Oscillations of nonlinear partial difference systems

This paper studies the following two-dimensional nonlinear partial difference systems {T(∇₁, ∇₂) (x_mn) + b_mn g(y_mn) = 0, {T(Δ₁, Δ₂)(y_mn) + a_mn f(x_mn) = 0, where m, n ε N₀ = {0, 1, 2,...}, T(Δ₁, Δ₂) = Δ₁ + Δ₂ + I, T(∇₁, ∇₂) = ∇₁ + ∇₂ + I, Δ₁ y_mn = y_m+1,n - y_mn, Δ₂y_mn = y_m,n+1 - y_mn, Iy_mn = y_mn, ∇₁y_mn = y_m-1,n - y_mn, ∇₂y_mn = y_m,n-1 - y_mn, {a_mn} and {b_mn} are real sequences, m, n ε N₀, and f, g: R → R are continuous with uf (u) > 0 and ug(u) > 0 for all u ≠ 0. A solution ({x_mn}, {y_mn}) of the system is oscillatory if both components are oscillatory. Some sufficient conditions for all solutions of this system to be oscillatory are derived. © 2002 Elsevier Science (USA). All rights reserved.