing provided that M
⊂
D with ϕ
(
T
(
M
)
)
≥
ϕ
(
M
); then M is relative compact; that is, M
- is compact.Remark 4.
Note that every set-valued mapping defined on a compact set isϕ-condensing for any measure of noncompactness ϕ. If X is locally convex, then a compact set-valued mapping (i.e., T
(
D
) is precompact) is ϕ-condensing for any measure of noncompactness ϕ. Obviously, if T
:
D
→
2
X is ϕ-condensing and T
′
:
D
→
2
X satisfies T
′
(
x
)
⊂
T
(
x
), for all x
∈
X, then T
′ is also ϕ-condensing.The following maximal element theorems will play key role in establishing existence results.Theorem 5 (see [8]).
For eachi
∈
I, let K
i be a nonempty convex subset of a topological vector space X
i and let S
i
,
T
i
:
K
→
2
K
i be the two set-valued mappings. For each i
∈
I, assume that the following conditions hold: (a)
for allx
∈
K, C
o
S
i
(
x
)
⊆
T
i
(
x
);
(b)
for allx
∈
K, x
i
∉
T
i
(
x
);
(c)
for ally
i
∈
K
i, S
i
-
1
(
y
i
) is compactly open K;
(d)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊆
K
i, for each i
∈
I, such that, for all x
∈
K
∖
D, there exists i
∈
I such that S
i
(
x
)
∩
E
i
≠
∅. Then, there exists x
-
∈
K such that S
i
(
x
-
)
=
∅ for each i
∈
I.We will use the following particular form of a maximal element theorem for a family of set-valued mappings due to Deguire et al. [5].Theorem 6 (see [5]).
LetI be any index set, for each i
∈
I, let K
i be a nonempty convex subset of a Hausdorff topological vector space X
i, and let S
i
:
K
=
∏
i
∈
I
K
i
→
2
K
i be a set-valued mapping. Assume that the following conditions hold: (i)
∀
i
∈
I and ∀
x
∈
K; S
i
(
x
) is convex;
(ii)
∀
i
∈
I and ∀
x
∈
K; x
i
∉
S
i
(
x
), where x
i is the ith component of x;
(iii)
∀
i
∈
I and ∀
y
i
∈
K
i; S
i
-
1
(
y
i
) is open K;
(iv)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊆
K
i
,
∀
i
∈
I such that ∀
x
∈
K
∖
D and there exists i
∈
I such that S
i
(
x
)
∩
E
i
≠
∅. Then, there exists x
-
∈
K such that S
i
(
x
-
)
=
∅ for each i
∈
I.Remark 7.
If∀
i
∈
I
,
K
i is nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i, then condition (iv) of Theorem 6 can be replaced by the following condition:
(
iv
)
1 the set-valued mapping S
:
K
→
2
K is defined as S
(
x
)
=
∏
i
∈
I
S
i
(
x
)
,
∀
x
∈
K, ϕ-condensing.


## 3. Main Result

Throughout this paper, unless otherwise stated, for any index setI and for each i
∈
I, let Y
i be a topological vector space and let K
=
∏
i
∈
I
K
i
,
C
i
:
K
→
2
Y
i be a set-valued mapping such that, for each f
∈
K, C
i
(
f
) is proper, solid, open, and convex cone such that 0
∉
C
i
(
f
) and P
i
=
⋂
f
∈
K
C
i
(
f
). We denote L
(
X
i
,
Y
i
), the space of all continuous linear operators from X
i into Y
i. We also assume that ∀
i
∈
I
,
A
i
:
K
→
2
K
i is a set-valued mapping such that ∀
f
∈
K
,
A
i
(
f
) is nonempty and convex, A
-
1
(
g
i
) is open in K
,
f
i
∈
K
i, and the set F
i
:
{
f
∈
K
:
f
i
∈
A
i
(
f
)
} is closed in K, where f
i is the ith component of f.Now, we have the following existence result for SOQEP (2).Theorem 8.
For eachi
∈
I, let K
i be nonempty and convex subset of a Hausdorff topological vector space X
i and let F
i
:
K
×
K
i
→
Y
i be a bifunction. Suppose that the following conditions hold: (i)
∀
i
∈
I and ∀
f
∈
K
,
F
i
(
f
,
f
i
)
∉
-
C
i
(
f
), where f
i is the ith component of f;
(ii)
∀
i
∈
I and ∀
f
∈
K; the vector-valued function g
i
↦
F
i
(
f
,
g
i
) is natural P
i-quasifunction;
(iii)
∀
i
∈
I and ∀
g
i
∈
K
i; the set {
f
∈
K
:
F
i
(
f
,
g
i
)
∉
-
C
i
(
f
)
} is closed in K;
(iv)
there exist a nonempty compact subsetN of K and a nonempty compact convex subset B
i of K
i, for each i
∈
I such that ∀
f
∈
K
∖
N; there exists i
∈
I and g
^
i
∈
B
i such that g
^
i
∈
A
i
(
f
) and F
i
(
f
,
g
^
i
)
∈
-
C
i
(
f
). Then SOQEP (2) has a solution.Proof.
Let us define, for each giveni
∈
I, a set-valued mapping T
i
:
K
→
2
K
i by
(5)
T
i
(
f
)
=
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
∈
-
C
i
(
f
)
}
,
∀
f
∈
K
.
