Oscillations of difference equations with non-monotone retarded arguments

Consider the first-order retarded difference equationΔx(n)+p(n)xτ(n)=0,n∈N0where (p(n))n⩾0 is a sequence of nonnegative real numbers, and (τ(n))n⩾0 is a sequence of integers such that τ(n)⩽n-1, n⩾0, and limn→∞τ(n)=∞. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving liminf, is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail.