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Arabian Journal of Mathematics

, Volume 1, Issue 4, pp 395-416

On nonlinear inclusions in non-smooth domains


  • Hichem Ben-El-Mechaiekh
    • Department of MathematicsBrock University

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 T X (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.

Mathematics Subject Classification

47H10 47H04 54C60 34K21 54C15 54C55 54C65 65K10


This paper was presented at the Conference on Fixed Point Theory held at the University of Tabuk, Kingdom of Saudi Arabia, May 2012. The author wishes to thank the University of Tabuk for its wonderful hospitality.

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© The Author(s) 2012