Approximation of nonlinear spectral problems in a Hilbert space
The article deals with the following eigenvalue problems \[ a(\lambda,u,v) = \lambda b(\lambda,u,v), \qquad 0 < \lambda < \infty, \;u,v \in V,\tag{1} \] in a Hilbert space \(V\). Here \(a: \;\Lambda \times V \times V \to {\mathbb R}\) is a symmetrical bounded positive definite form, \(b: \;\Lambda \times V \times V \to {\mathbb R}\) a compact symmetrical bounded positive definite form. It is assumed that the Rayleigh ratio \[ R(\mu,v) = \frac{a(\mu,v,v)}{b(\mu,v,v)} \] is a nonincreasing function from \(\lambda\). For a fixed \(\mu \in (0,\infty)\) each problem (1) has a countable set of eigenvalues \(\gamma_1(\mu) \leq \ldots \leq \gamma_n(\mu) \leq \ldots\), \(\gamma_n(\mu) \to \infty\) of finite multiplicity. In the first part of the article, the author proves that problem (1) has a sequence of eigenvalues \(\lambda_k\) of finite multiplicity; each of these eigenvalues is a solution of the scalar equation \(\mu = \gamma_i(\mu)\). In the second part of the article, the approximation of problem (1) by finite-dimensional problems \[ a(\lambda,u,v) = \lambda b(\lambda,u,v), \qquad 0 < \lambda < \infty, \;u,v \in V_h,\tag{2} \] is studied, where \(V_h\) is a family of finite-dimensional subspaces \(V\) satisfying the limit density condition \[ \varepsilon_h(v) = \bigg(\inf_{v_h \in V_h}\;\|v - v_h\|\bigg)\bigg|_{h \to 0} \to 0. \] The author proves the solvability of problem (2) and the convergence, as \(h \to 0\), of its solutions to solutions of (1). Moreover, estimates of type \[ 0 \leq \lambda_k^h - \lambda_k \leq c(\varepsilon^h)^2, \qquad \varepsilon^h = \sup \;\{\varepsilon_h(u):\;u \in V_h,\;\|u\| = 1\}, \] are obtained. It should be noted that some results of this article are well known in terms of self adjoint operators in Hilbert spaces.
nonlinear eigenvalue problems for quadratic forms
finite-dimensional approximations of linear problems in Hilbert spaces
scientific article; zbMATH DE number 6493791
scientific article