The Existence of Maximum and Minimum Solutions to General Variational Inequalities in the Hilbert Lattices

Abstract

We apply the variational characterization of the metric projection to prove some results about the solvability of general variational inequalities and the existence of maximum and minimum solutions to some general variational inequalities in the Hilbert lattices.

1. Introduction

The variational inequality theory and the complementarity theory have been studied by many authors and have been applied in many fields such as optimization theory, game theory, economics, and engineering [1–12]. The existence of solutions to a general variational inequality is the most important issue in the variational inequality theory. Many authors investigate the solvability of a general variational inequality by using the techniques of fixed point theory and the variational characterization of the metric projection in some linear normal spaces. Meanwhile, a certain topological continuity of the mapping involved in the considered variational inequality must be required, such as continuity and semicontinuity.

A number of authors have studied the solvability of general variational inequalities without the topological continuity of the mapping. One way to achieve this goal is to consider a linear normal space to be embedded with a partial order satisfying certain conditions, which is called a normed Riesz space. The special and most important cases of normed Riesz spaces are Hilbert lattices and Banach lattices [1, 2, 7, 13–15]. Furthermore, after the solvability has been proved for a general variational inequality, a new problem has been raised: does this general variational inequality have maximum and minimum solutions (with respect to the partial order)? (e.g., see [7]). In this paper, we study this theme and provide some results about the existence of maximum and minimum solutions to some general variational inequalities in Hilbert lattices.

This paper is organized as follows. Section 2 recalls some basic properties of Hilbert lattices, variational inequalities, and general variational inequalities. Section 3 provides some results about the existence of maximum and minimum solutions to some general variational inequalities defined on some closed, bounded, and convex subsets in Hilbert lattices. Section 4 generalizes the results of Section 3 to unbounded case.

2. Preliminaries

In this section, we recall some basic properties of Hilbert lattices and variational inequalities. For more details, the reader is referred to [1, 2, 7, 13–15].

We say that is a Hilbert lattice if is a Hilbert space with inner product and with the induced norm and is also a poset with the partial order satisfying the following conditions:

(i)the mapping is a -preserving self-mapping on (this definition will be recalled later) for every and positive number , where defines the identical mapping on ,

(ii) is a lattice,

• (iii) the norm on is compatible with the partial order , that is,

(2.1)

A nonempty subset of a Hilbert lattice is said to be a subcomplete - sublattice of , if for any nonempty subset of , and . Since every bounded closed convex subset of a Hilbert space is weakly compact, as an immediate consequence of Lemma 2.3 in [7], we have the following result.

Lemma 2.1.

Let be a Hilbert lattice and a bounded, closed, and convex -sublattice of . Then, is a subcomplete -sublattice of .

Now, we recall the -preserving properties of set-valued mappings below. A set-valued mapping is said to be upper -preserving, if , then for any , there exists such that . A set-valued mapping is said to be lower -preserving, if , then for any , there exists such that . is said to be -preserving if it is both of upper and lower -preserving. Similarly, we can define that is said to be strictly upper -preserving, if , then for any , there exists such that and is said to be strictly (lower -preserving if , then for any , there exists such that .

Observations

(1)If is upper -preserving, then implies .

(2)If is lower -preserving, then implies .

Let be a nonempty, closed, and convex sublattice of and a mapping. Let us consider the following variational inequality:

(2.2)

An element is called a solution to the variational inequality (2.2) if, for every . The problem to find a solution to variational inequality (2.2) is called a variational inequality problem associated with the mapping and the subset , which is denoted by .

Let be a set-valued mapping. The general variational inequality problem associated with the set-valued mapping and the subset , which is denoted by , is to find , with some , such that

(2.3)

Let be the metric projection. Then, we have the well-known variational characterization of the metric projection (e.g., see [7, Lemma  2.5]): if is a nonempty, closed, and convex sublattice of a Hilbert lattice , then an element is a solution to if and only if

(2.4)

Similarly, we can have the representation of a solution to a , defined by (2.3), by a fixed point as given by relation (2.4).

