The Hartman-Stampacchia Theorem and the Maximum Displacements of Nonvanishing Continuous Vector-Valued Functions

TL;DR

This paper investigates the behavior of nonvanishing continuous vector-valued functions in various normed spaces, providing new bounds on their maximum displacements by leveraging the Hartman-Stampacchia Theorem and novel geometric methods.

Contribution

It offers the first sharp lower estimates for maximum displacements of such functions, addressing open questions in finite and infinite-dimensional spaces.

Findings

Established sharp lower bounds for maximum displacements.

Extended the Hartman-Stampacchia Theorem to new contexts.

Provided solutions to open problems posed by Professor Ricceri.

Abstract

This paper aims at giving solutions to six interesting interconnected open questions suggested by Professor Biagio Ricceri. The questions focus on the behavior of nonvanishing continuous vector-valued functions in finite-dimensional normed spaces as well as in infinite-dimensional normed spaces. Using the celebrated Hartman-Stampacchia Theorem (1966) on the solution existence of variational inequalities, we establish sharp lower estimates for the maximum displacements of nonvanishing continuous vector-valued functions. Then, combining the obtained results with suitable tools from functional analysis and several novel geometrical constructions, we get the above-mentioned solutions.