Oscillations of nonlinear partial difference systems

  • Liu S
  • Chen G
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Abstract

This paper studies the following two-dimensional nonlinear partial difference systems {T(∇1, ∇2) (xmn) + bmn g(ymn) = 0, {T(Δ1, Δ2)(ymn) + amn f(xmn) = 0, where m, n ε N0 = {0, 1, 2,...}, T(Δ1, Δ2) = Δ1 + Δ2 + I, T(∇1, ∇2) = ∇1 + ∇2 + I, Δ1 ymn = ym+1,n - ymn, Δ2ymn = ym,n+1 - ymn, Iymn = ymn, ∇1ymn = ym-1,n - ymn, ∇2ymn = ym,n-1 - ymn, {amn} and {bmn} 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 ({xmn}, {ymn}) 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.

Author-supplied keywords

  • Nonlinear partial difference systems
  • Oscillation

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