Graph-theoretic conditions for injectivity of functions on rectangular domains

TL;DR

This paper establishes graph-theoretic criteria to determine when collections of differentiable functions on rectangular domains are injective, impacting the understanding of fixed points and flows in dynamical systems.

Contribution

It introduces general graph-theoretic conditions for injectivity that extend and simplify existing results on systems with signed Jacobians.

Findings

Provides sufficient conditions for injectivity based on graph properties

Shows implications for fixed points and flow behavior in dynamical systems

Derives known results as special cases of the new conditions

Abstract

This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of R^n. The results have implications for the possibility of multiple fixed points of maps and flows. Well-known results on systems with signed Jacobians are shown to be easy corollaries of more general results presented here.