Generalized Vector Complementarity Problems with Moving Cones

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

(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

(1.4)

Generalized positive vector complementarity problem (GPVCP): finding and such that

(1.5)

Generalized strong vector complementarity problem (GSVCP): finding 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

(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,

(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

(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 ,

(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

(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

(2.10)

If , then

(2.11)

and hence the conclusion holds. If , 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

(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

(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

(2.16)

Let

(2.17)

Consider the following generalized vector optimization problem (GVOP) 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

(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

(3.1)

We now consider the following five problems.

(i)The generalized vector optimization problem (GVOP): 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

(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

(3.4)

for all and .

Definition 3.3.

Let 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:

(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

(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

(3.8)

If is Fréchet differentiable at , then is -subdifferentiable at and .

If 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 ,

(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

(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,

(3.12)

Also, since 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

(3.16)

Since 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

(3.20)

Letting , 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

(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

(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

(3.26)

Taking 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

(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

(3.29)

On the other hand, since (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

(3.31)

We now consider the generalized positive vector complementarity problem (GPVCP). Finding 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

(3.34)

where is associated with in the definition of .

The generalized vector variational inequality problem (GVVIP): finding 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

(3.36)

Definition 3.19 (see [9]).

We say that 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

(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

(3.39)

which is a contradiction. If , by the -strict monotonicity of ,

(3.40)

Suppose . Since is -positive, and

(3.41)

By the assumption, we get and so

(3.42)

It follows that

(3.43)

and thus

(3.44)

for some . This implies

(3.45)

Since 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

(3.48)

If , 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].