A class of first-order nonlinear difference equations of neutral type
The authors classify all nonoscillatory solutions and find criteria for oscillation of all solutions of the first order nonlinear difference equations of neutral type \[ \Delta\Big(x(n)-p(n)x\big(\tau(n)\big)\Big)+ \delta\sum_{j=1}^nq_j(n)f_j\Big(x\big(\sigma_j(n)\big)\Big)=0, \] where \(\delta=\pm 1\), \(f_j:{\mathbb R}\to{\mathbb R}\), and for \({\mathbb N}(n_0)=\{n_0,n_0+1,\dots\}\), \(p,q_j:{\mathbb N}(n_0)\to{\mathbb R}_+\), \(\tau,\sigma_j:{\mathbb N}(n_0)\to{\mathbb N}\), and \(\lim\limits_{n\to\infty}\tau(n)=\infty\), \(\lim\limits_{n\to\infty}\sigma_j(n)=\infty\) for all \(j\in\{1,\dots,n\}\).
first-order nonlinear difference equations of neutral type
scientific article; zbMATH DE number 2137078