Theorems of the alternative for set-valued functions in infinite-dimensional spaces

Abstract

In a recent paper [4] a theorem of the alternative is stated for generalized systems of intersection type, namely where the set-valued function T is defined on a subset Ω of a Banach space and, for every x∈Ω, assigns a subset of a finite-dimensional space Y. In this paper we will extend the above result to the case where Y is a reflexive Banach space, and subsequently to the case where Y is a Haus-dorff vector space. The results are achieved by making use of the image space and of nonlinear separation theorems. The results obtained contain most of known theorems of the alternative; in particular they recover the result in [2]. Some applications to extremum and variational problems in Banach spaces are discussed.

Keywords:

• Separation theorems
• alternative (theorems of)
• multifunctions
• image space

AMS 1980 Subject Classification:

• Primary:90 C 25
• Secondary:65 K 05