Convergence of eigenelements of a Steklov-type problem in a half-band with a small hole

We consider a Steklov-type problem for the Laplace operator in a half-band containing a small hole. On the lateral boundaries and the boundary of the hole, the Dirichlet conditions are stated, and on the base of the half-band the Steklov spectral condition. We prove that eigenvalues of this problem tend to zero as the small parameter (the “diameter” of the hole) vanishes.