Calculating Covering Constants for Mappings in Euclidean Spaces Using Mordukhovich Coderivatives with Applications

TL;DR

This paper develops methods to calculate covering constants for mappings in Euclidean spaces using Mordukhovich derivatives, providing practical rules and solutions with applications to parameterized equations.

Contribution

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

Findings

Derived explicit formulas for Mordukhovich derivatives in Euclidean spaces

Established methods to estimate covering constants for specific mappings

Applied results to solve parameterized equations using coincidence point theorems

Abstract

In this paper, we calculate the covering constants for single-valued mappings in Euclidean space by using Mordukhovich derivatives (or coderivatives). 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 between Euclidean spaces. By using these solutions, we find or estimate the covering constants for the considered mappings. As applications of the results about the covering constants involved in the Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem, we solve some parameterized equations