In this paper we establish characterizations of the containment of the set { : , ( ) } { : ( ) 0}, x X x C g x K x X f x where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g X Y : is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. | Science & Technology Development, Vol 19, Asymptotic Farkas lemmas for convex systems Nguyen Dinh University International, VNU – HCM Tran Hong Mo Tien Giang University, Tien Giang (Received on June 5th 2015, accepted on November 21th 2016) ASTRACT In this paper we establish characterizations of the containment of the set {x X : x C, g(x) K} {x X : f (x) 0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g : X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization Keywords: Farkas lemma, sequential Farkas lemma, limit inferior, limit superior INTRODUCTION AND PRELIMINARIES Farkas-type results have been used as one of the main tools in the theory of optimization [8]. Typical Farkas lemma for cone-convex systems characterizes the containment of the set where is a closed convex subset of a locally convex Hausdorff topological vector space (brieftly, lcHtvs), is a closed convex cone in another lcHtvs and is a convex mapping, in a reverse convex set, define by the proper, lower semi-continuous, convex function. If the characterization holds under some constraint qualification condition or qualification condition then it is called non-asymptotic Farkas-type result (see [6], [10-12]). Otherwise .
