Schranken für Eigenwerte Nichtlinearer Eigenwertaufgaben

Zusammenfassung

In this paper eigenvalue problems of type

$$u - \lambda Au - {\lambda ^2}Bu + \sum\limits_{k = 1}^n {\frac{{{\lambda ^2}}}{{\lambda - {a_k}}}{H_k}u = 0}$$

(*)

are considered where A,B,H₁,...,H_n are compact selfadjoint operators in a complex separable Hilbert space, a_l,...,a_n, a_i ≠ a_l, i ≠ j, a_i ≠ 0, i,j = 1,2,...,n, are real numbers, B,a_l ^-1H₁,..., a_n ^-1H_n are positive and H₁,...,H_n are of finite rank. The method of orthogonal invariants as developed by Fichera is applied to the eigenvalue problem (*) giving lower bounds for the eigen-values or the absolute values of the eigenvalues of (*) whereas the Rayleigh-Ritz method yields upper bounds. For completion an inclusion theorem for the eigenvalues of (*) is given.