Existence of nonoscillatory solutions of nonlinear delay difference equations (Q2844401)

The authors study the existence of nonoscillatory solutions for the nonlinear delay difference equation of the form NEWLINE\[NEWLINE\Delta x(n)= p(n) x(n)- q(n) x(n) x(\tau(n)),\quad n\in\mathbb{N}_0= \{n_0,n_0+ 1,\dots\},\tag{1}NEWLINE\]NEWLINE where \(n_0\in\mathbb{N}\), \((p(n))_n\) and \((q(n))_n\) are positive real sequences, \(\tau(n)\leq n\) and \((\tau(n))_n\) is an increasing positive sequence of integer such that \(\lim_{n\to+\infty} \tau(n)=+\infty\).NEWLINENEWLINE A solution of (1) is a sequence \((x(n))_n\) defined for \(n\geq n_0-\theta\), and satisfying (1) for all \(n\in\mathbb{N}_0\), where \(\theta= \max_{n_0\leq n<+\infty} \tau(n)\).