Research Article
Open Access
Generalized Vector Complementarity Problems with Moving Cones
- Lu-Chuan Ceng1 and
- Yen-Cherng Lin2Email author
Journal of Inequalities and Applications20092009:617185
https://doi.org/10.1155/2009/617185
© L.-C. Ceng and Y.-C. Lin 2009
Received: 3 October 2009
Accepted: 30 November 2009
Published: 7 December 2009
Abstract
We introduce and discuss a class of generalized vector complementarity problems with moving cones. We discuss the existence results for the generalized vector complementarity problem under inclusive type condition. We obtain equivalence results between the generalized vector complementarity problem, the generalized vector variational inequality problem, and other related problems. The theorems presented here improve, extend, and develop some earlier and very recent results in the literature.
1. Introduction and Preliminaries
It is well known that vector variational inequalities were initially studied by Giannessi [1] and ever since have been widely studied in infinite-dimensional spaces see, for example, [2–8] and the references therein.
Very recently, Huang et al. [9] considered a class of vector complementarity problems with moving cones. They established existence results of a solution for this class of vector complementarity problems under an inclusive type condition. They also obtained some equivalence results among a vector complementarity problem, a vector variational inequality problem, a vector optimization problem, a weak minimal element problem, and a vector unilateral optimization problem in ordered Banach spaces. Their results generalized the main results in [4].
The purpose of this paper is to introduce and discuss a class of generalized vector complementarity problems with moving cones which is a variable ordering relation. We derive existence of a solution for this class of generalized vector complementarity problems under an inclusive type condition. This inclusive condition requires that any two of the family of closed and convex cones satisfy an inclusion relation so long as their corresponding variables satisfy certain conditions. We also obtain some equivalence results among a generalized vector complementarity problem, a generalized vector variational inequality problem, a generalized vector optimization problem, a generalized weak minimal element problem, and a generalized vector unilateral optimization problem under some monotonicity conditions and some inclusive type conditions in ordered Banach spaces. The theorems presented in this paper improve, extend, and develop some earlier and very recent results in the literature including [4, 9].
Let
be a Banach space, and
a subset of
. The topological interior of a subset
in
is denoted by
. A nonempty subset
in
is called a convex cone if
, and
for any
. The relations
and
in
are defined as
if
and
if
, for any
. Similarly, we can define the relations
and
if we replace the set
by
.
is called a pointed cone if
is a cone and
.
Let
be the space of all continuous linear mappings from
to
. We denote the value of
at
by
.
Let
be two Banach spaces, and
a set-valued mapping such that, for each
,
is a proper closed convex and pointed cone with apex at the origin and
, and
. Very recently, Huang et al. [9] introduced the following three kinds of vector complementarity problems.
(Weak) vector complementarity problem (VCP): finding
such that
such that
(1.1)
Positive vector complementarity problem (PVCP): finding
such that
(1.2)
Strong vector complementarity problem (SVCP): finding
such that
(1.3)
We remark that if
for all
, where
is a closed, pointed, and convex cone in
with nonempty interior
, then (VCP), (PVCP) and (SVCP) reduce to the problems considered in Chen and Yang [4]. In [9], they actually only studied the first two kinds complementarity problems. For the existence results of (SVCP), we refer the reader to our recent results [Submitted, On the
-implicit vector complementarity problem].
Motivated and inspired by the above three kinds of vector complementarity problems, in this paper we introduce three kinds of generalized vector complementarity problems. Let
be two Banach spaces, and
a set-valued mapping such that, for each
,
is a proper closed convex and pointed cone with apex at the origin and
, let
be a single-valued mapping, and
a set-valued mapping, where
is a collection of all nonempty subsets of
. We consider the following three kinds of generalized vector complementarity problems.
(Weak) generalized vector complementarity problem (GVCP): finding
and
such that
and
such that
(1.4)
Generalized positive vector complementarity problem (GPVCP): finding
and
such that
and
such that
(1.5)
Generalized strong vector complementarity problem (GSVCP): finding
and
such that
and
such that
(1.6)
We remark that if
the identity mapping of
, and
is a single-valued mapping, then three kinds of generalized vector complementarity problems reduce to three kinds of vector complementarity problems in Huang et al. [9], respectively.