First, we claim that ∀
i
∈
I and f
∈
K
,
T
i
(
f
) is convex. Fix an arbitrary i
∈
I and f
∈
K. Let g
i
,
1
,
g
i
,
2
∈
T
i
(
f
) and λ
∈
[
0,1
]; then we have
(
i
)
F
i
(
f
,
g
i
j
)
∈
-
C
i
(
f
)
,
for
j
=
1,2
.
Since F
i
(
f
,
·
) is natural P
i-quasifunction, there exists μ
∈
[
0,1
] such that
(
i
i
)
F
i
(
f
,
λ
g
i
,
1
+
(
1
-
λ
)
g
i
,
2
)
∈
μ
F
i
(
f
,
g
i
,
1
)
+
(
1
-
μ
)
F
i
(
f
,
g
i
,
2
)
-
P
i
.
From the inclusion of (
i
) and (
i
i
), we get
(6)
F
i
(
f
,
λ
g
i
,
1
+
(
1
-
λ
)
g
i
,
2
)
∈
-
C
i
(
f
)
-
C
i
(
f
)
-
P
i
⊆
C
i
(
f
)
.
Hence, λ
g
i
,
1
+
(
1
-
λ
)
g
i
,
2
∈
T
i
(
f
) and therefore T
i
(
f
) is convex. Since i
∈
I and f
∈
K are arbitrary, T
i
(
f
) is convex, ∀
f
∈
K and ∀
i
∈
I.
Hence, our claim is then verified.
Now∀
i
∈
I and ∀
g
i
∈
K
i; the complement of T
i
-
1
(
g
i
) in K can be defined as
(7)
[
T
i
-
1
(
g
i
)
]
c
=
{
f
∈
K
:
F
i
(
f
,
g
i
)
∉
-
C
i
(
f
)
}
.
From condition (iii) of the above theorem, [
T
i
-
1
(
g
i
)
]
c will be closed in K.
Suppose that∀
i
∈
I and ∀
f
∈
K; we define another set-valued mapping M
i
:
K
→
2
K
i by
(8)
M
i
(
f
)
=
{
A
i
(
f
)
∩
T
i
(
f
)
if
f
∈
F
i
A
i
(
f
)
;
if
f
∈
K
∖
F
i
.
Then, it is clear that∀
i
∈
I and ∀
f
∈
K
,
M
i
(
f
) is convex, because A
(
f
) and T
i
(
f
) are both convex. Now, by condition (i), f
i
∉
M
i
(
f
). Since ∀
i
∈
I and ∀
g
i
∈
K
i,
(9)
M
i
-
1
(
g
i
)
=
(
A
i
-
1
(
g
i
)
∩
T
i
-
1
(
g
i
)
)
⋃
‍
(
(
K
∖
F
i
)
∩
A
i
-
1
(
g
i
)
)
is open in K, because A
i
-
1
(
g
i
)
,
T
i
-
1
(
g
i
) and K
∖
F
i are open in K.
Condition (iv) of Theorem6 is followed from condition (iv). Hence, by fixed point Theorem 6, there exists f
-
∈
K such that M
i
(
f
-
)
=
∅
,
∀
i
∈
I. Since ∀
i
∈
I and ∀
f
∈
K
,
A
i
(
f
) is nonempty, we have A
i
(
f
-
)
∩
T
i
(
f
-
)
=
∅
,
∀
i
∈
I. Therefore, ∀
i
∈
I
,
f
-
i
∈
A
i
(
f
-
) and F
i
(
f
-
,
g
i
)
∈
-
C
i
(
f
-
)
,
∀
g
i
∈
A
i
(
f
-
).
This completes the proof.Now, we establish an existence result for SOQEP (2) involving ϕ-condensing maps.Theorem 9.
For eachi
∈
I, let K
i be a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i, suppose that F
i
:
K
×
K
i
→
Y
i is a bifunction, and let the set-valued mapping A
=
∏
i
∈
I
A
i
:
K
→
2
K defined as A
(
f
)
=
∏
i
∈
I
A
i
(
f
)
,
∀
f
∈
K be ϕ-condensing. Assume that conditions (i), (ii), and (iii) of Theorem 8 hold. Then SOQEP (2) has a solution.Proof.
In view of Remark7, it is sufficient to show that the set-valued mapping S
:
K
→
2
K defined as S
(
f
)
=
∏
i
∈
I
S
i
(
f
)
,
∀
f
∈
K, is ϕ-condensing, where S
is are the same as defined in the proof of Theorem 8. By the definition of S
i
,
S
i
(
f
)
⊆
A
i
(
f
)
,
∀
i
∈
I and ∀
f
∈
K and therefore S
(
f
)
⊆
A
(
f
)
,
∀
f
∈
K. Since A is ϕ-condensing, by Remark 7, we have S being also ϕ-condensing.
This completes the proof.


