On a method for solving the nonlinear eigenvalue problem for a differential algebraic system of equations

For a linear differential algebraic system of equations, a method for determining the number of eigenvalues in a neighborhood of a given complex scalar is proposed and examined. It is assumed that the coefficients of the system and the entries in the matrices of the boundary conditions depend analytically on the spectral parameter. Constructions typical for the argument principle are employed, although the functions arising in the proposed method are not analytic.