Unbounded increasing solutions of a system of difference equations with delays

We consider a homogeneous system of difference equations with deviating arguments in the formΔy(n)=∑k=1qβk(n)[y(n-pk)-y(n-rk)],where n⩾n0,n0∈Zn⩾n0,n0∈Z, pk,rkpk,rk are integers, rk>pk⩾0rk>pk⩾0, q   is a positive integer, y=(y1,…,ys)Ty=(y1,…,ys)T, y:{n0-r,n0-r+1,…}→Rsy:{n0-r,n0-r+1,…}→Rs is an unknown discrete vector function, s⩾1s⩾1 is an integer, r=max{r1,…,rq},Δy(n)=y(n+1)-y(n)r=max{r1,…,rq},Δy(n)=y(n+1)-y(n), and βk(n)=(βijk(n))i,j=1s are real matrices such that βijk:{n0,n0+1,…}→[0,∞), and ∑k=1q∑j=1sβijk(n)>0 for each admissible i   and all n⩾n0n⩾n0. The behavior of solutions of this system is discussed for n→∞n→∞. The existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given. The scalar case is discussed as well.