Oscillation criteria for second order difference equation. (English) Zbl 0773.39001

Consider the difference equation \(\Delta^ 2x_{k-1}+b_ kx_ k=0\), \(k\geq 1\), where \(\Delta x_ k=x_{k+1}-x_ k\) and \(b_ k\geq 0\) with infinitely many positive terms. This equation is said to be oscillatory if it admits a solution \(\{x_ k\}^ \infty_ 0\) with the property that for any \(N\geq 1\), there exists an \(n\geq N\), such that \(x_ kx_{k+1}\leq 0\); and the equation is said to be nonoscillatory otherwise. In this paper, several necessary, sufficient conditions for the equation being oscillatory or nonoscillatory are proved. One of the results looks like the following: If the difference is nonoscillatory, then \({\varliminf_{n\to\infty}n\sum^ \infty_{k=n+1}b_ k\leq 1/4}\); and if the difference equation is oscillatory, then \(\varlimsup_{n\to\infty}n\sum^ \infty_{k=n+1}b_ k\geq 1/4\).


39A10 Additive difference equations
39A12 Discrete version of topics in analysis