Some notes on oscillation of two-dimensional system of difference equations

Abstract

Oscillatory properties of solutions to the system of first-order linear difference equations $$ \begin {aligned} \Delta u_k & = q_k v_k \\ \Delta v_k & = -p_k u_{k+1}, \end {aligned} $$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. \endgraf We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).