2. Existence of a Solution for GVCP
Huang et al. [9] established some equivalence results between the positive vector complementarity problem and the vector extremum problem and also sufficient conditions for the existence of a solution of the vector extremum problem. In this section, we extend their results to the cases involving the set-valued mappings.
Let
be an arbitrary real Hausdorff topological vector spaces, and
a Banach space.
denotes the space of all continuous linear mappings from
to
. Let
be a nonempty set of
, and
a set-valued mapping such that, for each
,
is a proper closed convex and pointed cone with apex at the origin and
. Let
be a subset of
. For each
, a point
is called a minimal point of
with respect to the cone
if
;
is the set of all minimal points of
with respect to the cone
; a point
is called a weakly minimal point of
with respect to the cone
if
;
is the set of all weakly minimal points of
with respect to the cone
, we refer the reader to [10] for more detail.
Let
be a single-valued mapping and let
be a set-valued mapping. Now, we consider the following generalized vector complementarity problem (GVCP). Find
and
such that
be a single-valued mapping and let
be a set-valued mapping. Now, we consider the following generalized vector complementarity problem (GVCP). Find
and
such that
(2.1)
A feasible set of (GVCP) is
(2.2)
We consider the following generalized vector optimization problem (GVOP):
(2.3)
A point
is called a weakly minimal solution of (GVOP) with respect to the cone
, if
is a weakly minimal point of (GVOP) with respect to the cone
, that is,
. We denote the set of all weakly minimal solutions of (GVOP) with respect to the cone
by
and the set of all weakly minimal solutions of (GVOP) by
, that is,
is called a weakly minimal solution of (GVOP) with respect to the cone
, if
is a weakly minimal point of (GVOP) with respect to the cone
, that is,
. We denote the set of all weakly minimal solutions of (GVOP) with respect to the cone
by
and the set of all weakly minimal solutions of (GVOP) by
, that is,
(2.4)
Theorem 2.1.
If
and, for some
, there exists
such that
, then the generalized vector complementarity problem (GVCP) is solvable.
Proof.
Let
and
. Then
, and
and
. Then
, and
(2.5)
It follows that
is a solution of (GVCP). This completes the proof.
We remark that if
the identity mapping of
and
is a single-valued mapping from
to
, then Theorem 2.1 coincides with Theorem
in Huang et al. [9].
Definition 2.2.
Let
,
be two set-valued mappings with
for every
,
a single-valued mapping, and
a subset of
. We say that
is inclusive with respect to
if for any
,
,
be two set-valued mappings with
for every
,
a single-valued mapping, and
a subset of
. We say that
is inclusive with respect to
if for any
,
(2.6)
It is easy to see that, if
for all
, where
is a closed, pointed, and convex cone in
, then
is inclusive with respect to
.
Example 2.3.
Let
be the identity mapping of
. For each
, define
be the identity mapping of
. For each
, define
(2.7)
and, for each
,
,
(2.8)
Also, define
(2.9)
Then it is easy to see that
is inclusive with respect to
. Indeed, for any
with
, and
, if
, then
and so
. Therefore,
and
is inclusive with respect to
.
Theorem 2.4.
Suppose that
is inclusive with respect to
. If there exist at most a finite number of solutions for (GVCP), then (GVCP) is solvable if and only if
, and there exists
such that
.
Proof.
Let
be a solution of (GVCP). Then there exists
such that
be a solution of (GVCP). Then there exists
such that
(2.10)
If
, then
, then
(2.11)
and hence the conclusion holds. If
, by the definition of a weakly minimal solution, there exists
such that
, by the definition of a weakly minimal solution, there exists
such that
(2.12)
This implies that
(2.13)
Since
, and
is inclusive with respect to
, it follows that
and this implies that
, and
is inclusive with respect to
, it follows that
and this implies that
(2.14)
Thus,
is a solution of (GVCP) and
. Continuing this process, there exists
such that
is a solution of (GVCP) and
, since (GVCP) has at most a finite number of solutions. Thus,
and
is a solution of (GVCP) and
. Continuing this process, there exists
such that
is a solution of (GVCP) and
, since (GVCP) has at most a finite number of solutions. Thus,
and
(2.15)
Combining this result and Theorem 2.1, we have the conclusion of the theorem.
