Nonexistence and existence of positive solutions for difference equations with unbounded delay

Consider the delay difference equation xn+1−xn+pnxτ(n)=0, 0,1,2,…, where τ : N → Z, τ(n) < n for n N and limn→∞ τ(n) = ∞, pn R. Some existence results of positive solutions and oscillation criteria for this equation are obtained.