The Oscillation Of Some Delay Difference Equations And Distribution Of Zeros

The oscillation theory of functional differential equations has a rapid development in the recent 30 years, see [1-4]. The oscillation theory of functional differential equa-tions is different from that of ordinary differential equations. It is well known that homogeneous ordinary differntial equations(ODEs) of first order do not possess oscil-latory solutions[2]. But the presence of deviating arguments can cause the oscillation of solutions. The oscillation of first order difference equations with continuous vari-ables and the distribution of zeros for the delay differential equations on time scales are important component parts of oscillation theory of functional differential equations.This paper is composed of three sections:The first section is preface. The author introduces the application and the de-velopment of oscillation theory for functional differential equations and gives the main questions we study.The second section is the research about the oscillation of delay difference equations with continuous variables. [12] indicates the relation of oscillation between delay difference equations with continuous variables and discrete variables. The author studies the oscillation of delay difference equations with continuous variables by the improved method of [22] and improves previous results.The third section is the research about the distribution of zeros for the first order differential equations on time scales. Difference equations are derived by differential equations. Naturally, people ask if the property of difference equations is the same as that of differential equations. In 1990, Germany mathematician Hilger put forward "Analysis on measure chains - a unified approach to continuous and discrete calculus" which arose the extensive interest of mathematician. [26] studied the first order differential equation on time scales x(t) + p(t}x(r(t)} = 0, t T, where T is a time scale, and obtained the sufficient condition of oscillation for the equation. When the equation oscillates, if the distance between adjacent zeros of all oscillatory solutions is bounded and how to estimate the distance between adjacent zeros of all oscillatory solutions if it is bounded. In this section, the author studies the question and obtains some important results.