3. The Existence of Maximum and Minimum Solutions to Some General Variational Inequalities Defined on Closed, Bounded, and Convex Subsets in Hilbert Lattices

In this section, we apply the variational characterization of the metric projection in Hilbert spaces to study the solvability of general variational inequalities without the continuity of the mappings involved in the considered general variational inequalities. Then, we provide some results about the existence of maximum and minimum solutions to some general variational inequalities defined on some closed, bounded, and convex subsets in Hilbert lattices. Similar to the conditions used by Smithson [15], we need the following definitions.

Let be a nonempty subset of a Hilbert lattice and a set-valued correspondence. is said to be upper (lower) -bound if there exists , such that exists and

(3.1)

is said to have upper (lower) bound -closed values, if for all , we have

(3.2)

Remarks

Let be a nonempty subset of a Hilbert lattice , a set-valued correspondence. Then, we have the following.

(1)If subset is upper -bound -closed and is upper -preserving, then is upper -bound and

(3.3)

(2)If subset is lower -bound -closed and is lower -preserving, then is lower -bound and

(3.4)

(3)If  is strictly upper -preserving and has upper bound -closed values, then

(3.5)

(4)If  is strictly lower -preserving and has lower bound -closed values, then

(3.6)

Now, we state and prove the main theorem of this paper below, which provides the existence of maximum and minimum solutions to general variational inequalities in Hilbert lattices.

Theorem 3.1.

Let be a Hilbert lattice and a nonempty closed bounded and convex -sublattice of . Let be a set-valued correspondence. Then, one has

(1)if is upper -preserving with upper bound -closed values for some function , then the problem is solvable and there exists -maximum solution to ,

(2)if is lower -preserving with lower bound -closed values for some function , then the problem is solvable and there exists -minimum solution to ,

(3)if is -preserving with both of upper and lower bounds -closed values for some function , then the problem is solvable and there exist both of -minimum and -maximum solutions to .

Proof of Theorem 3.1.

Part (1)

From (2.4), the representations of the solutions to by fixed points of a projection , we have that is a solution to if, and only if, there exists such that

(3.7)

Lemma  2.4 in [7] shows that the projection is -preserving. As a composition of upper -preserving mappings, so is also an upper -preserving mapping. From Corollary  1.8 in Smithson [15] and the variational characterization of the metric projection  (3.7), we have that the problem is solvable. Let denote the set of solutions to the problem . Then, . Since is a nonempty closed bounded and convex -sublattice of a Hilbert lattice , it is weakly compact. From Corollary  2.3 in [7], is a subcomplete -sublattice of . Hence, . Denote

(3.8)

Let

(3.9)

Then, from (3.8) and (3.9), we have

(3.10)

The first -inequality in (3.10) is based on and the property that the correspondence is upper -preserving. The second -inequality in (3.10) follows from and the fact that is upper -preserving. The third -inequality in (3.10) follows from the fact that . Then, we define

(3.11)

From (3.10), , applying the upper -preserving property of the mapping again, we get

(3.12)

that is, . Denote

(3.13)

From the upper -preserving property of , we obtain

(3.14)

which implies

(3.15)

From (3.9)−(3.11), it is clear that , and therefore, . Define

(3.16)

It holds that

(3.17)

From the upper -preserving property of the mapping again, we have

(3.18)

Applying (3.16), it implies

(3.19)

It is obvious that , so . From (3.15), we have

(3.20)

Then, (3.20), (3.16), and (3.19) together imply

(3.21)

From the assumption that , we get

(3.22)

Hence, . Then, the relation and (3.8) imply . Thus,

(3.23)

It completes the proof of part (1) of this theorem.

