A note on intersection of lower semicontinuous multifunctions

Abstract

Let F 1 {F_1} and F 2 {F_2} be closed and convex valued multifunctions from a topological space X X to a normed space Y Y . Assume that the multifunctions are lower semicontinuous at x 0 {x_0} . We proof that the intersection multifunction F = F 1 ∩ F 2 F = {F_1} \cap {F_2} is lower semicontinuous at x 0 {x_0} provided F ( x 0 ) F({x_0}) is bounded and has nonempty interior.