 Solution

For a sequence trapped between the fixed barriers, let $M$ bound their absolute values and put $A={\sup }_{|s|\le M}|V^{\prime }(s)|$. The mean value theorem givesThe global Schauder estimate, including the fixed boundary data $\psi$, and $\Vert u_k{\Vert }_{\infty }\le M$ implyApply the supplied Hölder interpolation inequality with $KA\epsilon \le 1/2$. Its remaining supremum-norm term is bounded by $M$, yieldingIf $A=0$, 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 $k$.