# Bound-narrowing trajectory

This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold

## Definition

Let $M$ be a manifold and $u = u(t,x)$ be a function $\R \times M \to \R$, where:

• $t$ denotes the time parameter, and varies in $\R$
• $x$ denotes the spatial parameter, and varies in $M$

In other words, $u$ is a trajectory (or path) in the space of all functions from $M$ to $\R$.

Then, $u$ is said to be bound-narrowing if it is both max-decreasing and min-increasing or equivalently if it satisfies the following condition:

$C_1 \le u(0,x) \le C_2 \forall x \in M \implies C_1 \le u(t,x) \le C_2 \forall x \in M$