## 4. System of Generalized Quasi Operator Equilibrium Problem

Throughout this section, unless otherwise stated, letI be any index set. For each i
∈
I, let X
i be a Hausdorff topological vector space. We denote L
(
X
i
,
Y
i
), the space of all continuous linear operators from X
i into Y
i, where Y
i is topological vector space for each i
∈
I and for each i
∈
I; let P
i
⊂
Y
i be a closed, pointed, and convex cone with int
⁡
P
i
≠
∅, where int
⁡
P
i denotes the interior of set P
i
,
∀
i
∈
I. Consider a family of nonempty convex subsets {
K
i
}
i
∈
I with K
i in L
(
X
i
,
Y
i
). Let, for each i
∈
I, a bifunction F
i
:
K
×
K
i
→
Y
i and two set-valued mappings A
i
,
B
i
:
K
→
2
K
i be with nonempty values.Lete
i be the unit vector in Y
i, for each i
∈
I, and also α
i
e
i
,
β
i
e
i
∈
Y
i such that α
i
e
i
≮
P
i
β
i
e
i, where α
i
,
β
i
∈
R are two real numbers such that α
i
≤
β
i.Now, we consider thesystem of generalized quasi operator equilibrium problems (for short, SGQOEP). Find f
-
∈
K such that, for each i
∈
I,
(10)
f
-
i
∈
B
i
(
f
-
)
,
α
i
e
i
≮
P
i
F
i
(
f
-
,
g
i
)
≮
P
i
β
i
e
i
;
∀
g
i
∈
A
i
(
f
-
)
.


### 4.1. Special Cases

(I)
IfA
i
=
B
i
,
∀
i, then SGQOEP (10) reduces to finding of f
-
∈
K such that, for each i
∈
I,
(11)
α
i
e
i
≮
P
F
i
(
f
-
,
g
i
)
≮
P
β
i
e
i
;
∀
g
i
∈
A
i
(
f
-
)
.
(II)
If, in Case (I), we takeY
i
=
R, then P
i
=
R
+ and e
i
=
1; then problem (10) reduces to the system of generalized quasi operator equilibrium problems with lower and upper bounds (for short, SGQOEPLUB). Find f
-
∈
K such that, for each i
∈
I,
(12)
f
-
i
∈
B
i
(
f
-
)
,
α
i
≤
F
i
(
f
-
,
g
i
)
≤
β
i
;
∀
g
i
∈
A
i
(
f
-
)
.Now, we establish the existence result for SGQOEP (10).Theorem 10.
For eachi
∈
I, let K
i be a nonempty convex subset of a topological vector space X
i and F
i
,
P
i
,
Q
i
:
K
×
K
i
→
K
i are the bifunctions, B
i
:
K
→
2
K
i is a set-valued mapping such that the set F
i
=
{
f
∈
K
:
f
i
∈
B
i
(
f
)
} is compactly closed, A
i
:
K
→
2
K
i is a set-valued mapping with nonempty values such that, for each g
i
∈
K
i
,
A
i
-
1
(
g
i
) is compactly open in K, and ∀
i
∈
I
,
e
i
∈
Y
i are the unit vector such that α
i
e
i
≮
P
i
β
i
e
i, where α
i
,
β
i
∈
R are two real numbers such that α
i
≤
β
i. For each i
∈
I, assume that the following conditions hold: (i)
for allf
∈
K, C
o
A
i
(
f
)
⊆
B
i
(
f
);
(ii)
for allf
∈
K, α
i
e
i
>
P
i
P
i
(
f
,
f
i
) or Q
i
(
f
,
f
i
)
>
P
i
β
i
e
i;
(iii)
for allf
∈
K and for every nonempty finite subset N
i
⊆
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}, we have
(13)
C
o
N
i
⊆
{
g
i
∈
K
i
:
P
i
(
f
,
g
i
)
≮
P
i
α
i
e
i
a
n
d
Q
i
(
f
,
g
i
)
≯
P
i
β
i
e
i
}
;
(iv)
for allg
i
∈
K
i, the set {
f
∈
K
:
β
i
e
i
≮
P
i
F
i
(
f
,
g
i
)
≮
P
i
α
i
e
i is compactly closed in K;
(v)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊂
K
i, for each i
∈
I, such that, for all f
∈
K
∖
D, there exists i
∈
I such that g
^
i
∈
E
i satisfying g
^
i
∈
A
i
(
f
) and either F
i
(
f
,
g
^
i
)
P
i
<
α
i
e
i or F
i
(
f
,
g
^
i
)
>
P
i
β
i
e
i. Then the problem SGQOEP (10) has a solution.Proof.
For eachi
∈
I and for all f
∈
K, define two set-valued mappings G
i
,
H
i
:
K
→
2
K
i by
(14)
G
i
(
f
)
=
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}
,
H
i
(
f
)
=
{
g
i
∈
K
i
:
P
i
(
f
,
g
i
)
P
i
≮
α
i
e
i
and
Q
i
(
f
,
g
i
)
≯
P
i
β
i
e
i
}
.
Condition (iii) implies that, for each i
∈
I and for all f
∈
K, C
o
G
i
(
f
)
⊆
H
i
(
f
).
From condition (ii), we havef
i
∉
H
i
(
f
) for all f
∈
K and for each i
∈
I.
Thus, for eachi
∈
I and for all g
i
∈
K
i,
(15)
G
i
-
1
(
g
i
)
=
{
f
∈
K
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}
.
