Asymptotic convergence of the solutions of a discrete system with delays

A system of s discrete equationsΔy(n)=β(n)[y(n-j)-y(n-k)]is considered where k and j are integers, k>j⩾0, β(n) is a real s×s square matrix defined for n⩾n0-k,n0∈Z with non-negative elements βij(n),i,j=1,…,s such that ∑j=1sβij(n)>0, y=(y1,y2,…,ys)T:{n0-k,n0-k+1,…}→Rs and Δy(n)=y(n+1)-y(n) for n⩾n0. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions, i.e., for every solution y(n) defined for all sufficiently large n, there exists a finite limit limn→∞y(n). Moreover, it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates. Some discussion related to the so-called critical case known for scalar equations is given as well.