In this paper, we study the oscillatory behavior of first-order linear delay difference equations \(\begin{aligned} \Delta x(t)+\sum _{i=1}^mp_i(t)x(\tau _i(t))=0,\quad t\ge t_0, \end{aligned}\) where \(m\in {\mathbb {N}}\) , \(t_0\in {\mathbb {R}}\) , coefficient functions \(p_1,\ldots ,p_m:[t_0,\infty )\rightarrow {\mathbb {R}}\) are piecewise continuous, and delay arguments \(\tau _1,\ldots ,\tau _m:[t_0,\infty )\rightarrow {\mathbb {R}}\) are piecewise continuous, \(\tau _i(t)\le t\) for \(t\ge t_0\) , and \(\lim _{t\rightarrow \infty }\tau _i(t)=\infty \) , \(i=1,2,\ldots ,m\) . More precisely, we consider two types of delay arguments. One type is the delay arguments that are arithmetic quasi-periodic functions, i.e., their delays are periodic functions. The other type of delay arguments are more general nonmonotone functions but with intersecting graphs. We present new explicit sufficient conditions for oscillatory solutions of the considered equations. The conditions are given explicitly in terms of coefficients and delays and based on sufficient positivity of coefficients. The positivity of coefficients is measured by the limit inferior and limit superior of coefficients in the predefined points. The proposed criteria can be applied to difference equations with oscillating coefficients as well as nonnegative coefficients. The presented results are illustrated by examples.