We have complement of G
i
-
1
(
g
i
) in K:
(16)
[
G
i
-
1
(
g
i
)
]
c
=
{
β
i
e
i
≮
P
i
F
i
(
f
-
,
g
i
)
≮
P
i
α
i
e
i
}
,
which is compactly closed by virtue of condition (iv). Therefore, for each i
∈
I and for all g
i
∈
K
i
,
G
i
-
1
(
g
i
) is compactly open in K.
For eachi
∈
I, define two set-valued mappings S
i
,
T
i
:
K
→
2
K
i by
(17)
S
i
(
f
)
=
{
G
i
(
f
)
∩
A
i
(
f
)
;
if
f
∈
F
i
A
i
(
f
)
;
if
f
∈
K
∖
F
i
,
T
i
(
f
)
=
{
H
i
(
f
)
∩
B
i
(
f
)
;
if
f
∈
F
i
B
i
(
f
)
;
if
f
∈
K
∖
F
i
.
Thus, for each i
∈
I and for all f
∈
K
,
C
o
G
i
(
f
)
⊆
H
i
(
f
) and in view of condition (i), we obtain C
o
S
i
(
f
)
⊆
T
i
(
f
). It is easy to see that
(18)
S
i
-
1
(
g
i
)
=
(
A
i
-
1
(
g
i
)
∩
G
i
-
1
(
g
i
)
)
⋃
‍
(
(
K
∖
F
i
)
∩
A
i
-
1
(
g
i
)
)
for each i
∈
I and for all f
∈
K. Thus, for each i
∈
I and for all G
i
-
1
(
g
i
)
,
A
i
-
1
(
g
i
) and K
∖
F
i are compactly open in K. We have S
i
-
1
(
g
i
) being compactly open in K. Also f
i
∉
T
i
(
f
) for all f
∈
K and for each i
∈
I.
Then, by Theorem5, there exists f
-
∈
K such that S
i
(
f
-
)
=
∅ for each i
∈
I. If f
-
∈
K
∖
F
i, then A
i
(
f
-
)
=
S
i
(
f
-
)
=
∅, which contradicts the fact that A
i
(
f
) is nonempty for each i
∈
I and for all f
∈
X. Hence, f
-
∈
F
i, for each i
∈
I. Therefore, f
-
i
∈
B
i
(
f
-
) and G
i
(
f
-
)
∩
A
i
(
f
-
)
=
∅, for all i
∈
I. Thus, for each i
∈
I
,
f
-
i
∈
B
i
(
f
-
) and β
i
e
i
≮
P
i
F
i
(
f
-
,
g
i
)
≮
P
i
α
i
e
i for all g
i
∈
A
i
(
f
-
). This completes the proof.Now, we establish an existence result for SGQOEP (10) involving ϕ-condensing maps.Theorem 11.
For eachi
∈
I, assume that conditions (i)–(iv) of Theorem 10. hold. Let ϕ be a measure of noncompactness on ∏
i
∈
I
X
i. Further, assume that the set-valued mapping B
:
K
→
2
K defined as K
i is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i and F
i
:
K
×
K
i
→
Y
i is a bifunction and let the set-valued mapping A
=
∏
i
∈
I
A
i
:
K
→
2
K defined as B
(
f
)
=
∏
i
∈
I
B
i
(
f
)
,
∀
f
∈
K be ϕ-condensing. Then, there exists a solution f
-
∈
K of SGQOEP (10).Proof.
In view of Remark7, it is sufficient to show that the set-valued mapping T
:
K
→
2
K defined as T
(
f
)
=
∏
i
∈
I
T
i
(
f
)
,
∀
f
∈
K, is ϕ-condensing, where T
is are the same as defined in the proof of Theorem 10. By the definition of T
i
,
T
i
(
f
)
⊆
B
i
(
f
)
,
∀
i
∈
I and ∀
f
∈
K and therefore S
(
f
)
⊆
A
(
f
)
,
∀
f
∈
K. Since B is ϕ-condensing, by Remark 7, we have T being also ϕ-condensing.
This completes the proof.Next, we derive the existence result for the solution of SGQOEPLUB (12).Corollary 12.
For eachi
∈
I, let K
i be a nonempty convex subset of a topological vector space X
i and F
i
,
L
i
,
Q
i
:
K
×
K
i
→
R are the bifunctions, B
i
:
K
→
2
K
i is a set-valued mapping such that the set F
i
=
{
f
∈
K
:
f
i
∈
B
i
(
f
)
} is compactly closed, A
i
:
K
→
2
K
iis a set-valued mapping with nonempty values such that, for each g
i
∈
K
i
,
A
i
-
1
(
g
i
) is compactly open in K, and α
i
,
β
i
∈
R are two real numbers such that α
i
≤
β
i. For each i
∈
I, assume that the following conditions hold:(i)
for allf
∈
K, C
o
A
i
(
f
)
⊆
B
i
(
f
);
(ii)
for allf
∈
K, L
i
(
f
,
f
i
)
<
α
i or Q
i
(
f
,
f
i
)
>
β
i;
(iii)
for allf
∈
K and for every nonempty finite subset N
i
⊆
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
α
i
o
r
F
i
(
f
,
g
i
)
>
β
i
}, we have
(19)
C
o
N
i
⊆
{
g
i
∈
K
i
:
L
i
(
f
,
g
i
)
≥
α
i
a
n
d
Q
i
(
f
,
g
i
)
≤
β
i
}
;
(iv)
for allg
i
∈
K
i, the set {
f
∈
K
:
α
i
≤
F
i
(
f
,
g
i
)
≤
β
i
} is compactly closed in K;
(v)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊂
K
i, for each i
∈
I, such that, for all f
∈
K
∖
D, there exists i
∈
I such that g
^
i
∈
E
i satisfying g
^
i
∈
A
i
(
f
) and either F
i
(
f
,
g
^
i
)
<
α
i or F
i
(
f
,
g
^
i
)
>
β
i. Then the problem SGQOEPLUB (12) has a solution.Proof.
