In a recent paper a theorem of the alternative is stated for generalized systems of intersection type, namely T(x)ℝℝ=ab42/0ab21/, x∈¦[, where the set-valued function T is defined on a subset ¦[ of a Banach space and, for every x∈¦[, assigns a subset of a finite-dimensional space Y. This paper will extend the above result to the case where Y is a reflexive Banach space, and subsequently to the case where Y is a Hausdorff 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. Some applications to extremum and variational problems in Banach spaces are discussed.
