Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications

Abstract

By using a Dane' drop theorem in locally convex spaces we obtain a vectorial form of Ekeland-type variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk's fixed-point theorem, Takahashi's nonconvex minimization theorem, and Oettli-Théra's theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given.

1. Introduction and Preliminaries

A very important result in nonlinear analysis about the existence result for an approximate minimizer of a lower semicontinuous and bounded below function was first presented by Ekeland [1]. Known nowadays as Ekeland's variational principle (in short, EVP), it has significant applications in the geometry theory of Banach spaces, optimization theory, game theory, optimal control theory, dynamical systems, and so forth; see [1–11] and references therein. It is well known that EVP is equivalent to many famous results, namely, the Caristi-Kirk fixed-point theorem, the petal theorem, Phelp's lemma, Danês' drop theorem, Oettli-Théra's theorem and Takahashi's theorem, see, for example, [4, 6, 7, 10, 12–19]. Many authors have obtained EVP on complete metric spaces [1, 10, 19, 20] and in locally convex spaces [20–24]. Along with the development of vector optimization and motivated by the wide usefulness of EVP, many authors have been interested in obtaining this principle for vector-valued functions and set-valued mappings; see [3, 5, 8, 9, 11–15, 21, 22, 25]. Recently, this principle has been obtained for bifunctions and applied to solve equilibrium problem in nonconvex setting [10, 12, 14–16, 20, 26, 27]. Our goal in this paper is to obtain Ekeland's variational principle for vector-valued bifunctions in locally convex spaces. By using this result we derive the existence of solution of vector equilibrium problem in the setting of seminormed spaces. Also, we obtain vectorial Caristi-Kirk's fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and vectorial Oettli-Théra's theorem. Moreover, we show that these results and Dane' drop theorem are equivalent to each other. Let us, introduce some known definitions and results which will be used in the sequel.

Let be a Hausdorff locally convex real vector space. A subset of is said to be a disc, if is bounded and absolutely convex. Let be the vector subspace spanned by , and be the Minkowski functional of , then is a normed space. If is a Banach space, then is called a Banach disc. A sequence in is said to be locally convergent to an element if there is a disc in such that the sequence is convergent to in and is said to be locally Cauchy if there is a disc in such that is a Cauchy sequence in . We say that is a locally complete space if every locally Cauchy sequence is locally convergent. This is equivalent to that each bounded subset of is contained in a certain Banach disc. A nonempty subset of is said to be locally complete if every locally Cauchy sequence in is locally convergent to a point in . The subset is said to be locally closed if for any locally convergent sequence in , its local limit point belongs to . It is well known that every sequentially complete locally convex space is locally complete and the converse is not true; see [28, 29].

Let be a locally convex space ordered by the nontrivial closed convex cone as follows:

(1.1)

For every we write

(1.2)

Definition 1.1.

Let be a nonempty subset of a locally convex space , be a locally convex space ordered by the nontrivial closed convex cone . A vector-valued function is said to be

(1)-locally lower semicontinuous if for every the set is locally closed in ;

(2)-upper semicontinuous at if for any neighborhood of , there exists a neighborhood of such that , for all . If is -upper semicontinuous at each point of , then is said to be -upper semicontinuous on ;

(3)-bounded from below, if there exists such that for all .

Assume that the interior of () is nonempty and is a vector-valued function. The vector equilibrium problem (in short, VEP) is to find such that

(1.3)

It is well known that VEP includes fundamental mathematical problems like vector optimization, vector variational inequality, and vector complementarity problem. For further details on VEP, one can refer to [23, 24, 30–32].

Let . Recall the definition of the Gerstewitz function [33]:

(1.4)

The following lemma describes some properties of the Gerstewitz function and it will be used in the sequel. For its proof we refer the reader to [5, 6, 33].

Lemma 1.2.

For each and , the following statements are satisfied:

(i).

(ii).

(iii).

(iv).

(v) is positively homogeneous and continuous on .

(vi), for all .

(vii)is monotone, that is, if , then .

In order to obtain a vectorial form of Ekeland-type variational principle we need the following result.

Theorem 1.3 (see [17]).

Let be a locally closed subset of a locally convex space and a locally closed, bounded convex subset of with . If either or is locally complete, then for each , there exists such that , where denotes the convex hull of .

