The asymptotic and oscillatory behavior of neutral difference inclusions. (Q1428182)

The authors study the asymptotic and oscillatory properties of solutions of the first order neutral difference inclusion \[ -\Delta(x_n-r_n\,x_{n-\tau})\in p_n\,F(x_{n-\sigma_1},x_{n-\sigma_2},\dots,x_{n-\sigma_s}), \quad n=0,1,2,\dots, \tag{\(*\)} \] where \(F:{\mathbb R}^s\to2^{{\mathbb R}^1}\) is a compact convex upper semicontinuous multifunction. They establish sufficient criteria when every nonoscillatory solution of (\(*\)) converges to zero, and when every eventually positive/negative solution of (\(*\)) diverges to \(+/-\infty\). The main results concerning the (non)oscillation of (\(*\)) are a sufficient condition for the oscillation of (\(*\)), and a sufficient condition for the existence of a nonoscillatory solution of \((*)\), e.g. by comparing (\(*\)) with a certain difference equation. Two examples illustrate the application of the main theorems.