On some periodic systems of max-type difference equations

Abstract

We show that all positive solutions to the system of max-type difference equations x n(1)=max 1≤i≤m1{f1i(x n-ki,1(1)(1),x n-ki,2(1)(2),⋯, x n-ki,l(1)(l),n,x n-s(1),x n(2)=max 1≤i≤ m2f2i xn-ki,1(2)(1),x n-ki,2(2)(2), ⋯,x n-ki,l(2)(l),n,x n-s(2),⋮x n(l)=max 1≤i≤ ml{fli(x n-ki,1(l)(1),x n-ki,2(l)(2),⋯,x n-ki,l(l)(l),n),x n-s(l)},n∈ℕ 0, where s,l, mj,k i,t(j)∈ℕ, j,t∈{1,⋯,l}, and for a fixed j, i∈{1,⋯,m j}, and where the functions f ji:(0,∞) l×ℕ 0→(0,∞), j∈{1,⋯,l},i∈{1,⋯,m j}, 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.