Eventual periodicity of some systems of max-type difference equations

Abstract

The periodicity of the next system of max-type difference equations x n(1) = max1≤i1 ≤ m1{f 1i1(xn-k(1)i1,1, xn-k(1)i1,22, ⋯ xn-k(1)i1,ll, n), xn-t1s(σ (1))}, xn(2) = max 1≤i2 ≤ m2{f2i2(xn-k(2)i2,1, x n-k(2)i2,22, ⋯ xn-k(2)i2,ll, n), xn-t2s(σ (2))}, xn(l) = max1≤il ≤ ml{flil(xn-k(l)il,1, x n-k(l)il,22, ⋯ xn-k(l)il,ll, n), xn-tls(σ (l))}, n ε 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.