On the question of extended convexity of Green operator

Let $Q$ be a differential operator of order $m-1$, $2\leqslant m \leqslant n$, for which $(a, b)$ is the interval of nonoscillation, and the Green's operator $G\colon L[a, b]\to W^n[a, b]$ of boundary value problem $Lx=f$, $l_i(x)=0$, $i=1,\dots,n$ has the property of generalized convexity: $QGP>0$ for some linear homeomorphism $P$ of Lebesgue space $L[a,b]$. Under some conditions, we prove, that the perturbed boundary value problem $Lx=PVQx+f$, $l_i(x)=0$, $i=1,\dots,n$ is also uniquely solvable in the Sobolev space $W^n[a,b]$ and the Green's operator $\widehat G$ inherits the property of $G$, that is $Q\widehat GP>0$.