Abstract

Let $Q$ be a differential operator of order $m1$, $2\leqslant m \leqslant n$, for which $(a, b)$ is the interval of nonoscillation, and the Green's operator $G: L[a, b]\to W^n[a, b]$ of boundary value problem $Lx = f$, $l_i(x) = 0$, $i = 1, \ldots,n$ 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 $Lx = PVQx + f$, $l_i(x) = 0$, $i = 1,\ldots,n$ 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 $Q \widehat GP> 0.$

References

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