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Metrically differentiable set-valued functions and their local linear approximants

Alona Mokhov, Nira Dyn, Elza Farkhi

Abstract

A new notion of metric differentiability of set-valued functions at a point is introduced in terms of right and left limits of special set-valued metric divided differences of first order. A local metric linear approximant of a metrically differentiable set-valued function at a point is defined and studied. This local approximant may be regarded as a special realization of the set-valued Euler approximants of M.~S.~Nikolskii and the directives of Z.~Artstein. Error estimates for the local metric linear approximant are obtained. In particular, second order approximation is derived for a class of ``strongly'' metrically differentiable set-valued maps.

Metrically differentiable set-valued functions and their local linear approximants

Abstract

A new notion of metric differentiability of set-valued functions at a point is introduced in terms of right and left limits of special set-valued metric divided differences of first order. A local metric linear approximant of a metrically differentiable set-valued function at a point is defined and studied. This local approximant may be regarded as a special realization of the set-valued Euler approximants of M.~S.~Nikolskii and the directives of Z.~Artstein. Error estimates for the local metric linear approximant are obtained. In particular, second order approximation is derived for a class of ``strongly'' metrically differentiable set-valued maps.
Paper Structure (4 sections, 4 theorems, 37 equations)

This paper contains 4 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First metric divided differences and one-sided metric derivative
  4. The local metric linear approximant and $\alpha$-differentiability

Key Result

Theorem 3.11

Let $F:(a,b)\to {\cal K}_n$ be metrically differentiable from the right (left) at $x_0\in (a,b)$.

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Remark 3.7
  • Example 3.8
  • ...and 13 more