= Solution
For a sequence trapped between the fixed barriers, let $M$ bound their absolute values and put $A=\sup_{|s|\leq M}|V'(s)|$. The <mean value theorem> gives
$$
\|V(u_{k-1})\|_{C^{0,\alpha}}\leq C_0+A\|u_{k-1}\|_{C^{0,\alpha}}.
$$
The <global Schauder estimate>, including the fixed boundary data $\psi$, and $\|u_k\|_\infty\leq M$ imply
$$
\|u_k\|_{C^{2,\alpha}}\leq KA\|u_{k-1}\|_{C^{0,\alpha}}+C_1.
$$
Apply the supplied <Hölder interpolation inequality> with $KA\varepsilon\leq1/2$. Its remaining supremum-norm term is bounded by $M$, yielding
$$
\boxed{\|u_k\|_{C^{2,\alpha}}\leq\tfrac12\|u_{k-1}\|_{C^{2,\alpha}}+C.}
$$
If $A=0$, the stronger bound without a previous-iterate term holds directly. The constant depends on the fixed domain, exponent, barriers, boundary data and $V$, independently of $k$.
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