On the general nonlinear self-adjoint spectral problem for differential-algebraic systems

We consider a general self-adjoint spectral problem, nonlinear with respect to the spectral parameter, for linear differential-algebraic systems of equations. Under some assumptions, we present a method for reducing such a problem to a general self-adjoint nonlinear spectral problem for a system of differential equations. In turn, this permits one to pass to a problem for a Hamiltonian system of ordinary differential equations. In particular, in this way, one can obtain a method for computing the number of eigenvalues of the original problem lying in a given range of the spectral parameter.