Asymptotic Properties of Parametric Eigenvalue Problems in a Hilbert space

We study a parametric eigenvalue problem in an infinite-dimensional Hilbert space arisingin the mechanics of loaded thin-walled structures. Asymptotic properties of solutions depending onthe loading parameters are established. The initial infinite-dimensional problem is approximatedin a finite-dimensional subspace. Theoretical error estimates of approximate solutions areobtained. Effective numerical methods for calculating the main resonance frequency and thecorresponding resonance form of vibrations based on the asymptotic formulas are proposed.