Remark 2.5.
- (1)
If
the identity mapping of
,
is a single-valued mapping from
to
, and
for all
, where
is a closed, pointed, and convex cone in
, then
satisfies the inclusive condition with respect to
and Theorem 2.4 reduces to Theorem
of Chen and Yang [4]. - (2)
If
the identity mapping of
and
is a single-valued mapping from
to
, then Theorem 2.4 coincides with Theorem
of Huang et al. [9].
We next consider the generalized positive vector complementarity problem (GPVCP). Finding
and
such that
and
such that
(2.16)
Let
(2.17)
Consider the following generalized vector optimization problem (GVOP)
to be
to be
(2.18)
We denote the set of all minimal points of (GVOP)0 with respect to the cone
by
, that is,
, and denote the set of all minimal points of (GVOP)
by
by
, that is,
, and denote the set of all minimal points of (GVOP)
by
(2.19)
Using a similar argument of Theorem 2.1, we have the following results of solvability for (GPVCP).
Theorem 2.6.
If
and there exists
such that
, then (GPVCP) is solvable.
Theorem 2.7.
Suppose that
is inclusive with respect to
. If there exist at most a finite number of solutions of (GPVCP), then (GPVCP) is solvable if and only if
, and there exists
such that
.
One remarks that If
the identity mapping of
and
is a single-valued mapping from
to
, then Theorems 2.6 and 2.7 coincide with Theorems
and
of Huang et al. [9], respectively.
3. Equivalences between Generalized Vector Complementarity
3.1. Problems and Generalized Weak Minimal Element Problems
Let
be two Banach spaces, and
a set-valued mapping such that, for each
,
is a proper closed convex and pointed cone with apex at the origin and
, let
be a single-valued mapping, and
a set-valued mapping, where
is a collection of all nonempty subsets of
, and
a given operator.
Define the feasible set associated with
and
and
(3.1)
We now consider the following five problems.
(i)The generalized vector optimization problem (GVOP)
: for a given
, finding
such that
: for a given
, finding
such that
(3.2)
(ii)The generalized weak minimal element problem (GWMEP): finding
such that
.
(iii)The generalized vector complementarity problem (GVCP): finding
such that
where
is associated with
in the definition of
.
(iv)The generalized vector variational inequality problem (GVVIP): finding
and
such that
and
such that
(3.3)
(v)The generalized vector unilateral optimization problem (GVUOP): finding
such that
.
We remark that if
the identity mapping of
and
is a single-valued mapping from
to
, then the (GVOP)
, (GWMEP), (GVCP), (GVVIP), and (GVUOP) reduce to Huang, et al.'s problems (VOP)
, (WMEP), (VCP), (VVIP), and (VUOP), respectively; see [9] for more details.
Definition 3.1 (see [4]).
A linear operator
is called weakly positive if, for any
implies that
.
Definition 3.2.
Let
and
be two Banach spaces and
a linear operator from
to
. If the image of any bounded set in
is a self-sequentially compact set in
, then
is called completely continuous.
A mapping
is said to be convex if
is said to be convex if
(3.4)
for all
and
.
Definition 3.3.
Let
and
be two mappings.
is said to be
-subdifferentiable at
if there exists
such that
and
be two mappings.
is said to be
-subdifferentiable at
if there exists
such that
(3.5)
If
is
-subdifferentiable at
, then we define the
-subdifferential of
at
as follows:
is
-subdifferentiable at
, then we define the
-subdifferential of
at
as follows:
(3.6)
If
is
-subdifferentiable at each
, then we say that
is
-subdifferentiable on
.
