On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

Oscillation Criteria for First Order Linear Difference Equations with Several Delay Arguments

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

Oscillatory mixed difference systems

The aim of this paper is to discuss the oscillatory behavior of difference systems of mixed type. Several criteria for oscillations are obtained. Particular results are included in regard to scalar equations. Copyright © 2006 J. M. Ferreira and S. Pinelas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited. 1