Quasi-adjoint third order difference equations: Oscillatory and asymptotic behavior (Q1089531)

It is shown that the linear difference equation \(\Delta^ 3V_ n+P_{n-1}V_{n+1}=0\) has oscillatory solutions if and only if any non- oscillatory solution \(V_ n\) satisfies the relation \(V_ n\Delta V_ n\Delta^ 2V_ n\neq 0\), \(sgnV_ n=sgn\Delta^ 2V_ n\neq sgn\Delta V_ n\), \(n\geq 1\). This is a discrete analogue of a result of \textit{A. C. Lazer} [Pac. J. Math. 17, 435-466 (1966; Zbl 0143.315)] for the differential equation \(y\prime''+py'+qy=0\). An example is given for an equation of the above form without oscillatory solutions.