Remark 3.4.
We note that as the mentions in [9], if
and
are two Banach spaces, a mapping
is Fréchet differentiable at
if there exists a linear bounded operator
such that
and
are two Banach spaces, a mapping
is Fréchet differentiable at
if there exists a linear bounded operator
such that
(3.7)
where
is said to be the Fréchet derivative of
at
. The mapping
is said to be Fréchet differentiable on
if
is Fréchet differentiable at each point of
. If
is convex and Fréchet differentiable on
, then
is said to be the Fréchet derivative of
at
. The mapping
is said to be Fréchet differentiable on
if
is Fréchet differentiable at each point of
. If
is convex and Fréchet differentiable on
, then
(3.8)
If
is Fréchet differentiable at
, then
is
-subdifferentiable at
and
.
If
is
-subdifferentiable on
, then for each
we have
is
-subdifferentiable on
, then for each
we have
(3.9)
Definition 3.5.
Let
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
. The norm
in
is called strictly monotonically increasing on
[9] if, for each
,
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
. The norm
in
is called strictly monotonically increasing on
[9] if, for each
,
(3.10)
For the example of the strictly monotonically increasing property, we refer the reader to [9, Example
].
Theorem 3.6.
Let
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
a set-valued mapping with closed, convex, and pointed cones values such that
for all
. Suppose that
(1)
is the
-subdifferential of a convex operator
;
(2)
is a weakly positive linear operator;
(3)there exists
such that
is one to one and completely continuous, where
is associated with
in the definition of
;
(4)
is a topological dual space of a real normed space and the norm
in
is strictly monotonically increasing on
.
If (GVVIP) is solvable, then (GVOP)
, (GWMEP), (GVCP), and (GVUOP) are also solvable.
Corollary 3.7 (see [9, Theorem
]).
Let
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
that a family of closed, pointed, and convex cones in
such that
for all
. Suppose that
(1)
is the Fréchet derivative of a convex operator
;
(2)
is a weakly positive linear operator;
(3)there exists
such that
is one to one and completely continuous, where
;
(4)
is a topological dual space of a real normed space and the norm
in
is strictly monotonically increasing on
.
If (VVIP) is solvable, then (VOP)
, (WMEP), (VCP), and (VUOP) are also solvable.
Proof.
Since
the identity mapping of
and
is a single-valued mapping from
to
, we have
the identity mapping of
and
is a single-valued mapping from
to
, we have
(3.11)
Utilizing Theorem 3.6, we immediately obtain the desired conclusion.
Remark 3.8.
If
for all
, where
is a closed, pointed, and convex cone in
, then Corollary 3.7 coincides with Theorem
of Chen and Yang [4].
We need the following propositions to prove Theorem 3.6.
Proposition 3.9.
Let
and
be two mappings, and let
be the
-subdifferential of
. Then
solves (GVUOP) which implies that
solves (GVVIP). If in addition,
is a convex mapping, then conversely,
solves (GVVIP) which implies that
solves (GVUOP).
Proof.
Let
be a solution of (GVUOP). Then
and
, that is,
for all
. Since
is a convex cone,
be a solution of (GVUOP). Then
and
, that is,
for all
. Since
is a convex cone,
(3.12)
Also, since
is
-subdifferentiable on
, it follows that for all
is
-subdifferentiable on
, it follows that for all
(3.13)
This implies that
(3.14)
and hence
(3.15)
Thus,
solves (GVVIP).
Conversely, let
solve (GVVIP). Then there exists
such that
solve (GVVIP). Then there exists
such that
(3.16)
Since
is
-subdifferentiable on
, we have for all
is
-subdifferentiable on
, we have for all
(3.17)
and hence
(3.18)
This implies that
(3.19)
Consequently,
solves (GVUOP). This completes the proof.
Proposition 3.10.
If
solves (GVVIP), then
also solves (GVCP). Conversely, if
for all
, then
solves (GVCP) which implies that
solves (GVVIP).
Proof.
