New Oscillation and Nonoscillation Criteria for a Class of Linear Delay Differential Equation

In this article the authors established sufficient condition for the first order delay differential equation in the form , ( ) where , = and is a non negative piecewise continuous function. Some interesting examples are provided to illustrate the results. Keywords: Oscillation, delay differential equation and bounded. AMS Subject Classification 2010: 39A10 and 39A12.

Hille-Nehari type oscillation and nonoscillation criteria for linear and half-linear differential equations

Differential equations attract considerable attention in many applications. In particular, it was found out that half-linear differential equations behave in many aspects very similar to that in linear case. The aim of this contribution is to investigate oscillatory properties of the second-order half-linear differential equation and to give oscillation and nonoscillation criteria for this type of equation. It is also considered the linear Sturm- Liouville equation which is the special case of the half-linear equation. Main ideas used in the proof of these criteria are given and Hille-Nehari type oscillation and nonoscillation criteria for the Sturm-Liouville equation are formulated. In the next part, Hille-Nehari type criteria for the half-linear differential equation are presented. Methods used in this investigation are based on the Riccati technique and the quadratic functional, that are very useful instruments in proving oscillation/nonoscillation both for linear and half-linear equation. Conclude that there are given further criteria which guarantee either oscillation or nonoscillation of linear and half-linear equation, respectively. These criteria can be used in the next research in improving some conditions given in theorems of this paper.

A new oscillation criterion for two-dimensional dynamic systems on time scales

Consider the linear dynamic system on time scales\begin{equation}u^\Delta=pv, \quad\quad v^\Delta=-qu^\sigma\end{equation}where $p>0$ and $q$ are rd-continuous functions on a time scale $\mathbb T$ such that $\sup\mathbb T=\infty$. When $p(t)$ is allowed to take on negative values, we establish an oscillation criterion for system (0.1). Our result improves a main result of Fu and Lin [S. C. Fu and M. L. Lin, Oscillation and nonoscillation criteria for linear dynamic systems on time scales, Computers and Mathematics with Applications, 59(2010), 2552-2565].