The authors study the oscillatory properties of the delay difference equation \[ x(n+1)-x(n)+p(n)g(x(n-k))=0, \] where \(p(n)\) is a nonnegative discrete function for \(n\geq0\), \(g(x)\) is continuous function and \(xg(x)>0\) for \(x\neq 0\), \(k\) is a positive integer. A nontrivial solution \(x(n)\) is said to be oscillatory if for every \(N\) there is \(n\geq N\) such that \(x(n)x(n+1)\leq0\). By putting \(y(n)=x(n+1)\) the equation is transformed into a discrete system of the Liénard type.
Investigating the behavior of this system’s trajectories on the phase plane, the authors establish sufficient conditions for all solutions to be oscillatory. Sufficient conditions for the existence of a nonoscillatory solution are also established. Illustrative examples accompany the abstract reasonings. These examples demonstrate that the oscillatory properties of solutions depend on delay essentially.