Part (2)

Very similar to the proof of part (1), we can prove the second part of this theorem. Denote

(3.24)

From the proof of part (1), we see that . We need to prove . Let

(3.25)

Then, we have

(3.26)

The first-order inequality in (3.26) is based on (piecewise) and the property that the correspondence is lower -preserving, which is the composition of the -preserving map and a lower -preserving map (condition 2 in this theorem). The second-order inequality in (3.26) follows from the definition of in (3.24) and the fact that ; it is because . Then, we define

(3.27)

From (3.26), , the lower -preserving of , and the Observation part in last section, we get

(3.28)

that is, . Denote

(3.29)

From the lower -preserving property of , we obtain

(3.30)

which implies

(3.31)

From (3.24)−(3.27), it is clear that , and therefore, . Define

(3.32)

that is,

(3.33)

From the lower -preserving property of the mapping again, we have

(3.34)

Applying (3.32), it implies

(3.35)

It is obvious that , so . From (3.35), we have

(3.36)

Then, (3.36), (3.32), and (3.35) together imply

(3.37)

From the assumption that , we get

(3.38)

Hence, . Then, the relation and (3.24) imply . Thus,

(3.39)

It completes the proof of part (2) of this theorem. Part (3) is an immediate consequence of parts (1) and (2). It completes the proof of Theorem 3.1.

If is a single-valued mapping, then it can be considered as a special case of set-valued mapping with singleton values. The result below follows immediately from Theorem 3.1.

Corollary 3.2.

Let be a Hilbert lattice and a nonempty closed, bounded, and convex -sublattice of . Let be a single-valued mapping such that is -preserving, for some function . Then, one has

(1)the problem is solvable,

(2)there are both of -maximum and -minimum solutions to .

For a bounded and convex -sublattice of a Hilbert lattice , the behavior of its maximum and minimum solutions to a problem should be noticeable. The following corollary can be obtained from the proof of Theorem 3.1.

Corollary 3.3.

Let be a Hilbert lattice and a nonempty, closed, bounded, and convex -sublattice of . Let be a set-valued correspondence. Then, the following properties hold.

(1)Assume that is upper -preserving for some function , and has upper bound -closed values. Let be the set of solutions to , then

(3.40)

(2)Assume that is lower -preserving for some function , and has lower bound -closed values. Then,

(3.41)

Proof of Corollary 3.3.

Part  (1)

In the proof of part (1) of Theorem 3.1, we have

(3.42)

It implies

(3.43)

From the definition of in (3.8), we get

(3.44)

Similar to the proof of part (2) of Theorem 3.1, we can prove Part (2) of this corollary.

The following corollary is an immediate consequence of Corollary 3.3.

Corollary 3.4.

Let be a Hilbert lattice and a nonempty, closed, bounded, and convex -sublattice of . Let be a set-valued correspondence. Then, the following properties hold.

(1)Assume that is upper -preserving for some function , and has upper bound -closed value at point . If is a solution to , then

(3.45)

(2)Suppose that is lower -preserving for some function , and has lower bound -closed value at point . If is a solution to , then

(3.46)

Proof of Corollary 3.4.

Part (1)

If is a solution to , then we must have

(3.47)

Substituting it into part (1) of Corollary 3.3, we get

(3.48)

The first part is proved. Similarly, the second part can be proved.

In Theorem 3.1, without the upper bound -closed condition for the values of the mapping , Theorem 3.1 may be failed, that is, if is upper -preserving that has no upper bound -closed values for some function , then, there may not exist a -maximum solution to . The following example demonstrates this argument.

Example 3.5.

Take . Define the partial order as follows:

(3.49)

Then, is a Hilbert lattice with the normal inner product in and the above partial order .

Let be the closed rhomb with vertexes , , , and . Then, is a compact (of course weakly compact) and convex -sublattice of .

Take and define as follows:

(3.50)

Then, is a set-valued mapping with compact values. From the definitions of and , we have

(3.51)

We can see that is an upper -preserving correspondence (in fact, it is both of upper -preserving and lower -preserving) and has no upper bound -closed values. One can check that the mapping has the set of fixed points below

(3.52)

which is the set of solutions to . It is clear that

(3.53)

But, the point is not a solutions to , which shows that there does not exist a -maximum solution to this problem .

Similarly, in Theorem 3.1, without the lower bound -closed condition for the values of the mapping , then Theorem 3.1 (part (2)) may be failed. That is, if is lower -preserving that has no lower bound -closed values for some function , then there may not exist a -minimum solution to . This can be demonstrated by the following example.

