On asymptotic behaviour and oscillation of forced first-order nonlinear neutral difference equations (Q2770151)

The authors study the difference equation \(\Delta(y_{n}\pm y_{n-m})+q_{n}G(y_{n-k})=f_{n}\), \(n\geq 0\), where \(\Delta\) is the forward difference operator, \(\{q_{n}\}\) and \(\{f_{n}\}\) are sequences of real numbers, \(\sum_{n=0}^{\infty}|f_{n}|<\infty\), \(q_{n}\geq 0\) or \(q_{n}\leq 0\), \(G\) is a continuous nondecreasing function such that \(xG(x)>0\) for \(x\neq 0\), \(m,k\in\{0,1,2,\ldots\}\). The equation \(\Delta (y_{n}+p_{n}y_{n-m})+q_{n}G(y_{n-k})=f_{n}\), where \(p_{n}\) is allowed to change sign, is investigated as well. Eight theorems on oscillation and asymptotic behaviour of these equations are proved. Six examples illustrate the results.