Quadratic Integrability And Boundedness Of Solutions Of A Class Of Second Order Differential Equations

Qian Xuesen and Song Jian have pointed out that it is an urgent mission to give strict, simple and convenient criteria in order to determine the stability of linear differ-ential systems with variable cofficients.Since linear differential Equations are often met with in engineering,elasticity mechanics,electricity and various oscillation problems,it is of immediate significance to study its stability.In this paper we obtain some new results of second order nonhomogeneous func-tional differential equation by making up auxiliary function and using integrability and employing the second method of Lyapunov, and instruct these results by some examples.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 In this paper,we consider the following second order nonhomogeneous functional differential equation (r(t)x1(t))1+p(t)x1(t)+q1(t)x(t)+q2(t)x(h(t))=f(t,x(t)), (2.1.1) wherer(t)> Ofort∈[a,∞]and r(t)is absolutely continuous on [a,∞),p(t),q1(t),q2(t)are real functions defined on [a,∞]and f(t)is a real function defined on[a,∞]×(-∞,∞).With the aid of auxiliary function and two important inequalities,we obtain some new conclusion.Chapter 3 By virtue of an auxiliary function with parameters m,n∈R, we obtain some new result for quadratic integrability and boundedness of all solution of second order nonhomogeneous delay differential equation. (r(t)x'(t))'+p(t)x'(t)+q1(t)x(t)+q2(t)x(t-τ)=f(t,x(t)). (3.1.1)Chapter 4 At the basis of the former two chapters, in this section we obtained some sufficient conditions,necessary conditions,necessary and sufficient conditions of the zero solution to the the type of equation x"(t)+p(t)x'(t)+q(t)x(t)=0, (4.1.1) and the following 2-order differential equations by transformations and the second method of Lyapunov.