a positive real number (see Proposition \[prop:TimeTranslation\]).

The central technique we use to determine whether $(a)$ or $(b)$ holds for a given cube is the Krawczyk method [@neumaier1990interval; @moore2009introduction; @hansen2003global; @moore1977test]. For a function $f\in C^1({\mathbb{R}}^n,{\mathbb{R}}^n)$ the Krawczyk operator takes as input a rectangular set $X\subseteq  {\mathbb{R}}^n$ and produces as output a rectangular set $K(X) \subseteq {\mathbb{R}}^n$. This set $K(X)$ has the properties that, $(i)$ if $K(X) \subseteq X$, then there exists a unique point $ \hat{x} \in X$ for which $f(\hat{x})=0$, and $(ii)$ if $\hat{x} \in X$ and $f(\hat{x}) =0$, then $ \hat{x} \in K(X)$. Clearly $(i) $ implies $(a)$, and if $ X \cap K(X) = \emptyset$ then $(b)$ follows. Additionally, even if we can prove neither $(a)$ nor $(b)$ our situation could still improve; we can replace $X \mapsto X \cap K(X)$ without losing any solutions.

Adjustments are needed to generalize the Krawczyk operator to infinite dimensional systems. In [@galias2007infinite] a Krawczyk operator is defined in Hilbert space to study fixed points and period-2 orbits in an infinite dimensional map. In Section \[sec:KrawczykBanach\] we present a generalization of the Krawczyk operator to Banach spaces.

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To determine whether $(a)$ or $(b)$ holds the Krawczyk operator by itself 