Example 3.6.

Take as in Example 3.5. Let be the closed rhomb with vertexes , , , and . Then, is a compact (of course weakly compact) and convex -sublattice of .

Take and define exactly the same as that in the proof of part (1)

(3.54)

We also have

(3.55)

which is the set of solutions to . It is clear that

(3.56)

But, is not a solutions to , which shows that there does not exist a -minimum solution to this problem .

Suppose that is upper (lower) -preserving. The condition that has upper (lower) bound -closed values for some function , is not necessary for the problem to have a -maximum (minimum) solution to . The following example was given by Nishimura and Ok.

Example 3.7.

Take as in Example 3.5. Let . Define

(3.57)

Take . Then, is upper -preserving. has a unique solution , which is also the -maximum solution to . But does not have upper bound -closed values (except at point ).

Example 3.7 leads us to consider some conditions on the mapping that are weaker than that in Theorem 3.1 which still guarantees the existence of a -maximum (minimum) solution to . To achieve this goal, we have the following notations. Let be a Hilbert lattice and a bounded and convex -sublattice of . Let be a set-valued correspondence. An element is said to be nondescending (nonascending) with respect to the mapping if

(3.58)

for some function .

Applying the -preserving property of , for every , we have

(3.59)

If is upper -preserving (lower -preserving), then from the upper (lower) -preserving property of the mapping , we have

(3.60)

The properties in (3.60) imply that, under the condition is upper -preserving (lower -preserving), if an element is nonascending (nonascending) with respect to the mapping , then is nondescending (nonascending) with respect to the mapping .

Definition 3.8.

For every , we denote

(3.61)

A point is said to be an upper (lower) absorbing point with respect to a set-valued mapping , if there exists with such that .

The following proposition describes some existence and uniqueness properties of upper (lower) absorbing point with respect to a set-valued mapping .

Theorem 3.9.

Let be a Hilbert lattice and a subcomplete -sublattice of . Let be a set-valued correspondence. Then, the set of upper (lower) absorbing point with respect to the mapping is not empty. In addition, if is upper (lower) -preserving, for some function , then upper (lower) absorbing point with respect to the mapping is unique.

Proof.

Since is a subcomplete -sublattice of , it contains minimum (maximum ). It is clear that ), which implies . Let . Since is a subcomplete -sublattice of , so . It implies that is an upper (lower) absorbing point with respect to the mapping .

In addition, suppose that is upper (lower) -preserving, for some function , we prove that is the unique upper (lower) absorbing point with respect to the mapping . Assume that is an upper (lower) absorbing point with respect to the mapping , such that ), for some . It is clear that ) which implies

(3.62)

On the other hand, similar to the proof of (3.10), we have

(3.63)

where the second -inequality in (3.63) follows from the definitions of and , that contain all nonascending with respect to the mapping greater than , , respectively. The -inequalities (3.63) implies is nonascending with respect to the mapping . It is clear that , and, therefore, . The definition implies

(3.64)

Combining (3.62) and (3.64), we get (similar to (3.62) and (3.64), we can prove ). It shows the uniqueness of upper (lower) absorbing point with respect to the mapping . The proposition is proved.

Now, we apply the concepts of absorbing points with respect to a mapping to extend Theorem 3.1 to the following theorem with conditions that are weaker than those in Theorem 3.1.

Theorem 3.10.

Let be a Hilbert lattice and a nonempty, closed, bounded, and convex - sublattice of . Let be a set-valued correspondence. Then, one has

(1)if is upper -preserving with upper bound -closed value at the unique upper absorbing point with respect to the mapping , for some function , then the problem is solvable and the unique upper absorbing point with respect to the mapping is a solution to ,

(2)if is lower -preserving with lower bound -closed value at the unique lower absorbing points with respect to the mapping , for some function , then the problem is solvable and the unique lower absorbing point with respect to the mapping is a solution to .

Proof of Theorem 3.10.