For eachi
∈
I and for all f
∈
K, define two set-valued mappings G
i
,
H
i
:
K
→
2
K
i by
(20)
G
i
(
f
)
=
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
α
i
or
F
i
(
f
,
g
i
)
>
β
i
}
,
H
i
(
f
)
=
{
g
i
∈
K
i
:
L
i
(
f
,
g
i
)
≥
α
i
and
Q
i
(
f
,
g
i
)
≤
β
i
}
.
Condition (iii) implies that, for each i
∈
I and for all f
∈
K, C
o
G
i
(
f
)
⊆
H
i
(
f
).
From condition (ii), we havef
i
∉
H
i
(
f
) for all f
∈
K and for each i
∈
I.
Thus, for eachi
∈
I and for all g
i
∈
K
i,
(21)
G
i
-
1
(
g
i
)
=
{
f
∈
K
:
F
i
(
f
,
g
i
)
<
α
i
or
F
i
(
f
,
g
i
)
>
β
i
}
.
We have complement of G
i
-
1
(
g
i
) in K:
(22)
[
G
i
-
1
(
g
i
)
]
c
=
{
α
i
≤
F
i
(
f
,
g
i
)
≤
β
i
}
,
which is compactly closed by virtue of condition (iv). Therefore, for each i
∈
I and for all g
i
∈
K
i
,
G
i
-
1
(
g
i
) is compactly open in K.
For eachi
∈
I, define two set-valued mappings S
i
,
T
i
:
K
→
2
i
K by
(23)
S
i
(
f
)
=
{
G
i
(
f
)
∩
A
i
(
f
)
if
f
∈
F
i
A
i
(
f
)
;
if
f
∈
K
∖
F
i
,
T
i
(
f
)
=
{
H
i
(
f
)
∩
B
i
(
f
)
if
f
∈
F
i
B
i
(
f
)
;
if
f
∈
K
∖
F
i
.
Thus, for each i
∈
I and for all f
∈
K
,
C
o
G
i
(
f
)
⊆
H
i
(
f
) and in view of condition (i), we obtain C
o
S
i
(
f
)
⊆
T
i
(
f
). It is easy to see that
(24)
S
i
-
1
(
g
i
)
=
(
A
i
-
1
(
g
i
)
∩
G
i
-
1
(
g
i
)
)
⋃
‍
(
(
K
∖
F
i
)
∩
A
i
-
1
(
g
i
)
)
for each i
∈
I and for all f
∈
K. Thus, for each i
∈
I and for all G
i
-
1
(
g
i
)
,
A
i
-
1
(
g
i
) and K
∖
F
i are compactly open in K. We have S
i
-
1
(
g
i
) being compactly open in K. Also f
i
∉
T
i
(
f
) for all f
∈
K and for each i
∈
I.
Then, by Theorem5, there exists f
-
∈
K such that S
i
(
f
-
)
=
∅ for each i
∈
I. If f
-
∈
K
∖
F
i, then A
i
(
f
-
)
=
S
i
(
f
-
)
=
∅, which contradicts the fact that A
i
(
f
) is nonempty for each i
∈
I and for all f
∈
X. Hence, f
-
∈
F
i, for each i
∈
I. Therefore, f
-
i
∈
B
i
(
f
-
) and G
i
(
f
-
)
∩
A
i
(
f
-
)
=
∅, for all i
∈
I. Thus, for each i
∈
I
,
f
-
i
∈
B
i
(
f
-
) and α
i
≤
F
i
(
f
-
,
g
i
)
≤
β
i for all g
i
∈
A
i
(
f
-
). This completes the proof.Now, we establish an existence result for SGQOEPLUB (12) involving ϕ-condensing maps.Theorem 13.
For eachi
∈
I, assume that conditions (i)–(iv) of Corollary 12 hold. Let ϕ be a measure of noncompactness on ∏
i
∈
I
X
i. Further, assume that the set-valued mapping B
:
K
→
2
K defined as K
i is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i and F
i
:
K
×
K
i
→
R is a bifunction and let the set-valued mapping A
=
∏
i
∈
I
A
i
:
K
→
2
K defined as B
(
f
)
=
∏
i
∈
I
B
i
(
f
)
,
∀
f
∈
K, be ϕ-condensing. Then, there exists a solution f
-
∈
K of SGQOEPLUB (12).Proof.
