Asymptotic behaviour and existence of nonoscillatory solutions of second-order neutral delay difference equations
The authors investigate asymptotic properties of nonoscillatory solutions of the second-order neutral delay difference equation \[ \Delta^2 (x_n-c_nx_{n-\tau})=f(n,x_{g_1(n)},\dots,x_{g_m(n)}). \tag{*} \] It is supposed that \(\tau\) is a positive integer, \(c_n\in (0,1-\delta)\) for some (sufficiently small) \(\delta>0\), \(\lim_{n\to \infty}g_i(n)=\infty\), \(i=1,\dots,m\), and the nonlinearity \(f\) satisfy certain (more or less natural) restrictions. The solutions of (*) are classified according to the values of the limits \(\lim_{n\to \infty}x_n\), \(\lim_{n\to \infty}y_n\), where \(y_n = x_n-c_nx_{n-\tau}\), and of \(\lim_{n\to \infty} \Delta y_n\). Additional conditions on the sequence \(c_n\) and the nonlinearity \(f\) are given which guarantee that (*) possesses at least one solution with a prescribed asymptotic.
second-order neutral delay difference equation
scientific article; zbMATH DE number 1907725