Abstract
This paper reviews the old and new landmark extensions of the famous intermediate value theorem (IVT) of Bolzano and Poincare 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. Open image in new window
Highlights
N (x) ∩ TX (x) = ∅, where TX (x) is the tangent cone of convex analysis to X at x ∈ ∂ X
This work contained among other things the first modern criterion of convergence and the celebrated Bolzano-Weierstrass theorem,1 which he used as a lemma to provide the first “purely analytic proof of the theorem that between any two values, which give results of opposite sign, there lies at least one real root of the equation.”. This is known as the intermediate value theorem (IVT) in dimension one: the first existence theorem in a freshmen calculus course
This paper aims at reviewing, in some detail, the most important extensions of the Bolzano IVT to spaces of arbitrary dimensions and point to set mappings defined on compact domains that may or may not be convex
Summary
In 1817, the Bohemian philosopher, theologian and mathematician Bernhard Bolzano published a series of investigations consecrating him as one of the forefathers of modern analysis [19]. This work contained among other things the first modern criterion of convergence and the celebrated Bolzano-Weierstrass theorem, which he used as a lemma to provide the first “purely analytic proof of the theorem that between any two values, which give results of opposite sign, there lies at least one real root of the equation.” This is known as the intermediate value theorem (IVT) in dimension one: the first existence theorem in a freshmen calculus course. Cauchy was the first to contribute significantly (in 1831 and 1837) to the study of the planar case ( f1(x1, x2) = 0, f2(x1, x2) = 0) with scalar functions fi “defined inside a domain on whose boundary they do not vanish simultaneously.” He defined the Cauchy’s index, precursor of the notion of degree of a mapping.
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