Oscillatory and asymptotic behavior in certain third order difference equations (Q1095322)

The difference equation (E): \(\Delta^ 3x_ n-a_ nx_{n+2}=0\) for \(n\in {\mathbb{N}}\), where \(\Delta\) denotes the forward difference operator, is studied subject to the condition \(a_ n>0\) for any n. The main result is that equation (E) has an oscillatory solution if and only if for every nonoscillatory solution \((x_ n)\) of (E), there exists an integer N for which \(x_ n>0\), \(\Delta x_ n>0\) and \(\Delta^ 2x_ n>0\) for all \(n\geq N\). This is a discrete analogue of a theorem about differential equations [cf. \textit{W. E. Taylor jun.}, ibid. 12, 97-102 (1982; Zbl 0484.34020)]. Within the proof the concept of solutions of a ``quasi- adjoint'' equation of (E) is used.