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

Abstract

We consider the following system: Δmui(k) + fi(k, u1(k), u2(k), . . . , un(k)) = 0, k ∈ {0, 1, . . . , T}, Δjui(0) = 0, 0 ≤ j ≤ m - 3, αiΔm-2ui(0) - βiΔm-1ui(0) = 0, γiΔm-2ui(T + 1) + δiΔm-1ui(T + 1) = 0, i = 1, 2, . . . , n, where m ≥ 2, T ≥ m - 1, and for each 1 ≤ i ≤ n, αi > 0, γi > 0, βi ≥ 0, δi ≥ γi and αiγi(T + 1) + αiδi + βiγi > 0. Criteria are offered for the existence of single- and double-fixed-sign solutions of the system. © 2000 Elsevier Science B.V. All rights reserved.