Eventual periodicity of some systems of max-type difference equations

The periodicity of the next system of max-type difference equationsxn(1)=max1⩽i1⩽m1f1i1xn-ki1,1(1)(1),xn-ki1,2(1)(2),…,xn-ki1,l(1)(l),n,xn-t1s(σ(1)),xn(2)=max1⩽i2⩽m2f2i2xn-ki2,1(2)(1),xn-ki2,2(2)(2),…,xn-ki2,l(2)(l),n,xn-t2s(σ(2)),⋮xn(l)=max1⩽il⩽mlflilxn-kil,1(l)(1),xn-kil,2(l)(2),…,xn-kil,l(l)(l),n,xn-tls(σ(l)),n∈N0, where s,l,mj,tj,kij,h(j)∈N, j,h∈{1,…,l}, (σ(1),σ(2),…,σ(l)) is a permutation of (1,2,…,l), and fjij:(0,∞)l×N0→(0,∞), j∈{1,…,l}, ij∈{1,…,mj}, is studied. It is shown that under some conditions posed on functions fjij all positive solutions of the system are eventually periodic with period Ts, for some T∈N.