Sequential Farkas lemmas for convex systems

In this paper we introduce two new versions of Farkas lemma for two kinds of convex systems in locally convex Hausdorff topological vector spaces which hold without any constraint qualification conditions. These versions hold in the limits and will be called sequential Farkas lemmas. Concretely, we establish sequential Farkas lemmas for cone-convex systems and for systems which are convex with respect to a sublinear function. The first result extends some known ones in the literature while the second is a new one.


INTRODUCTION
Farkas lemma is one of the most important results from fundamental mathematics.It is equivalent to the Hahn-Banach theorem [10] and has had many applications in economics [9], in finance [8], in mechanics, and in many fields of applied mathematics such asmathematical programming and optimal control.For more details, see the recent survey paper [7].
The first correct version of Farkas lemma for a linear system was introduced by the physicist Gyula Farkas in 1902.Since then, many generalized versions of this "lemma" have been proved.Most of these extensions are nonasymptotic versions and hold under some qualification conditions [5,7].In the recent years, several asymptotic versions for generalized Farkas lemma have been established and found many applications in optimization problems [4,6,11].
In this paper, we first introduce an asymptotic version of Farkas lemma for a coneconvex system which extends some known results in the literature [6,11].Imitating the idea in [5], we then establish the corresponding asymptotic version of Farkas lemma for systems which are convex with respect to a sublinear function, which appears for the first time and may pay the way for applications to some areas in applied mathematics.

Trang 21 NOTATIONS AND PRELIMINARIES
Let X and Y be locally convex Hausdorff topological vector spaces (l.c.H.t.v.s.), with their topological dual spaces  X and  Y , endowed with  w -topologies, respectively.Given a set * X A  , we denote by cl A the closure of A (with respect to the The function . The set of all proper, lower semi-continuous (lsc) and convex functions on X is denoted by  .

X 
The conjugate function regarding the set The indicator function of the set We add to Y a greatest element with respect to K  , denoted by K  , which does not belong to Y and let for every .

 Y y
We shall use the following conventions on subset in the product space Y X  , and in this case, the set Here the relation " Given an extended sublinear function } { : , we adapt the notion S - convex (i.e., convex with respect to a sublinear function) (see [13]) and introduce the one corresponding to an extended sublinear function Some properties of limit inferior and limit superior of nets of extended real numbers will be quoted in the next lemma.
) ( be nets of extended real numbers.Then the following statements hold: provided that the right hand side of the inequalities are well-defined.Moreover, the equalities hold whenever one of the nets is convergent.

SEQUENTIAL FARKAS LEMMA FOR K - CONVEX SYSTEMS
In this section we will introduce a version of Farkas lemma for cone-convex systems which holds in the limit form without any qualification condition.The result extends the ones in [6], [11] and [4] in some senses.

Theorem 2 [Approximate Farkas lemma 1]
Let Y X , be locally convex Hausdorff topological vector spaces, Then the following statements are equivalent: Thus, there are nets * * * ( ) ,( ) , , . Note also that for any and so, it follows from the definition of Taking the limit superior both sides of (2) we get 0 = ( ) limsup Assume that (ii) holds, i.e., there exists a net for all .
The proof is complete.

Remark 3 The equivalence between (i) and
(ii) in Theorem 2 was established in [6], [11] under the assumption that X C = and g is a continuous, K -convex mapping with values in Y .This equivalence was also established in [4] by another approach, called dual approach, for the case where Y X g  : , which is much stronger than our assumption that g is K -epi-closed (see [2]).Our result extends all the results in the mentioned papers.

SEQUENTIAL FARKAS LEMMA FOR SUBLINEAR-CONVEX SYSTEMS
In this section, we will establish a sequential version of Farkas lemma for systems of inequalities involving sublinear-convex mappings.The key tools used here are the technique of switching a sublinear-convex system to a cone-convex system used in the recent paper [5] and Theorem 2 from the previous section.
(5) Then the following statements are equivalent: 1 When this condition holds, it is also said that the function is (S, g)-compatible [13] Now let K ~ be the closed convex cone defined by 0} ) , ( : Then it is easy to see that g ~ is K ~-convex as well. The assumption (4) and K  playing the roles of X , Y , C , g , f , and K , respectively, and with 0. =  From (a) and the definitions of f ~, g ~, we have which shows that (i) from Theorem 2 holds, and hence, by this theorem there exists a net Since Therefore, (7) can be rewritten as .