Upper bounds for eigenvalues of a nonlinear eigenvalue problem

Abstract

with compact, positive definite operators A, B in a complex separable Hilbert space. Turner [ll, 121 has studied this problem (see also Weinberger [13], Eisenfeld [4], W erner [14], Langer [S]). He describes the spectrum and obtains variational principles for the eigenvalues; furthermore he gives some estimates and bounds for the eigenvalues. Among others he proposes a method introduced by Aronszajn [l] to construct an operator less than B in order to obtain upper bounds for the positive eigenvalues of (1). This method has been applied to linear problems by Aronszajn [l] and Bazley and Fox [2, 31. In this paper we extend the proposal of Turner and likewise apply some methods to (I) developed by the author in [9, 101 for some other nonlinear eigenvalue problems. We describe several methods which give upper bounds to the eigenvalues of (1). The basic idea is the folIowing: We choose problems of the type (I), whose eigenvalues are upper bounds for the eigenvalues of (1) and which can be computed as the eigenvalues of similar eigenvalue problems for symmetric matrices. Thereby we use special choices for vectors and truncationoperators (see Bazley and Fox [3] for the application to linear eigenvalue problems).