| The fixed point theory as a new tool for the stability of differential equation, the advantages of it has causing more and more people to concern. And many scholars has being interesting in the research of it. Among the much research work, Burton, who made the most representative work. This article is inspired by Burton, Using the fixed point theory to discuss the stability of two classes of second order differential equations,and Receiving new sufficient conditions and generalizing the results of the literature.This paper is divided into three chapters.The first chapter introduces the background and the practical significance of fixed point theory. Briefly introduced the practices of treating stability theory of equation using the fixed point by some examples. At the same time includes some basic theoretical knowledge relevant to the second chapter and third chapt as the nature and terms of stability theory, the fixed point principle of law and so on. Finally,gives the structure and the associated main content of the article.The second chapter discussed second order differential equation x"+f(t,x,x')x'+a(t)g(x(q(t)))= 0. The stability of the solution is proved by using fixed point theory. When f(t,x,x')= f(x) The equation is converted to the equation when Burton in 2004 discussed the direct method using Liapunov function. By comparing,we find that using fixed point technique to obtain the conditions is better than the Liapunov direct method.The third chapter discussed second-order differential equation which the constant containing integral form, the stability of the solution is proved by using the fixed point theory,and we gained new sufficient conditions, in 2004, Burton has been discussed a special type of the equation and researched it by using the Liapunov direct method, received extremely harsh conditions... |
