Oscillations of nonlinear partial difference systems

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This paper studies the following two-dimensional nonlinear partial difference systems {T(∇1, ∇2) (xmn) + bmng(ymn) = 0, {T(Δ1, Δ2)(ymn) + amnf(xmn) = 0, where m, n ε N0= {0, 1, 2,...}, T(Δ1, Δ2) = Δ1+ Δ2+ I, T(∇1, ∇2) = ∇1+ ∇2+ I, Δ1ymn= 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.




Liu, S. T., & Chen, G. (2003). Oscillations of nonlinear partial difference systems. Journal of Mathematical Analysis and Applications, 277(2), 689–700. https://doi.org/10.1016/S0022-247X(02)00620-0

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