Convergence and structure theorems for order-preserving dynamical systems with mass conservation

Abstract

We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which 0 is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987 and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.1. Introduction. In this paper we study order-preserving dynamical systems having a certain kind of "mass conservation property" and establish a general theory on the convergence of bounded orbits and the existence of fixed points. We will then apply our general theory to a number of problems including reaction-diffusion equations possibly with nonlocal terms and some delay-differential equations. Our theory implies the existence of multiple stationary and/or time-periodic solutions as an immediate corollary.Our present study was originally motivated by the analysis of mathematical models for transportation in molecular motor systems and those for chemical reversible