2. Vectorial Ekeland-Type Variational Principle

Recently, Qiu [18] obtained some versions of Ekeland's variational principle in locally convex spaces, which only need to assume local completeness of some related sets. Motivated by this paper we obtain some versions of EVP for vector-valued bifunctions in locally convex spaces. These results extend Qiu's results to vector-valued bifunctions.

Throughout this section is a locally convex space, is locally closed subset of , is a family of seminorms generating the locally convex topology on , is a Hausdorff locally convex space ordered by a closed convex cone with and . We consider a vector-valued bifunction , a family of positive real numbers and the following assumptions:

(A1) for all .

(A2) for any .

(A3) is C-bounded from below for all .

(A4) is -locally lower semicontinuous for any .

(A5)There exists such that the set is locally complete.

(A6)The set is locally complete.

Notice that if assumptions (A1) and (A2) hold, then is called half distance. The following result is a vectorial form of Ekeland-type variational principle.

Theorem 2.1.

Suppose that assumptions (A1)–(A4) are satisfied. If either assumption (A5) or assumption (A6) holds, then for any , there exists such that

(i), for any ;

(ii)For any , there exists such that .

Proof.

Without loss of generality, we may assume that and put with the product topology, then the topology can be generated by a family of seminorms, where , for all . If and , then since is -bounded from below for all we have . Take any fixed real number and put . Then is exactly the set , for all . If the set is locally complete, then is locally complete and if is locally complete, then is locally complete. Furthermore, is bounded closed convex subset of and . Hence, by Theorem 1.3, there exists

(2.1)

such that

(2.2)

According to (2.1), we have , so

(2.3)

and for each

(2.4)

Therefore, by (2.3) and (2.4), we have

(2.5)

Hence, the part (i) holds. We show that the point satisfies in the part (ii). Let . Since and is monotone, then . On the other hand we have . Hence,

(2.6)

But from (2.5) we have

(2.7)

Therefore, . Thus, Also, clearly . Hence, we have

(2.8)

Therefore, by (2.1), and so .

Supposing that and , we consider the following two cases.

Case 1.

If , then . Since is monotone, then for all we have

(2.9)

But is half distance and is sublinear, thus

(2.10)

Hence, by the part (ii) of Lemma 1.2;

(2.11)

Case 2.

Let , we will show that . If not, we assume that , that is,

(2.12)

Since and separates points in , we conclude that there exists such that thus . Put

(2.13)

Since is a cone,

(2.14)

that is,

(2.15)

By (2.1),

(2.16)

Since is a convex cone, by (2.15) and (2.16) we have

(2.17)

so

(2.18)

It is easy to verify that

(2.19)

Hence,

(2.20)

Therefore, and so , which it is a contradiction. This shows that . Thus, there exists such that

(2.21)

On the other hand by (A2) we have

(2.22)

Hence,

(2.23)

Therefore,

(2.24)

Remark 2.2.

In the above theorem, if assumption (A5) holds, then instead of assumption (A3), we can assume that is -bounded from below. Also, if assumption (A6) holds, assumption (A3) can be replaced by the following assumption: is -bounded from below for some .

As a consequence of the above theorem we can obtain the following result which is a vectorial version of Theorem 3.1 of [18].

Corollary 2.3.

Let be a function such that is -bounded from below and is -locally lower semicontinuous. Furthermore, let assumption (A6) holds or there exists such that the set is locally complete. Then there exists such that

(i), for any ;

(ii)for any , there exists such that .

Proof.

It is enough in Theorem 2.1 to consider for all .

In the following theorem we show that the previous results are equivalent to each other.

Theorem 2.4.

Corollary 2.3 implies Theorem 2.1.

Proof.

Let be defined as follows:

(2.25)

It is an easy task to derive the assumptions of Corollary 2.3 for the above function from the assumptions of Theorem 2.1. Therefore, there exists which satisfies the conditions (i) and (ii) of Corollary 2.3. Hence,

(i), for any ;

(ii)for any , there exists such that .

Also, by assumption (A2) we have . Thus,

(2.26)

Let be a convex subset of containing 0. The Minkowski functional of is defined as follows:

(2.27)

We extend by an additional element such that for all and for all .

By using Theorem 2.1 we obtain another version of vectorial form of Ekeland-type variational principle in which the perturbation function is the Minkowski functional of a bounded set.

