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." />
On nonlinear inclusions in non-smooth domains
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  • 作者：Hichem Ben-El-Mechaiekh (1)
  • 关键词：47H10 ; 47H04 ; 54C60 ; 34K21 ; 54C15 ; 54C55 ; 54C65 ; 65K10
  • 刊名：Arabian Journal of Mathematics
  • 出版年：2012
  • 出版时间：December 2012
  • 年：2012
  • 卷：1
  • 期：4
  • 页码：395-416
  • 全文大小：675KB
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  • 作者单位：Hichem Ben-El-Mechaiekh (1)

    1. Department of Mathematics, Brock University, St. Catharines, ON, L2S 3A1, Canada
  • ISSN：2193-5351
文摘
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 ?em class="a-plus-plus">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.

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