Necessary and Sufficient Conditions for Oscillation of Nonlinear Neutral Difference Systems of dim-2

Abstract

This work is concerned about the necessary and sufficient conditions for oscillation of solutions of 2-dimensional nonlinear neutral delay difference systems of the form: Δ[x(n)+p(n)x(n−m)y(n)+p(n)y(n−m)]=[a(n)b(n)c(n)d(n)] [f(x(n−α))g(y(n−β))],\Delta \left[ {\matrix{ {x\left( n \right) + p\left( n \right)x\left( {n - m} \right)} \hfill \cr {y\left( n \right) + p\left( n \right)y\left( {n - m} \right)} \hfill \cr } } \right] = \left[ {\matrix{ {a\left( n \right)} \hfill & {b\left( n \right)} \hfill \cr {c\left( n \right)} \hfill & {d\left( n \right)} \hfill \cr } } \right]\,\,\left[ {\matrix{ {f\left( {x\left( {n - \alpha } \right)} \right)} \hfill \cr {g\left( {y\left( {n - \beta } \right)} \right)} \hfill \cr } } \right], where m > 0, α ≥ 0, β ≥ 0 are integers, a(n), b(n), c(n), d(n), p(n) are real sequences and f, g ∈ 𝒞(ℝ, ℝ).

Keywords

• oscillation
• nonoscillation
• nonlinear
• system of neutral equations
• krasnoselskii’s fixed point theorem
• 34k11
• 34c10
• 39a13