Theorem 2.5.

Suppose that assumptions (A1)–(A4) are satisfied. Let be a locally closed, bounded convex set containing 0 and α be a positive real number. Let be locally complete or assumption (A5) holds. Then, for any , there exists such that:

(i);

(ii)For any , .

Proof.

Suppose that is the absolutely convex hull of the set , then is a normed space. Assume that

(2.28)

Since , then . Also, is -locally lower semicontinuous and is locally lower semicontinuous, then is closed in . Suppose that is restricted to . If is locally complete then is a Banach disk and is a Banach space. If the set is locally complete, then is a complete set in . Therefore, by Theorem 2.1 there exists such that:

(a).

(b)For any and ,

(2.29)

Since , then the part (a) holds. Now, we show that the part (b) holds. If and , then (2.29) becomes

(2.30)

Let and , then , so the part (b) holds. Let , , and . Since , then

(2.31)

Therefore, which is a contradiction. Hence,

(2.32)

Assuming that is a locally complete locally convex space, the condition on local completeness of some related subsets is automatically satisfied. However, we give the following examples of spaces which are not locally complete but the condition on local completeness of some related subsets is satisfied.

Example 2.6.

Let be the space of all continuous functions defined on . By Corollary 11-7-3, 11-7-4 of [29], with weak-topology is quasi barreled but it is not barreled. Therefore, by Proposition 11-2-5 of [29], with weak*-topology is not locally complete.

Moreover,

(2.33)

But by Banach-Alaoglu theorem this set is weak*-compact. Also is separable, so weak*-topology on unit ball is metrizable. Hence, this set is locally complete.

Example 2.7.

Let be the space of all differentiable functions whose derivative is continuous. Then is not a complete space. Therefore, by Proposition [28] is not a locally complete space.

Also, the set is not locally complete. Suppose that is defined as follows:

(2.34)

where is ordered by the cone . If we choose , then the set is locally complete.

3. Caristi-kirk's Fixed-Point Theorem, Takahashi's Nonconvex Minimization Theorem, and Oettli-Théra's Theorem and Equilibrium Problem

In this section, we obtain an existence result for solution of vector equilibrium problem in nonconvex setting. Also, some new versions of the vectorial Caristi-Kirk fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and the vectorial Oettli-Théra theorem are given.

Theorem 3.1.

Let be a weakly compact subset of a semi normed space . Suppose that is a function satisfying assumptions (A1)–(A5) together with some and is -upper semicontinuous for every . Then, there exists such that

(3.1)

Proof.

Assume that , then for all . Taking and , from Theorem 2.5, we find a sequence , such that

(3.2)

By the weakly compactness of , we can assume that weakly converges to . Suppose that for a suitable . Take a neighborhood of such that . By -upper semicontinuity of , there exists a natural number such that , for . Moreover, if is big enough, , thus

(3.3)

This is a contradiction.

In the above theorem, when is not necessarily weakly compact we have the following result. Since its proof is similar to Theorem 4 of [20], we omit it.

Theorem 3.2.

Let be a nonempty subset of a reflexive semi normed space . Suppose that is a function satisfying assumptions (A1)–(A5) together with some and is -upper semicontinuous for every . Let the following coercivity condition holds:

There exists a nonempty closed bounded subset of such that for all there exists with satisfying .

Then, there exists such that

(3.4)

As a consequence of Theorems 2.1 and 2.5 we can obtain two versions of vectorial Caristi-Kirk's fixed-point theorem.

Theorem 3.3.

Suppose that all of the conditions of Theorem 2.1 are satisfied. Assume that is a set-valued mapping with nonempty valued and the following property holds:

(3.5)

Then, there exists such that .

Proof.

Let be a point which satisfies in the parts (i) and (ii) with . We show that . If there exists and , then by Theorem 2.1 there is a such that and it is a contradiction.

The proof of the following results is similar to that of Theorem 3.3 and we omit it.

Theorem 3.4.

Suppose that all of the conditions of Theorem 2.5 are satisfied and is a set-valued mapping with nonempty valued and the following condition holds:

(3.6)

Then, there exists such that .

In the following we give two versions of vectorial Takahashi's nonconvex minimization.

Theorem 3.5.

Suppose that all of conditions of Theorem 2.1 are satisfied and for each with , there exists such that for any . Then, there exists such that that is .