In view of Remark7, it is sufficient to show that the set-valued mapping T
:
K
→
2
K defined as T
(
f
)
=
∏
i
∈
I
T
i
(
f
)
,
∀
f
∈
K, is ϕ-condensing, where T
i
s are the same as defined in the proof of Theorem 10. By the definition of T
i
,
T
i
(
f
)
⊆
B
i
(
f
)
,
∀
i
∈
I and ∀
f
∈
K and therefore S
(
f
)
⊆
A
(
f
)
,
∀
f
∈
K. Since B is ϕ-condensing, by Remark 7, we have T being also ϕ-condensing.
This completes the proof.


## 4.1. Special Cases

(I)
IfA
i
=
B
i
,
∀
i, then SGQOEP (10) reduces to finding of f
-
∈
K such that, for each i
∈
I,
(11)
α
i
e
i
≮
P
F
i
(
f
-
,
g
i
)
≮
P
β
i
e
i
;
∀
g
i
∈
A
i
(
f
-
)
.
(II)
If, in Case (I), we takeY
i
=
R, then P
i
=
R
+ and e
i
=
1; then problem (10) reduces to the system of generalized quasi operator equilibrium problems with lower and upper bounds (for short, SGQOEPLUB). Find f
-
∈
K such that, for each i
∈
I,
(12)
f
-
i
∈
B
i
(
f
-
)
,
α
i
≤
F
i
(
f
-
,
g
i
)
≤
β
i
;
∀
g
i
∈
A
i
(
f
-
)
.Now, we establish the existence result for SGQOEP (10).Theorem 10.
For eachi
∈
I, let K
i be a nonempty convex subset of a topological vector space X
i and F
i
,
P
i
,
Q
i
:
K
×
K
i
→
K
i are the bifunctions, B
i
:
K
→
2
K
i is a set-valued mapping such that the set F
i
=
{
f
∈
K
:
f
i
∈
B
i
(
f
)
} is compactly closed, A
i
:
K
→
2
K
i is a set-valued mapping with nonempty values such that, for each g
i
∈
K
i
,
A
i
-
1
(
g
i
) is compactly open in K, and ∀
i
∈
I
,
e
i
∈
Y
i are the unit vector such that α
i
e
i
≮
P
i
β
i
e
i, where α
i
,
β
i
∈
R are two real numbers such that α
i
≤
β
i. For each i
∈
I, assume that the following conditions hold: (i)
for allf
∈
K, C
o
A
i
(
f
)
⊆
B
i
(
f
);
(ii)
for allf
∈
K, α
i
e
i
>
P
i
P
i
(
f
,
f
i
) or Q
i
(
f
,
f
i
)
>
P
i
β
i
e
i;
(iii)
for allf
∈
K and for every nonempty finite subset N
i
⊆
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}, we have
(13)
C
o
N
i
⊆
{
g
i
∈
K
i
:
P
i
(
f
,
g
i
)
≮
P
i
α
i
e
i
a
n
d
Q
i
(
f
,
g
i
)
≯
P
i
β
i
e
i
}
;
(iv)
for allg
i
∈
K
i, the set {
f
∈
K
:
β
i
e
i
≮
P
i
F
i
(
f
,
g
i
)
≮
P
i
α
i
e
i is compactly closed in K;
(v)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊂
K
i, for each i
∈
I, such that, for all f
∈
K
∖
D, there exists i
∈
I such that g
^
i
∈
E
i satisfying g
^
i
∈
A
i
(
f
) and either F
i
(
f
,
g
^
i
)
P
i
<
α
i
e
i or F
i
(
f
,
g
^
i
)
>
P
i
β
i
e
i. Then the problem SGQOEP (10) has a solution.Proof.
For eachi
∈
I and for all f
∈
K, define two set-valued mappings G
i
,
H
i
:
K
→
2
K
i by
(14)
G
i
(
f
)
=
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}
,
H
i
(
f
)
=
{
g
i
∈
K
i
:
P
i
(
f
,
g
i
)
P
i
≮
α
i
e
i
and
Q
i
(
f
,
g
i
)
≯
P
i
β
i
e
i
}
.
Condition (iii) implies that, for each i
∈
I and for all f
∈
K, C
o
G
i
(
f
)
⊆
H
i
(
f
).
From condition (ii), we havef
i
∉
H
i
(
f
) for all f
∈
K and for each i
∈
I.
Thus, for eachi
∈
I and for all g
i
∈
K
i,
(15)
G
i
-
1
(
g
i
)
=
{
f
∈
K
:
F
i
(
f
,
g
i
)
<
P
i
α
i
e
i
or
F
i
(
f
,
g
i
)
>
P
i
β
i
e
i
}
.
We have complement of G
i
-
1
(
g
i
) in K:
(16)
[
G
i
-
1
(
g
i
)
]
c
=
{
β
i
e
i
≮
P
i
F
i
(
f
-
,
g
i
)
≮
P
i
α
i
e
i
}
,
which is compactly closed by virtue of condition (iv). Therefore, for each i
∈
I and for all g
i
∈
K
i
,
G
i
-
1
(
g
i
) is compactly open in K.
For eachi
∈
I, define two set-valued mappings S
i
,
T
i
:
K
→
2
K
i by
(17)
S
i
(
f
)
=
{
G
i
(
f
)
∩
A
i
(
f
)
;
if
f
∈
F
i
A
i
(
f
)
;
if
f
∈
K
∖
F
i
,
T
i
(
f
)
=
{
H
i
(
f
)
∩
B
i
(
f
)
;
if
f
∈
F
i
B
i
(
f
)
;
if
f
∈
K
∖
F
i
.
