The authors investigate oscillatory properties of the first order nonlinear delay difference equation \[ \Delta y_n+p_nf(y_{n-k}),\quad \Delta y_n:=y_{n+1}-y_n, \tag{*} \] where \(p_n\geq 0\), \(k\) is a positive integer and \(f\) is a continuous function satisfying \(uf(u)>0\) for \(u\neq 0\). Recently, several papers dealing with the linear case \(f(u)\equiv u\) have been published, see e.g. X. H. Tang and J. S. Yu [Comput. Math. Appl. 37, 11-20 (1999; Zbl 0937.39012)], and the references given therein. In the present paper the authors extend some of these results to (*) under appropriate assumptions on the nonlinearity \(f\). A typical result is the following statement.
Theorem. Assume that \(\lim_{n\to \infty} \sum_{i=n-k}^{n-1}p_i>0\), \[ \sum_{n=0}^\infty \left[\sum_{i=n}^{n+k}p_i\lg \left(\sum_{i=n}^{n+k} p_i\right)-\sum_{i=n+1}^{n+k}p_i\lg \left(\sum_{i=n+1}^{n+k} p_i\right)\right]=\infty. \] If there exists a positive nondecreasing continuous function \(g\) such that \(\int_0^\infty g(\text{e}^{-u}) du<\infty\) and \(|f(u)/u-1|<g(u)\) for \(|u|\) sufficiently small, then every solution of (*) oscillates.
Some applications of the basic results of the paper, e.g. to the equation \(y_{n+1}=y_n\exp\{r_n(1-y_{n-k}/k)\}\), are offered.