Proof.

By Theorem 2.1 there exists such that for each there exists such that . If , then by assumption there exists such that , and this is a contradiction.

The proof of the following results is similar to that of Theorem 3.5 and we omit it.

Theorem 3.6.

Suppose that all of the conditions of Theorem 2.5 are satisfied and for each with , there exists such that . Then, there exists such that .

In the final step we obtain a vectorial form of Oettli-Théra type theorem.

Theorem 3.7.

Assume that all of the conditions of Theorem 2.1 are satisfied and . Let such that for every there exists such that and . Then .

Proof.

By Theorem 2.1 there exists such that satisfies the condition (ii) in Theorem 2.1. It is easy to see that .

4. Equivalences

In this section, we show that Dane' drop theorem (Theorem 1.3), two versions of vectorial form of Ekeland-type variational principle, vectorial Caristi-Kirk's fixed-point theorem and vectorial Takahashi's nonconvex minimization theorem are equivalent. In order to show that Theorems 1.3 and 2.1 are equivalent to each other we need the following definition which was introduced by Cheng et al. [34].

Definition 4.1.

Two nonempty subsets and of the locally convex space are said to be strongly Minkowski separated if and only if there exist a continuous seminorm and such that either

(4.1)

or

(4.2)

Theorem 4.2.

Theorems 2.1 and 1.3 are equivalent to each other.

Proof.

It is only enough to show that Theorem 2.5 implies Theorem 1.3. Since , then there exist and such that

(4.3)

Therefore, by Lemma 1 of [7] and are strongly Minkowski separated. Hence, there exist a continuous seminorm and such that

(4.4)

Without loss of generality we may assume that and put

(4.5)

Also, is bounded, thus is finite. Now, we apply Theorem 2.5 for the set , function , and a positive number α which . It is easy to see that the assumptions (A1)–(A4) are satisfied. If or is locally complete, then is locally complete, so is locally complete. Therefore, by Theorem 2.5 there exists a point such that

(4.6)

Let , then , where and . Hence,

(4.7)

Therefore, and so by (4.6), we conclude that .

Theorem 4.3.

Theorems 2.1, 2.5, 3.3, 3.5 and 3.7 are mutually equivalent.

Proof.

• (1) Theorem 2.5  ⇔ Theorem 2.1.

It is enough to show that Theorem 2.5  ⇒ Theorem 2.1. Choose

(4.8)

Then is a bounded, closed absolutely convex set. If is the Minkowski functional of , then

(4.9)

Now, if we apply Theorem 2.5 for the set , then we obtain Theorem 2.1.

• (2) Theorem 3.3  ⇔ Theorem 2.1.

It is enough to show that Theorem 3.3  ⇒ Theorem 2.1. Define as follows:

(4.10)

Obviously, for any , . And for each and ,

(4.11)

By Theorem 3.3, there exists such that . Therefore, for any , . Thus, there exists , such that . Hence, the part (ii) in Theorem 2.1 holds.

• (3) Theorem 3.5  ⇔ Theorem 2.1.

It is enough to show that Theorem 3.5  ⇒ Theorem 2.1. Without loss of generality we assume that . Let

(4.12)

Since , then is nonempty. Also, for any , is -locally lower semi continuous, thus is locally closed. Hence, if is locally complete, then is locally complete. Suppose that the part (ii) of Theorem 2.1 does not hold, that is, for all there exists such that

(4.13)

Therefore,

(4.14)

Hence, , so by Theorem 3.5, there exists such that .

However, there exists such that which satisfies (4.13). Therefore, and so , which is a contradiction.

• (4) Theorem 3.7  ⇔ Theorem 2.1.

It is enough to show that Theorem 3.7  ⇒ Theorem 2.1. Suppose that is defined as follows:

(4.15)

Choose . If , then there exists such that . Therefore, assumption of Theorem 3.7 is satisfied. Hence, there exists . From the definition of the results (i) and (ii) of Theorem 2.1 are satisfied.

Remark 4.4.

• (a) By the same proof as that of Theorem 4.3, one can show that Theorems 2.5, 3.4, and 3.6 are equivalent to each other.

• (b) By the same proof as that of Theorems 5.2 and 5.3 in [18] one can prove that Theorem 2.1 and the Phelp's lemma ([18, Theorem 5.1]) are equivalent.