Consider the equation \[ \Delta ^2(\,x_{n}-p_nx_{n-k})+q_nf(x_{n-\ell })=0, \quad n\geq n_0,\tag{\(*\)} \] where \(\{q_n\}\,\)is a sequence of real numbers such that \(q_n\geq 0,\) \((n\geq n_0)\) and not identically equal to zero for many values of \(n,\) \(f:\mathbb{R}\to \mathbb{R}\) is continuous and nondecreasing such that \(uf(u)>0\) for \(u\neq 0,\) and \(0<p_n\leq p<1, k,\) and \(\ell\) are positive integers. The authors first propose an interesting example, which illustrates that there exist some false assertions in the paper by E. Thandapani and K. Mahalingam [Tamkang J. Math. 34, No. 2, 137–145 (2003; Zbl 1044.39016)], then offer some sufficient criteria for the oscillatory and nonoscillatory behavior of \((*)\).