Thus, for each i
∈
I and for all f
∈
K
,
C
o
G
i
(
f
)
⊆
H
i
(
f
) and in view of condition (i), we obtain C
o
S
i
(
f
)
⊆
T
i
(
f
). It is easy to see that
(18)
S
i
-
1
(
g
i
)
=
(
A
i
-
1
(
g
i
)
∩
G
i
-
1
(
g
i
)
)
⋃
‍
(
(
K
∖
F
i
)
∩
A
i
-
1
(
g
i
)
)
for each i
∈
I and for all f
∈
K. Thus, for each i
∈
I and for all G
i
-
1
(
g
i
)
,
A
i
-
1
(
g
i
) and K
∖
F
i are compactly open in K. We have S
i
-
1
(
g
i
) being compactly open in K. Also f
i
∉
T
i
(
f
) for all f
∈
K and for each i
∈
I.
Then, by Theorem5, there exists f
-
∈
K such that S
i
(
f
-
)
=
∅ for each i
∈
I. If f
-
∈
K
∖
F
i, then A
i
(
f
-
)
=
S
i
(
f
-
)
=
∅, which contradicts the fact that A
i
(
f
) is nonempty for each i
∈
I and for all f
∈
X. Hence, f
-
∈
F
i, for each i
∈
I. Therefore, f
-
i
∈
B
i
(
f
-
) and G
i
(
f
-
)
∩
A
i
(
f
-
)
=
∅, for all i
∈
I. Thus, for each i
∈
I
,
f
-
i
∈
B
i
(
f
-
) and β
i
e
i
≮
P
i
F
i
(
f
-
,
g
i
)
≮
P
i
α
i
e
i for all g
i
∈
A
i
(
f
-
). This completes the proof.Now, we establish an existence result for SGQOEP (10) involving ϕ-condensing maps.Theorem 11.
For eachi
∈
I, assume that conditions (i)–(iv) of Theorem 10. hold. Let ϕ be a measure of noncompactness on ∏
i
∈
I
X
i. Further, assume that the set-valued mapping B
:
K
→
2
K defined as K
i is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i and F
i
:
K
×
K
i
→
Y
i is a bifunction and let the set-valued mapping A
=
∏
i
∈
I
A
i
:
K
→
2
K defined as B
(
f
)
=
∏
i
∈
I
B
i
(
f
)
,
∀
f
∈
K be ϕ-condensing. Then, there exists a solution f
-
∈
K of SGQOEP (10).Proof.
In view of Remark7, it is sufficient to show that the set-valued mapping T
:
K
→
2
K defined as T
(
f
)
=
∏
i
∈
I
T
i
(
f
)
,
∀
f
∈
K, is ϕ-condensing, where T
is are the same as defined in the proof of Theorem 10. By the definition of T
i
,
T
i
(
f
)
⊆
B
i
(
f
)
,
∀
i
∈
I and ∀
f
∈
K and therefore S
(
f
)
⊆
A
(
f
)
,
∀
f
∈
K. Since B is ϕ-condensing, by Remark 7, we have T being also ϕ-condensing.
This completes the proof.Next, we derive the existence result for the solution of SGQOEPLUB (12).Corollary 12.
For eachi
∈
I, let K
i be a nonempty convex subset of a topological vector space X
i and F
i
,
L
i
,
Q
i
:
K
×
K
i
→
R are the bifunctions, B
i
:
K
→
2
K
i is a set-valued mapping such that the set F
i
=
{
f
∈
K
:
f
i
∈
B
i
(
f
)
} is compactly closed, A
i
:
K
→
2
K
iis a set-valued mapping with nonempty values such that, for each g
i
∈
K
i
,
A
i
-
1
(
g
i
) is compactly open in K, and α
i
,
β
i
∈
R are two real numbers such that α
i
≤
β
i. For each i
∈
I, assume that the following conditions hold:(i)
for allf
∈
K, C
o
A
i
(
f
)
⊆
B
i
(
f
);
(ii)
for allf
∈
K, L
i
(
f
,
f
i
)
<
α
i or Q
i
(
f
,
f
i
)
>
β
i;
(iii)
for allf
∈
K and for every nonempty finite subset N
i
⊆
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
α
i
o
r
F
i
(
f
,
g
i
)
>
β
i
}, we have
(19)
C
o
N
i
⊆
{
g
i
∈
K
i
:
L
i
(
f
,
g
i
)
≥
α
i
a
n
d
Q
i
(
f
,
g
i
)
≤
β
i
}
;
(iv)
for allg
i
∈
K
i, the set {
f
∈
K
:
α
i
≤
F
i
(
f
,
g
i
)
≤
β
i
} is compactly closed in K;
(v)
there exist a nonempty compact subsetD of K and a nonempty compact convex subset E
i
⊂
K
i, for each i
∈
I, such that, for all f
∈
K
∖
D, there exists i
∈
I such that g
^
i
∈
E
i satisfying g
^
i
∈
A
i
(
f
) and either F
i
(
f
,
g
^
i
)
<
α
i or F
i
(
f
,
g
^
i
)
>
β
i. Then the problem SGQOEPLUB (12) has a solution.Proof.
