Fixed-sign solutions of a system of higher order difference equations

We consider the following system: Δmui(k)+fi(k,u1(k),u2(k),…,un(k))=0, k∈{0,1,…,T},Δjui(0)=0, 0⩽j⩽m−3,αiΔm−2ui(0)−βiΔm−1ui(0)=0,γiΔm−2ui(T+1)+δiΔm−1ui(T+1)=0,i=1,2,…,n,where m⩾2, T⩾m−1, and for each 1⩽i⩽n, αi>0, γi>0, βi⩾0, δi⩾γi and αiγi(T+1)+αiδi+βiγi>0. Criteria are offered for the existence of single- and double-fixed-sign solutions of the system.