Let
be a solution of (GVVIP). Then there exists
such that
be a solution of (GVVIP). Then there exists
such that
(3.20)
Letting
, we get
. For
with any
, we have
, we get
. For
with any
, we have
(3.21)
Thus,
is a solution of the (GVCP).
Conversely, let
solve the (GVCP). Then there exists
such that
solve the (GVCP). Then there exists
such that
(3.22)
This implies
(3.23)
and so
(3.24)
This completes the proof.
Proposition 3.11.
Let
be a weakly positive linear operator. Then
solves (GWMEP) which implies that
solves (GVOP)
.
Proof.
Let
be a solution of (GWMEP). Then
and
, that is, for any
,
. Since
is a weakly positive linear operator, it follows that
and so
be a solution of (GWMEP). Then
and
, that is, for any
,
. Since
is a weakly positive linear operator, it follows that
and so
(3.25)
hence
solves (GVOP)
. This completes the proof.
Definition 3.12 (see [9]).
Let
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
,
a nonempty subset of
.
(1)If, for some
, then
is called a section of the set
.
(2)
is called weakly closed if
for all
, then
.
(3)
is called bounded below if there exists a point
in
such that
.
Lemma 3.13 (see [11]).
Let
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
,
a nonempty subset of
and
the topological dual space of a real normed space
. Suppose there exists
such that the section
is weakly closed and bounded below and the norm
in
is strictly monotonically increasing, then the set
has at least one weakly minimal point.
Lemma 3.14.
If (GVVIP) is solvable, then the feasible set
is nonempty.
Proof.
Let
be a solution of (GVVIP). Then there exists
such that
be a solution of (GVVIP). Then there exists
such that
(3.26)
Taking
with any
, we know that
and
with any
, we know that
and
(3.27)
Thus,
. This completes the proof.
Let
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
. For any
,
is called an order interval.
Lemma 3.15 (see [4]).
Let
be a Banach space,
a proper closed convex and pointed cone with apex at the origin and
. If the norm
in
is strictly monotonically increasing, then the order intervals in
are bounded.
Proposition 3.16.
Suppose that (GVVIP) is solvable and
(1)there exists
in
such that
is one to one and completely continuous, where
is associated with
in the definition of
;
(2)
is the topological dual space of a real normed space
and the norm
in
is strictly monotonically increasing.
Then (GWMEP) has at least one solution.
Proof.
By the assumption and Lemma 3.14,
. Let
be a point such that
is one to one and completely continuous, where
is associated with
in the definition of
, and let
with
(weakly). Since
. Let
be a point such that
is one to one and completely continuous, where
is associated with
in the definition of
, and let
with
(weakly). Since
(3.28)
by Lemma 3.15,
is bounded and so is
. Since
is completely continuous,
is a self-sequentially compact set and so
implies that there exists a subsequence
which converges to
. We get a point
such that
is bounded and so is
. Since
is completely continuous,
is a self-sequentially compact set and so
implies that there exists a subsequence
which converges to
. We get a point
such that
(3.29)
On the other hand, since
(weakly) and
is completely continuous,
(weakly) and
is completely continuous,
(3.30)
By the uniqueness of the limit, we get
. Since
is one to one,
, and so
. Since
is weakly closed, it follows from Lemma 3.13 that
has a weakly minimal point
such that
for all
. Therefore, (GWMEP) has at least one solution. This completes the proof.
Definition 3.17.
Let
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
a set-valued mapping with closed, convex and pointed cones values such that
for all
. Let
be a single-valued mapping and
a set-valued mapping.
is called
-positive if
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
a set-valued mapping with closed, convex and pointed cones values such that
for all
. Let
be a single-valued mapping and
a set-valued mapping.
is called
-positive if
(3.31)
We now consider the generalized positive vector complementarity problem (GPVCP). Finding
and
such that
and
such that
(3.32)
The feasible set related to (GPVCP) is defined as
(3.33)
Let us consider the following problems.
The generalized vector optimization problem
: finding
such that
.
The generalized weak minimal element problem
: finding
such that
.
