On nonlinear inclusions in non-smooth domains

This paper reviews the old and new landmark extensions of the famous intermediate value theorem (IVT) of Bolzano and Poincaré to a set-valued operator Φ : E ⊃ X ⇒ E defined on a possibly non- convex, non-smooth, or even non-Lipschitzian domain X in a normed space E. Such theorems are most general solvability results for nonlinear inclusions: ∃ x_{0} ∈ X with 0 ∈ ф (x_{0})Naturally, the operator Φ must have continuity properties (essentially upper semi- or hemi-continuity) and its values (assumed to be non-empty closed sets) may be convex or have topological properties that extend convexity. Moreover, as the one-dimensional IVT simplest formulation tells freshmen calculus students, to have a zero, the mapping must also satisfy “direction conditions” on the boundary ∂X which, when X = [a,b] ⸦ E =ℝ, Φ (x) = f(x) is an ordinary single-valued continuous mapping, consist of the traditional “sign condition” f (a) f (b) ≤ 0. When X is a convex subset of a normed space, this sign condition is expressed in terms of a tangency boundary condition ф (x) ∩ TX (x) ≠ø, where TX(x) is the tangent cone of convex analysis to X at x ∈ ∂ X. Naturally, in the absence of convexity or smoothness of the domain X, the tangency condition requires the consideration of suitable local approximation concepts of non-smooth analysis, which will be discussed in the paper in relationship to the solvability of general dynamical systems. © 2012, The Author(s).