The Hartman-Stampacchia Theorem and the Maximum Displacements of Nonvanishing Continuous Vector-Valued Functions
Nguyen Nang Thieu, Nguyen Dong Yen
TL;DR
The paper addresses six open questions on the maximum and generalized maximum displacements of nonvanishing continuous vector-valued functions across finite- and infinite-dimensional spaces. It centers on the Hartman-Stampacchia variational-inequality framework to derive sharp lower bounds and extends these results to arbitrary convex sets and norm changes, revealing the geometry-norm interplay. It provides a complete resolution of Question 1 (no nonvanishing counterexample) and sharp displacement estimates, plus a spectrum of results for the remaining questions, including negative results in general infinite-dimensional settings and constructive finite-displacement examples in Hilbert spaces. Overall, the work demonstrates the versatility of variational-inequality methods in nonlinear functional analysis and clarifies how displacements behave under dimension, convexity, and norm changes.
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.