Variational approach to solution of nonlinear vector spectral problem for the case of self-adjoint positively semi-defined operators

The variational approach to solution of generalized eigenvalue problem with nonlinear occurrence of the vector spectral parameter is presented. The equivalence of spectral problem and corresponding variational problem are proved. The idea of coordinate-wise descent is the basis of numerical algorithm for minimization of the functional that is the square of the norm of misclosure of homogenous operator equation. The relaxation properties of the constructed iteration process are proved.