Oscillation of neutral difference equations with continuous arguments (Q2720068)

This paper is concerned with difference equations of the form: NEWLINE\[NEWLINE\Delta\bigl[x(t) -px(t-\tau) \bigr]+q(t)x (t-\sigma)=0,\tag{1}NEWLINE\]NEWLINE where \(\tau, \sigma >0\). The unknown function is supposed to be real and has continuous argument \(\in [t_0,\infty)\). The number \(p\) satisfies \(0<p<1\), and the function \(q\) is nonnegative on \([t_0,\infty)\) and for any \(t\geq t_0\), \(\{q(t+i \tau) \}^\infty_{i=1}\) has a positive subsequence. Oscillation criteria are derived. For instance, it is shown that if \(\sigma= k\tau\) for some positive integer \(k\) and NEWLINE\[NEWLINE(1+p)\varliminf_{t\to\infty} {1\over k}\sum^k_{i=1} q(t-i\tau)>{k^k \over (1+k)^{k+1}},NEWLINE\]NEWLINE then every solution of (1) is oscillatory.