Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Application (II)

TL;DR

This paper computes precise derivatives and covering constants for a specific norm-reserved mapping in Euclidean space, extending previous work and applying these results to solve parameterized equations.

Contribution

It provides exact solutions for derivatives and covering constants of a particular mapping, extending the analysis to higher dimensions and applying the results to solve equations.

Findings

Covering constant is exactly 1 at all points except the origin.

Derived explicit formulas for Frechet and Mordukhovich derivatives.

Applied derivatives and covering constants to solve parameterized equations.

Abstract

We continue the study in part I for calculating the Frechet derivatives and Mordukhovich derivatives (coderivatives) and covering constants for single-valued mappings in Euclidean spaces (It is part I). In this paper, we particularly consider a norm-reserved mapping f: R^2 to R^2 that is defined by (1.1) in Section 1. We will find the precise solutions of Frechet derivative and Mordukhovich derivative at every point in R^2. By using these solutions, we will find the covering constant for this mapping f is exact 1 at every point in R^2 except the origin. Then we extend this mapping to R^4. Finally, by using the covering constant for f and by applying the Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem, we will solve some parameterized equations.