Discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter

In this paper we establish new oscillation theorems for discrete symplectic eigenvalue problems with general boundary conditions. We suppose that the symplectic coefficient matrix of the system and the boundary conditions are nonlinear functions of the spectral parameter and that they satisfy certain natural monotonicity assumptions. In our new theory we admit possible oscillations in the coefficients of the symplectic system and the boundary conditions by incorporating their nonconstant rank with respect to the spectral parameter. We also prove necessary and sufficient conditions for boundedness of the real part of spectrum of these eigenvalue problems.