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

摘要

We consider the following system: Δ~m u_i(k) + f_i(k,u_1(k),u_2(k),…,u_n(k)) = 0, k ∈ {0,1,…,T}, Δ~j u_i(0) = 0, 0 ≤ j ≤ m - 3, α_iΔ~(m - 2)u_i(0) - β_iΔ~(m-1)u_i(0) = 0, γ_iΔ~(m-2)u_i(T + 1) + δ_iΔ~(m-1)u_i(T + 1) = 0, i = 1,2,…,n, where m ≥ 2, T ≥ m - 1, and for each 1 ≤ i ≤ n, α_i > 0, γ_i > 0, β_i ≥ 0, δ_i ≥ γ_i and α_iγ_i(T + 1) + α_iδ_i + β_iγ_i > 0. Criteria are offered for the existence of single- and double-fixed-sign solutions of the system.