Oscillation of Second Order Dynamic Equations with Mixed Arguments on Time Scales

Abstract

Studing the dynamic equations on time scales was introduced by Stefan Hilger [28]. It is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is a notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, studying time scales lead to several important applications, e.g., insect population models, neural networks, and heat transfer. A time scale T is a nonempty closed subset of the real numbers. When the time scale equals the set of real numbers, the obtained results yield results of ordinary di erential equations, while when the time scale equals the set of integers, the obtained results yield results of di erence equations. The new theory of the so - called "dynamic equation" is not only unify the theories of di erential and di erence equations, but also extends the classical cases to the so - called q - di erence equations (when T = qN0 := fqt : t 2 N0; q > 1g or T = qZ = qZ [ f0g) which have important applications in quantum theory (see [31]). A neutral di erential equation with deviating arguments is a di erential equation in which the highest order derivative of the unknown function appears with and without deviating arguments. In recent years, there has been an increasing interest in studying oscillation and nonoscillation of solutions of neutral dynamic equations on time scales which seek to harmonize the oscillation of continuous and discrete mathematics, however the study was restricted to speci c equations under certain conditions. The oscillation conditions of their equations are not applicable when these conditions change. For this reason we aim to generalize these equations. So we select the title of our thesis to be "Oscillation of Second Order Dynamic Equations with Mixed Arguments