Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$, since every open ball of $(H,d_{H})$ is a subset of $H$; accordingly $V\cap H=V$ in the notation of Hilbert Triples: Standing Notation and Background (v). Let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$ that is locally strictly proper, satisfies the second-order structure condition, satisfies the shift-continuity condition and satisfies the tail-insensitivity condition.

Let $C\in\mathbb{R}$ and let $u_{1},u_{2}:H\to\mathbb{R}$ be viscosity solutions of $F$ on $H$ that are continuous on $H$ and satisfy

$$|u_{1}(x)|\le C\qquad\text{and}\qquad |u_{2}(x)|\le C\qquad\text{for every }x\in H .$$

(i) Then $u_{1}(x)=u_{2}(x)$ for every $x\in H$.