Arabian Journal of Mathematics

, Volume 1, Issue 4, pp 395–416

On nonlinear inclusions in non-smooth domains

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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


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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© The Author(s) 2012

This article is published under license to BioMed Central Ltd. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of MathematicsBrock UniversitySt. CatharinesCanada