A differential operator $\mathscr L$, arising from the differential expression $$ lv(t)\equiv(-1)^rv^{[n]}(t)+\sum_{k=0}^{n-1}p_k(t)v^{[k]}(t)+Av(t),\quad0\le t\le1 $$ and system of boundary value conditions $$ P_\nu[v]=\sum_{k=0}^{n_\nu}\alpha_{\nu k}v^{[k]}(1)=0,\quad\nu=1,\dots,\mu,\quad0\le\mu<n $$ is considered in a Banach space $E$. Here $v^{[k]}(t)=\bigl(\alpha(t)\frac d{dt}\bigr)^kv(t)$, $\alpha(t)$ being continuous for $t\ge0$, $\alpha(t)>0$ for $t>0$ and $\int_0^1\frac{dz}{\alpha(z)}=+\infty$; the operator $A$ is strongly positive in $E$. The estimates, are obtained for $\mathscr L$: $$ \|A(\mathscr L+\lambda)^{-1}\|_{C_{01}^\alpha([0,1];E)}+\sum_{k=0}^n(1+|\lambda|)^{(n-k)/n}\Bigl\|\frac{d^{[k]}}{dt^k}(\mathscr L+\lambda)^{-1}\Bigr\|_{C_{01}^\alpha([0,1];E)}\le M, $$ $n$ even, $\lambda$ varying over a half plane.