THE RELATIONSHIP BETWEEN EXTREMAL PRINCIPLE WITH FARKAS LEMMA IN INFINITE DIMENSONS BANACH SPACE

Abstract

In the previous article (Nguyen Van Manh-2016), we introduced the concept of non-convex normal cone and three extremal principles of variational analysis, researched the relationship of extremal principles and Farkas lemma. By using the fact that in Asplund space, all extremal systems always satisfy exact extremal principle and by introducing of  Propositon 3.1-3.2 and Theorem 3.1-3.2, we gave the  method to prove Farkas lemma in infinite dimensions Asplund space. In the general Banach space, the fact that all extremal systems always satisfy the exact extremal principle is not hold. Therefore, in this article, we propose Proposition 3.3 thereby extending Theorem 3.1 in Banach space, thereby giving method to prove Farkas's Lemma in infinite dimensions Banach space.