On some periodic systems of max-type difference equations

摘要

We show that all positive solutions to the system of max-type difference equationsxn(1)=max1⩽i⩽m1f1ixn-ki,1(1)(1),xn-ki,2(1)(2),…,xn-ki,l(1)(l),n,xn-s(1),xn(2)=max1⩽i⩽m2f2ixn-ki,1(2)(1),xn-ki,2(2)(2),…,xn-ki,l(2)(l),n,xn-s(2),⋮xn(l)=max1⩽i⩽mlflixn-ki,1(l)(1),xn-ki,2(l)(2),…,xn-ki,l(l)(l),n,xn-s(l),n∈N0, where s,l,mj,ki,t(j)∈N, j,t∈{1,…,l}, and for a fixed j, i∈{1,…,mj}, and where the functions fji:(0,∞)l×N0→(0,∞), j∈{1,…,l},i∈{1,…,mj}, satisfy some conditions, are eventually periodic with (not necessarily prime) period s. A related result for the corresponding system of min-type difference equations is also proved.