Qualitative Behavior of the Solutions to Delay and Difference Euations

It is noteworthy to observe that a first-order linear ordinary differential equation without delay does not possess oscillatory solutions. Therefore the investigation of oscillatory solutions is of interest for equations with delays or for the discrete analogue difference equations. Furthermore, the mathematical modelling of several real-world problems leads to differential equations that depend on the past history rather than only the current state. In this article conditions are presented such that all solutions of delay and difference equations are oscillatory while all solutions of the corresponding ordinary differential equations without delay are, for example, decreasing and tend to zero. Equations with constant and variable arguments are investigated. Several examples of delay and difference equations with applications to many sectors of life are presented.