Oscillation of certain second order difference equations (Q2726137)

Oscillatory criteria for the nonlinear second order difference equation \(\Delta(r_n\Delta x_n)+f(n,x_n)=0 \quad\) (1) are presented, where \(r_n>0\), \(xf(n,x)>0\) for \(x\neq 0\), and the nonlinearity \(f\) is strongly superlinear/sublinear. For each of the two cases \(R_0:=\sum_{k=0}^\infty 1/r_k=\infty\) or \(R_0<\infty\), the authors prove the necessary and sufficient conditions for equation (1) to be oscillatory. A typical result is the following. NEWLINENEWLINENEWLINE\textbf{Theorem~1.} Suppose \(R_0=\infty\) and \(f\) is strongly superlinear. Then equation (1) is oscillatory if and only if NEWLINE\[NEWLINE \sum_{k=n_0}^\infty \frac{1}{r_k} \sum_{i=k+1}^\infty |f(i,c)|=\infty \quad\text{for all } c\neq 0. NEWLINE\]NEWLINE \smallskip The results of this paper complement the study of nonoscillatory solutions of (1) by \textit{X. He} [J.~Math. Anal. Appl. 175, No. 2, 482-498 (1993; Zbl 0780.39001)].