A number of necessary and/or sufficient conditions for the existence of oscillatory and nonoscillatory solutions of the nonlinear difference equation (*) \(\Delta^ mx_ n+(-1)^{m+1}p_ nf(x_{n-k})=0\), \(n=0\), \(1,\ldots,\) is formulated.
From the basic result with \(f\equiv 1\) and comparison results of two such equations with \(f\) eventually replaced by \(g\) further results are obtained. One of them says:
If \(p_ n\geq 0\), \(f\in C[R,R]\), \(uf(u)>0\) for \(u\neq 0\), \(\liminf p_ n>0\), \(\liminf_{u\to 0}f(u)/u\geq 1\) and every bounded solution of the linearized equation (with \(f\equiv 1)\) oscillates, then every bounded solution of equation (*) also oscillates.