Oscillation criteria of first order linear difference equations with delay argument

This paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument Δu(k)+p(k)u(τ(k))=0,k∈N, where Δu(k)=u(k+1)−u(k), p:N→R+p:N→R+, τ:N→Nτ:N→N and limk→+∞τ(k)=+∞limk→+∞τ(k)=+∞. Examples illustrating the results are given. It is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument τ(k)τ(k).