Part (1)

Since (as in the proof of Theorem 3.1) is a subcomplete -sublattice of . From Theorem 3.9, is the unique upper absorbing point with respect to the mapping , where is the minimum of . The assumptions of Part (1) imply

(3.65)

From the upper -preserving property of , the equation , and the definition of , we get

(3.66)

which implies

(3.67)

Since , the -inequality (3.67) implies that is nonascending with respect to the mapping , and, therefore,

(3.68)

Applying the property (3.60) that if is nonascending with respect to a mapping satisfying that is upper -preserving, for some function , then so is , from (3.67) and (3.68), it yields

(3.69)

The definition , the above relation, and (3.67) together imply

(3.70)

From (3.65) and the above equation, we obtain that the unique upper absorbing point with respect to the mapping is a solution to . Then, the solvability of the problem is proved. It completes the proof of part (1).

Similar to the proof of part (1), we can prove Part (2).

Theorem 3.11.

Let be a Hilbert lattice and a nonempty, closed, bounded, and convex -sublattice of . Let be a set-valued correspondence. Then, one has

(1)if is upper -preserving with upper bound -closed value at the unique upper absorbing point with respect to the mapping , for some function , then the unique upper absorbing point is the -maximum solution to ,

(2)if is lower -preserving with lower bound -closed value at the unique lower absorbing point with respect to the mapping , for some function , then the unique lower absorbing point is the -minimum solution to ,

(3)if is -preserving with both of upper and lower bounds -closed value at the unique upper absorbing point and the unique lower absorbing point with respect to the mapping in , for some function , then the unique upper absorbing point and the unique lower absorbing point are the -maximum and -minimum solutions to , respectively.

Proof of Theorem 3.11.

Part (1)

Let denote the set of solutions to the problem . From Theorem 3.10, we have . Since is a subcomplete -sublattice of , . Denote

(3.71)

Then, similar to (3.10) in the proof of Theorem 3.1, we can show

(3.72)

It implies . Let

(3.73)

Then, is the unique upper absorbing point in with respect to the mapping . From the Part (1) of Theorem 3.10 that the unique upper absorbing point with respect to the mapping is a solution to the problem , we obtain that . Combining the definition and the -inequality (3.73), we get . Hence, is a solution to , and therefore the problem has a maximum solution. Part (1) of this theorem is proved. Part (2) can be similarly proved. Part (3) is an immediate consequence of Part (1) and Part (2). This theorem is proved.

Notice that in Example 3.7, the conditions of Theorem 3.1 are not satisfied, that is, does not have upper bound -closed value at every point (except at point ). But there exists a -maximum solution to in Example 3.7. On the other hand, in Example 3.7, there is a unique upper absorbing point for the mapping , which is . It satisfies

(3.74)

that is,

(3.75)

Hence, Example 3.7 satisfies the conditions of Theorem 3.10 (Part (1)), and therefore there exists a -maximum solution to , which coincides with the result of Example 3.7.

Remark 3.12.

Theorem 3.1 is a special case of Theorem 3.10.

4. The Existence of Maximum and Minimum Solutions to Some General Variational Inequalities Defined on Unbounded Subsets in Hilbert Lattices

The difficulty to extend the results in bounded subsets in Hilbert lattices to unbounded subsets in Hilbert lattices is that the subcomplete property of unbounded closed convex -sublattice of a Hilbert lattice does not hold. All the proofs of Theorems 3.1–3.10 in last section are based on the property that any bounded closed convex -sublattice in Hilbert lattices is a subcomplete -sublattice. So, the techniques in those proofs are not applicable in the unbounded case. Hence, except new techniques are developed in unbounded case, we have to apply the results in last section to investigate the solvability of some problems. The following results are similar to Theorem  3.3 in [7] and Smithson Theorem  1.1 in [15].

Theorem 4.1.

Let be a Hilbert lattice and a closed convex -sublattice of . Let be a set-valued correspondence. Suppose that is upper -preserving with upper bound -closed values at all points in , for some function . In addition, assume

(i) is -preserving and there exist , with and , or

(ii) has a -minimum and there exists with .

Then, is solvable.

Proof of Theorem 4.1.

Part (i)

Set

(4.1)

Then, is -bounded closed convex -sublattice of . Then, from [7, Lemma  2.2]), it is a subcomplete -sublattice of . Similar to the proof of Theorem  3.3 in [7] or the proof of Theorem  1.1 in [15], we can show that under the conditions of part (i), the following -inequalities hold:

(4.2)

which imply

(4.3)

It is clear that satisfies all conditions of Theorem 3.1 and is a subcomplete -sublattice of . Notice that the only application of the nonempty closed bounded condition in Theorem 3.1 in the proof of Theorem 3.1 is to guarantee that the subset is a subcomplete -sublattice of . Here, the subset has been showed to be a subcomplete -sublattice of . So, applying Theorem 3.1, the problem is solvable and it has a maximum solution. Let be a solution to . Then,

(4.4)

Since , we have (piecewise)

(4.5)

From (4.4), there exists ) such that . Since , so . Using (4.5), from , we get

(4.6)

Since (from (4.3)), the above inequality implies , that is, . Hence, is a solution to . Part (i) is proved.

Part (ii)

Take to be the minimum of . The inequality can be proved by the condition in part (ii). It is obvious that the inequality holds, because is the minimum of . Then, (4.3) can be proved for part (ii) and the rest of the proof will be the same as that in the proof of part (i). It completes the proof of this theorem.

Theorem 4.2.

Let be a Hilbert lattice and a closed convex -sublattice of . Let be a set-valued correspondence. Suppose that is upper (lower) -preserving with upper (lower) bound -closed values at all points in , for some function . In addition, assume that is a -bounded closed -sublattice of . Then, is solvable and it has a maximum (minimum) solution.

Proof.

Let and . Then, the -preserving property of implies

(4.7)

Define

(4.8)

that is,

(4.9)

It is easy to see that is a -bounded -sublattice of . Then, from Lemma  2.2 in [7], is a subcomplete -sublattice of containing the set . It is clear that .We see that satisfies all conditions of Theorem 3.1. Similar to the proof of Theorem 4.1, from Theorem 3.1, the problem is solvable. Let be a solution to . Then, the proof that is a solution to is exactly the same as that in Theorem 4.1. Moreover, the maximum solution to the problem ) is also a solution to the problem .

Since , from the variational characterization of the metric projection, it yields that all solutions to the problem must be contained in . Hence, the maximum solution to the problem ) in is the maximum solution to the problem in . This theorem is proved.

Next, we consider a special type of mappings which has been used by number of authors in the fields of variational inequality theory and complementarity theory (see [4–6]). Let be a closed convex -sublattice of a Hilbert lattice . A set-valued correspondence is said to be a -completely continuous mapping if is a -bounded and closed -sublattice of . A set-valued correspondence is said to be a -completely continuous field if has the representation: , for some -completely continuous mapping . With these concepts, we provide an immediate consequence of Theorem 4.2 below.

Corollary 4.3.

Let be a Hilbert lattice and a closed convex -sublattice of . Let be a set-valued -completely continuous mapping with the representation , for some -completely continuous mapping . In addition, if f is upper (lower) -preserving with upper (lower) bound -closed values at all points in , then is solvable and it has a maximum (minimum) solution.

Proof.

Taking in Theorem 4.2, we get

(4.10)

From the condition of -completely continuous mapping, it implies that -bounded closed -sublattice of . So, satisfies all conditions of Theorem 4.2. Then, this corollary follows immediately.

The solvability of a general variational inequality in Theorem 4.2 can be extended as below. But, the existence of maximum or minimum solution will be failed.

Theorem 4.4.

Let be a Hilbert lattice and a closed convex -sublattice of . Let be a set-valued correspondence. Suppose that is upper (lower) -preserving with upper (lower) bound -closed values at all points in , for some function . In addition, if there exists a nonempty, closed, bounded, and convex -sublatticesuch that is a nonempty closed bounded and convex -sublattice in, then is solvable.

Proof.

As restricting to the mapping , the proof of this theorem is very similar to the proof of the solvability in Theorem 4.2. It is omitted.