The generalized positive vector complementarity problem (GPVCP): finding
such that
such that
(3.34)
where
is associated with
in the definition of
.
The generalized vector variational inequality problem (GVVIP): finding
and
such that
and
such that
(3.35)
The generalized vector unilateral optimization problem (GVUOP): for a given mapping
, finding
such that
.
Definition 3.18.
A set-valued mapping
is said to be
-strictly monotone where
is single-valued, if
is said to be
-strictly monotone where
is single-valued, if
(3.36)
Definition 3.19 (see [9]).
We say that
satisfies an inclusive condition if, for any
,
satisfies an inclusive condition if, for any
,
(3.37)
It is easy to see that, if
for all
, where
is a closed, pointed, and convex cone in
, then
satisfies the inclusive condition.
Example 3.20.
Let
, and
, and
(3.38)
for all
. Then it is easy to check that
satisfies the inclusive condition.
Proposition 3.21.
Let
be
-strictly monotone and
a solution of (GPVCP). If
satisfies the inclusive condition, then
is a weakly minimal point of
(i.e.,
solves (GWMEP)
).
Proof.
It is easy to see that
. If
(where
denotes the boundary of
), then
solves (GWMEP)
. Otherwise, there exists
such that
and so
. If
(where
denotes the boundary of
), then
solves (GWMEP)
. Otherwise, there exists
such that
and so
(3.39)
which is a contradiction. If
, by the
-strict monotonicity of
,
, by the
-strict monotonicity of
,
(3.40)
Suppose
. Since
is
-positive,
and
. Since
is
-positive,
and
(3.41)
By the assumption, we get
and so
and so
(3.42)
It follows that
(3.43)
and thus
(3.44)
for some
. This implies
. This implies
(3.45)
Since
and
,
and
,
(3.46)
and so
(3.47)
which leads to a contradiction. Therefore,
does not hold, that is,
for all
. It follows that
solves (GWMEP)
. This completes the proof.
Proposition 3.22.
If
solves (GPVCP), then
also solves (GVVIP).
Proof.
Suppose that
solves (GPVCP). Then
and there exists
such that
solves (GPVCP). Then
and there exists
such that
(3.48)
If
, then
, then
(3.49)
and so
(3.50)
which is a contradiction. It follows that
(3.51)
and
solves (GVVIP). This completes the proof.
Similarly, we can obtain other equivalence conditions. We have the following theorem.
Theorem 3.23.
Let
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
a family of closed, pointed, and convex cones in
such that
for all
. Suppose that
satisfies the inclusive condition and
(1)
is the
-subdifferential of the convex operator
;
(2)
is a weakly positive linear operator;
(3)
is
-strictly monotone.
If (GPVCP) is solvable, then (GVOP)
, (GWMEP)
, (GPVCP), (GVVIP), and (GVUOP) have at least a common solution.
Corollary 3.24 (see [9, Theorem
]).
Let
be two Banach spaces,
a proper closed convex and pointed cone with apex at the origin and
, and
a family of closed, pointed, and convex cones in
such that
for all
. Suppose that
satisfies the inclusive condition and
(1)
is the Fréchet derivative of the convex operator
;
(2)
is a weakly positive linear operator;
(3)
is strictly monotone.
If (PVCP) is solvable, then (VOP)
, (WMEP)
, (PVCP), (VVIP), and (VUOP) have at least a common solution.
Proof.
Note that
the identity mapping of
and
is a single-valued mapping from
to
. From Theorem 3.23, we immediately obtain the desired conclusion.
Remark 3.25.
If
for all
, where
is a closed, pointed, and convex cone in
, then Corollary 3.24 coincides with Theorem
of Chen and Yang [4].
Declarations
Acknowledgments
This research of the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality Grant (075105118). The second author would like to thank the support of NSC97-2115-M-039-001- Grant from the National Science Council of Taiwan.
Authors’ Affiliations
(1)
Department of Mathematics, Shanghai Normal University
(2)
Department of Occupational Safety and Health, China Medical University
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