Arabian Journal of Mathematics

, Volume 1, Issue 4, pp 395–416

On nonlinear inclusions in non-smooth domains

Authors

    • Department of MathematicsBrock University
Open AccessSurvey Paper

DOI: 10.1007/s40065-012-0048-8

Cite this article as:
Ben-El-Mechaiekh, H. Arab. J. Math. (2012) 1: 395. doi:10.1007/s40065-012-0048-8

Abstract

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 \({\Phi : E \supset X \rightrightarrows E}\) 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: \({\exists x_{0} \in X}\) with \({0 \in \Phi (x_{0}).}\) 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 \({X = [a,b] \subset E = \mathbb{R}}\), 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 \({\Phi (x) \cap T_{X}(x) \neq \emptyset}\), where TX(x) is the tangent cone of convex analysis to X at \({x \in \partial X}\). 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.
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Mathematics Subject Classification

47H1047H0454C6034K2154C1554C5554C6565K10
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© The Author(s) 2012

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