Applications of a theorem concerning sets with connected sections
Biagio Ricceri · Topological Methods in Nonlinear Analysis · 1995
Dedicated to Professor Ky Fan, with my greatest admiration and esteemAs the reader can notice, the title of the present paper differs from that of [3] only because the term connected replaces the term convex.This is not casual.Indeed, it remains our aim to show, by means of a series of further applications, the usefulness of our recent Theorem 2.3 of [6] which, in a certain sense, can be regarded as a "connected" version of the famous Theorems 1 and 2 of [3].In the sequel, given a product space X × Y , we denote by p X and p Y the projections from X × Y onto X and Y , respectively.Moreover, if A ⊆ X × Y , then for every x ∈ X and y ∈ Y , we put A x = {v ∈ Y : (x, v) ∈ A} and A y = {u ∈ X : (u, y) ∈ A}.Also, when, in proper settings, they will appear, the symbols B, int(B), ∂B, aff(B), and ri(B) will denote, respectively, the closure, the interior, the boundary, the affine hull, and the relative interior (that is, the interior in aff(B)) of the set B.For the reader's convenience, we recall the statement of Theorem 2.3 of [6]:Theorem 1 ([6], Theorem 2.3).Let X, Y be two topological spaces, with Y admitting a continuous bijection onto [0, 1], and let S, T be two subsets of X × Y , with S connected and, for each x ∈ X, T x connected.Moreover, assume 1991