Oscillatory and asymptotic behaviour of some difference equations (Q2774631)

The authors consider a nonlinear difference equation NEWLINE\[NEWLINE \Delta(r_n\Delta(u_n+p_nu_{n-k}))=q_nf(u_{n-l}),\qquad n=0,1,\ldots, NEWLINE\]NEWLINE where \(\Delta\) denotes the standard forward difference operator, i.e., \(\Delta v_n=v_{n+1}-v_n\) for any sequence \((v_n)\) of real numbers, \(k\) and \(l\) are nonnegative integers, \((p_n)\) and \((q_n)\) are sequences of real numbers with \(q_n\geq 0\) eventually, \((r_n)\) is a sequence of positive numbers with \(\sum_{n=0}^\infty(1/r_n)=\infty\), and \(f\) is a real valued function satisfying \(uf(u)>0\) for \(u\neq 0\). They analyze the oscillation and asymptotic behaviour of nonoscillatory solutions of the displayed equation.