Two-point boundary value problems for quasi-monotone dynamical systems
Lorena Bociu, Madhumita Roy, Khai T. Nguyen
TL;DR
This work develops a supersolution-based framework to prove the existence of a minimal Carathéodory solution for two-point boundary value problems arising from quasi-monotone dynamical systems, assuming the supersolution set is uniformly bounded below. It shows that the infimum over supersolutions yields a genuine minimal solution and provides a broad structural condition, expressed in terms of ${\bf f}_{\min},{\bf f}_{\max},{\bf g}_{\min},{\bf g}_{\max}$, guaranteeing boundedness and thus existence for all boundary data. The framework is applied to mean field games with a continuum of players, yielding nonuniqueness of stable strong solutions: a trivial solution and a nontrivial minimal solution, illustrating multiplicity of stable equilibria in a simple MFG. Overall, the paper bridges forward-backward characteristic analysis with monotone dynamical systems and MFG theory, delivering concrete criteria for existence and stability of minimal solutions.
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.