For eachi
∈
I and for all f
∈
K, define two set-valued mappings G
i
,
H
i
:
K
→
2
K
i by
(20)
G
i
(
f
)
=
{
g
i
∈
K
i
:
F
i
(
f
,
g
i
)
<
α
i
or
F
i
(
f
,
g
i
)
>
β
i
}
,
H
i
(
f
)
=
{
g
i
∈
K
i
:
L
i
(
f
,
g
i
)
≥
α
i
and
Q
i
(
f
,
g
i
)
≤
β
i
}
.
Condition (iii) implies that, for each i
∈
I and for all f
∈
K, C
o
G
i
(
f
)
⊆
H
i
(
f
).
From condition (ii), we havef
i
∉
H
i
(
f
) for all f
∈
K and for each i
∈
I.
Thus, for eachi
∈
I and for all g
i
∈
K
i,
(21)
G
i
-
1
(
g
i
)
=
{
f
∈
K
:
F
i
(
f
,
g
i
)
<
α
i
or
F
i
(
f
,
g
i
)
>
β
i
}
.
We have complement of G
i
-
1
(
g
i
) in K:
(22)
[
G
i
-
1
(
g
i
)
]
c
=
{
α
i
≤
F
i
(
f
,
g
i
)
≤
β
i
}
,
which is compactly closed by virtue of condition (iv). Therefore, for each i
∈
I and for all g
i
∈
K
i
,
G
i
-
1
(
g
i
) is compactly open in K.
For eachi
∈
I, define two set-valued mappings S
i
,
T
i
:
K
→
2
i
K by
(23)
S
i
(
f
)
=
{
G
i
(
f
)
∩
A
i
(
f
)
if
f
∈
F
i
A
i
(
f
)
;
if
f
∈
K
∖
F
i
,
T
i
(
f
)
=
{
H
i
(
f
)
∩
B
i
(
f
)
if
f
∈
F
i
B
i
(
f
)
;
if
f
∈
K
∖
F
i
.
Thus, for each i
∈
I and for all f
∈
K
,
C
o
G
i
(
f
)
⊆
H
i
(
f
) and in view of condition (i), we obtain C
o
S
i
(
f
)
⊆
T
i
(
f
). It is easy to see that
(24)
S
i
-
1
(
g
i
)
=
(
A
i
-
1
(
g
i
)
∩
G
i
-
1
(
g
i
)
)
⋃
‍
(
(
K
∖
F
i
)
∩
A
i
-
1
(
g
i
)
)
for each i
∈
I and for all f
∈
K. Thus, for each i
∈
I and for all G
i
-
1
(
g
i
)
,
A
i
-
1
(
g
i
) and K
∖
F
i are compactly open in K. We have S
i
-
1
(
g
i
) being compactly open in K. Also f
i
∉
T
i
(
f
) for all f
∈
K and for each i
∈
I.
Then, by Theorem5, there exists f
-
∈
K such that S
i
(
f
-
)
=
∅ for each i
∈
I. If f
-
∈
K
∖
F
i, then A
i
(
f
-
)
=
S
i
(
f
-
)
=
∅, which contradicts the fact that A
i
(
f
) is nonempty for each i
∈
I and for all f
∈
X. Hence, f
-
∈
F
i, for each i
∈
I. Therefore, f
-
i
∈
B
i
(
f
-
) and G
i
(
f
-
)
∩
A
i
(
f
-
)
=
∅, for all i
∈
I. Thus, for each i
∈
I
,
f
-
i
∈
B
i
(
f
-
) and α
i
≤
F
i
(
f
-
,
g
i
)
≤
β
i for all g
i
∈
A
i
(
f
-
). This completes the proof.Now, we establish an existence result for SGQOEPLUB (12) involving ϕ-condensing maps.Theorem 13.
For eachi
∈
I, assume that conditions (i)–(iv) of Corollary 12 hold. Let ϕ be a measure of noncompactness on ∏
i
∈
I
X
i. Further, assume that the set-valued mapping B
:
K
→
2
K defined as K
i is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space X
i and F
i
:
K
×
K
i
→
R is a bifunction and let the set-valued mapping A
=
∏
i
∈
I
A
i
:
K
→
2
K defined as B
(
f
)
=
∏
i
∈
I
B
i
(
f
)
,
∀
f
∈
K, be ϕ-condensing. Then, there exists a solution f
-
∈
K of SGQOEPLUB (12).Proof.
In view of Remark7, it is sufficient to show that the set-valued mapping T
:
K
→
2
K defined as T
(
f
)
=
∏
i
∈
I
T
i
(
f
)
,
∀
f
∈
K, is ϕ-condensing, where T
i
s are the same as defined in the proof of Theorem 10. By the definition of T
i
,
T
i
(
f
)
⊆
B
i
(
f
)
,
∀
i
∈
I and ∀
f
∈
K and therefore S
(
f
)
⊆
A
(
f
)
,
∀
f
∈
K. Since B is ϕ-condensing, by Remark 7, we have T being also ϕ-condensing.
This completes the proof.



---

*Source: 848206-2014-06-19.xml*