In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims
$$ -3\alpha\left(\begin{array}{cc} I\\ & I \end{array}\right)\preceq\left(\begin{array}{cc} X\\ & -Y \end{array}\right)\preceq3\alpha\left(\begin{array}{cc} I & -I\\ -I & I \end{array}\right). $$
However, it seems as though the $\epsilon$ term associated with $\epsilon(|x|^2+|y|^2)$ does not make an appearance. Where did this term go?
Even if this is a typo, it is not particularly concerning, since we can still derive the same bound with $4\alpha$ instead of $3\alpha$ by taking $\alpha$ large enough and $\epsilon$ small enough (in fact, I believe $\epsilon \leq 1$ and $\alpha \geq 3$ does the trick). Purely out of curiosity, however, I am wondering whether this is a typo or if there is some serious cleverness underneath it.
Here is what I believe is going on. In their use of Theorem 3.2, they bundle the $\varepsilon (|x|^2 + |y|^2)$ term with $u(x)-v(y)$, so $\varphi(x,y) = \frac{\alpha}{2}|x-y|^2$, $u_1(x) = u(x)-\varepsilon |x|^2$ and $u_2(y) = -v(y) + \varepsilon |y|^2$. This is why you see $X+2\varepsilon I$ and $Y - 2\varepsilon I$ in the unnumbered equation directly above (5.17). The usual choice of $\varepsilon = 1/\alpha$ still works here, and there is no typo.