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 
We now consider the following five problems.
(i)The generalized vector optimization problem (GVOP)
: for a given
, finding
such that
(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
(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
for all
and
.
Definition 3.3.
Let
and
be two mappings.
is said to be
-subdifferentiable at
if there exists
such that
If
is
-subdifferentiable at
, then we define the
-subdifferential of
at
as follows:
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
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
If
is Fréchet differentiable at
, then
is
-subdifferentiable at
and
.
If
is
-subdifferentiable on
, then for each
we have
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
,
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
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,
Also, since
is
-subdifferentiable on
, it follows that for all 
This implies that
and hence
Thus,
solves (GVVIP).
Conversely, let
solve (GVVIP). Then there exists
such that
Since
is
-subdifferentiable on
, we have for all 
and hence
This implies that
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
Letting
, we get
. For
with any
, we have
Thus,
is a solution of the (GVCP).
Conversely, let
solve the (GVCP). Then there exists
such that
This implies
and so
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
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
Taking
with any
, we know that
and
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
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
On the other hand, since
(weakly) and
is completely continuous,
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
We now consider the generalized positive vector complementarity problem (GPVCP). Finding
and
such that
The feasible set related to (GPVCP) is defined as
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
where
is associated with
in the definition of
.
The generalized vector variational inequality problem (GVVIP): finding
and
such that
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
Definition 3.19 (see [9]).
We say that
satisfies an inclusive condition if, for any
,
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
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
which is a contradiction. If
, by the
-strict monotonicity of
,
Suppose
. Since
is
-positive,
and
By the assumption, we get
and so
It follows that
and thus
for some
. This implies
Since
and
,
and so
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
If
, then
and so
which is a contradiction. It follows that
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].