Two-point boundary value problems for quasi-monotone dynamical systems

TL;DR

This paper investigates minimal solutions to two-point boundary value problems in quasi-monotone dynamical systems, revealing that the infimum of supersolutions equals the minimal solution and applying this to non-uniqueness in mean field games.

Contribution

It establishes the equivalence of the infimum of supersolutions with the minimal solution and applies this to demonstrate non-uniqueness in mean field games.

Findings

Infimum of supersolutions equals the minimal solution.

Non-uniqueness of strong stable solutions in certain mean field games.

Method for analyzing boundary value problems in quasi-monotone systems.

Abstract

This paper studies the existence of minimal solutions to two-point boundary value problems for quasi-monotone dynamical systems. Specifically, the pointwise infimum of all supersolutions is shown to coincide with the minimal solution. This result is then applied to establish a non-uniqueness result for strong stable solutions to a class of mean field games with a continuum of players.