A generalized eigenvalue problem of the form \((A_ 0+K)u=\lambda L_ 0u,\quad u\in D(A_ 0)\) is considered, where \(A_ 0\), K and \(L_ 0\) are linear differential operators and \(D(A_ 0)\) is the domain of definition of \(A_ 0\). The operators \(A_ 0\) and \(L_ 0\) are assumed to be symmetric and positive definite; K is supposed to be of lower order than \(A_ 0\), but not necessarily symmetric. J. D. P. Donnelly treated this problem [J. Inst. Math. Appl. 13, 249-261 (1974; Zbl 0287.65049)] for the case \(L_ 0=I.\)
Assuming the knowledge of the eigenvalues and corresponding eigenfunctions of the eigenvalue problem \(A_ 0v=\lambda L_ 0v\), \(v\in D(A_ 0)\) the original problem can be transformed into an eigenvalue problem with infinite matrices. After a suitable transformation of a finite part of this matrix eigenvalue problem employing the QZ-algorithm, a generalization of the Gerschgorin theorem can be used for eigenvalue inclusion. A single Gerschgorin disk being isolated from the other inclusion sets for eigenvalues furnishes an inclusion for a single eigenvalue. As an application domains of stability and instability were calculated for the Orr-Sommerfeld equation in the case of a plane Poiseuille flow.