Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications
© S. Eshghinezhad and M. Fakhar. 2010
Received: 23 June 2010
Accepted: 2 November 2010
Published: 21 November 2010
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 . 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].
(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 ;
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].
In order to obtain a vectorial form of Ekeland-type variational principle we need the following result.
Theorem 1.3 (see ).
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  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:
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 .
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
In the following theorem we show that the previous results are equivalent to each other.
Corollary 2.3 implies Theorem 2.1.
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,
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.
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:
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:
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.
Let be the space of all continuous functions defined on . By Corollary 11-7-3, 11-7-4 of , with weak-topology is quasi barreled but it is not barreled. Therefore, by Proposition 11-2-5 of , with weak*-topology is not locally complete.
Let be the space of all differentiable functions whose derivative is continuous. Then is not a complete space. Therefore, by Proposition  is not a locally complete space.
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.
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 , we omit it.
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:
As a consequence of Theorems 2.1 and 2.5 we can obtain two versions of vectorial Caristi-Kirk's fixed-point theorem.
The proof of the following results is similar to that of Theorem 3.3 and we omit it.
In the following we give two versions of vectorial Takahashi's nonconvex minimization.
The proof of the following results is similar to that of Theorem 3.5 and we omit it.
In the final step we obtain a vectorial form of Oettli-Théra type theorem.
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. .
Theorems 2.1 and 1.3 are equivalent to each other.
Theorems 2.1, 2.5, 3.3, 3.5 and 3.7 are mutually equivalent.
Theorem 2.5 ⇔ Theorem 2.1.
Theorem 3.3 ⇔ Theorem 2.1.
Theorem 3.5 ⇔ Theorem 2.1.
Theorem 3.7 ⇔ Theorem 2.1.
M. FAKHAR was partially supported by the Center of Excellence for Mathematics, University of Isfahan.
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