Oscillation and nonoscillation for difference equations with variable delays (Q1431980)

The paper deals with difference equation with variable delays \[ x_{n+1} - x_n + \sum_{i=1}^m p_i(n) x_{n-k_i(n)} = 0, \quad n \geq n_0, \tag{1} \] where \(p_i(n)\) are positive real numbers, \(m, n_0\) and \(k_i(n)\), \(n \geq n_0,\) are positive integers, \(\lim_{n\to 0}(n - k_i(n)) = \infty\), \(i=1,\dots, m.\) Sufficient conditions for oscillation of all solutions and conditions for the existence of non-oscillatory solutions are given. If \(p_i(n) \equiv p_i, k_i(n) = k_i, i=1,\dots,m,\) are constants, then all solutions of (1) oscillate if and only if \(\sum_{i=1}^m p_i/(\lambda(1-\lambda)^{k_i}) > 1\) for any \(\lambda \in (0,1).\)