Oscillation criteria for first order linear difference equations with several delay arguments

Abstract

The difference equation with delayed arguments ∆u(k) + Σpₗ(k) u(τₗ(k)) = 0, is considered, where ∆u(k) = u(k + 1) − u(k), pₗ : N → R, τₗ : N → N, lim k→+∞ τₗ (k) = +∞, i = 1, ..., m. In the paper sufficient conditions are established for all proper solutions of the above equation to be oscillatory.

The difference equation with delayed arguments ∆u(k) + Σpₗ(k) u(τₗ(k)) = 0, is considered, where ∆u(k) = u(k + 1) − u(k), pₗ : N → R, τₗ : N → N, lim k→+∞ τₗ (k) = +∞, i = 1, ..., m. In the paper sufficient conditions are established for all proper solutions of the above equation to be oscillatory. Розглянуто рiзницеве рiвняння з запiзненнями в аргументах ∆u(k) + Σpₗ(k) u(τₗ(k)) = 0, де u(k) = u(k + 1) − u(k), pₗ : N → R, τₗ : N → N, lim k→+∞ τₗ (k) = +∞, i = 1, ..., m. Знайдено достатнi умови для того, щоб всi правильнi розв’язки рiвняння були осцилюючими.