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

TL;DR

This paper develops methods to calculate Frechet and Mordukhovich derivatives of single-valued mappings in Euclidean spaces, derives rules for their computation, and applies these to find covering constants and solve parameterized equations.

Contribution

It introduces new rules for computing derivatives of single-valued mappings in Euclidean spaces and applies these to determine covering constants and solve equations.

Findings

Derived rules for Mordukhovich derivatives in Euclidean spaces

Calculated explicit derivatives for specific mappings in R^2

Solved parameterized equations using covering constants and derivatives

Abstract

In this paper, we study Frechet derivatives and Mordukhovich derivatives (or coderivatives) of single-valued mappings in Euclidean spaces. At first, we prove the guideline for calculating the Frechet derivatives of single-valued mappings by their partial derivatives. Then, by using the connections between Frechet derivatives and Mordukhovich derivatives (or coderivatives) of single-valued mappings in Banach spaces, we derive the useful rules for calculating the Mordukhovich derivatives of single-valued mappings in Euclidean spaces. For practicing these rules, we find the precise solutions of the Frechet derivatives and Mordukhovich derivatives for some single-valued mappings in Euclidean spaces (in R^2, it can be extended to R^n). By using these solutions, we will find the covering constants for the considered mappings. As applications of the results about the covering constants and by applying the Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem, we solve some parameterized equations.