In the first part of the paper the nonlinear difference equation \[ y(t-\tau)-y(t)+\sum_{i=1}^{m}p_if_i(y(t+\sigma_i))=0 \tag{*} \] is considered, where \(\tau>0\), \(\sigma_m\geq\dots\geq\sigma_1>0\), \(p_i>0\), \(f_i\in C(\mathbb{R},\mathbb{R})\), \(uf_i(u)>0\) for \(u\not=0\) and \(\lim_{u\to\infty} f_i(u)/u=1\) \((i=1,\dots,m)\). Under some additional assumptions on \(f_i\), a necessary and sufficient condition for oscillation of (\(\ast\)) (i.e., every solution oscillates) is given in terms of oscillation of the associated linear difference equation \[ y(t-\tau)-y(t)+\sum_{i=1}^{m}p_iy(t+\sigma_i)=0. \] Knaster’s fixed point theorem plays an important role in the proof of this statement. In the second part of the paper, various sufficient conditions for oscillation of the forced difference equation \[ y(t)-a(t)y(t-\tau)+G(t,y(t-\sigma))=f(t) \] are given, where \(\tau,\sigma>0\) and \(a\in C(\mathbb{R}_+,\mathbb{R}_+)\), \(G\in C(\mathbb{R}_+\times\mathbb{R},\mathbb{R})\) and \(f\in C(\mathbb{R}_+,\mathbb{R})\).