Comparison Principle for Slope-Based Solutions of the Eikonal Equation on a Complete Metric Space with Interpolation Points

In the setting of Slope-Based Viscosity Solutions on a Metric Space: Standing Notation, suppose that $(X,d)$ is complete and has interpolation points. Let $K\subseteq X$ be nonempty and such that $\Omega=X\setminus K$ is open, and write $D(x)=\operatorname{dist}(x,K)$ for $x\in X$. Let $f:\Omega\to\mathbb{R}$ be uniformly continuous on $\Omega$, and suppose there is a positive real $c_{0}$ with $c_{0}\le f(x)$ for every $x\in\Omega$. Let $u,v:X\to\mathbb{R}$ be bounded above and below, with $u$ upper semicontinuous on $X$ and $v$ lower semicontinuous on $X$, such that the restriction of $u$ to $\Omega$ is an s-subsolution and the restriction of $v$ to $\Omega$ is an s-supersolution of the eikonal equation $|\nabla u|=f$ in $\Omega$.

¶Suppose moreover that for every positive real $\beta$ there is a positive real $\sigma$ such that

¶Then $u(x)\le v(x)$ for every $x\in X$.