Oscillation Criteria Of Second-order Neutral Differential Equations

The development and research of ordinary differential equations have lasted for a long time and are still prosperous. In the early20th century mathematicians began to study the qualitative properties of ordinary differential equations as well as the characteristics of differential equations and their solutions such as uniqueness, stability, oscillation, periodicity, permanence and so on. Later due to a tremendous number of the fact and phenomena from physics, biology and chemistry, a large group of mathematicians worked on the analysis of differential equations with delay and nonlinear differential equations.In this paper we mainly discuss the oscillation of second-order forced delay neutral differential equations with mixed nonlinearities, and furthermore we specially are here con-cerned with the oscillation of equations when the range of p(t) is different.First of all, in chapter two by employing some inequalities we study the interval oscilla-tion criteria of the following second-order nonlinear forced neutral delay differential equations with mixed nonlinearities:(r(t)|(x(t)+p(t)x(r(t)))’|α-1(x(t)+p(t)x(τ(t))’)’+q0(t)|x(Τ0(t)|α-1x(Τ0(t))) And we obtain a new sufficient condition for oscillation of this equation.Then, by using some inequalities and comparison theorem in chapter three we argue the oscillation of second-order nonlinear forced neutral differential equations with mixed nonlinearities as follows:(r(t)|(x(t)+p(t)x(r(t)))’|α-1(x(t)+p(t)x(τ(t))’)’+q0(t)|x(Τ0(t)|α-1x(Τ0(t))) where0<p(t)≤p0<∞. We mainly use and improve the method used in [6] and discuss the oscillation of more general equation. At last the results in this chapter are more general.Finally to conclude in this paper our conclusions are based on former authors’works and some inequalities. And moreover our new results mainly are based on the inequality in [15] and the methods offered in Hassan, Erbe and Petcrson[3], Zhong, Ouyang and Zou[5], Baculikova and Dzurina[6].