Nonexistence of eventually positive solutions of a difference inequality with multiple and variable delays and coefficients (Q1591923)

The authors consider the difference inequality \[ x_{n+1}-x_n+\sum^n_{i=1} p_i(n)x_{n-k_i (n)}\leq 0,\quad n=0,1,2, \dots, \] where, for each \(i=1,\dots, m,\{p_i(n)\}^\infty_{n=0}\) and \(\{k_i(n)\}^\infty_{n=0}\) are sequences of nonnegative real numbers and positive integers, respectively. They prove a sufficient condition under which the above inequality has no eventually positive solutions. As a consequence, an oscillation criterion is given for the corresponding difference equation.