### 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].