Almost Periodic Solutions Of Second Order Delay Differential Equations With Piecewise Constant Argument

This article mainly studied Almost periodic solutions of two class of two order delaied differential equations with piecewise constant argument. The first category equation is (x(t＋1)＋px(t))"＝qx([t])＋f(t). and the second category equation is (x(t+1)+px(t))"=qx(2[t+1/2])+f(t). Here [·] denotes the integral function, p, q are nonzero real numbers,f(t) is an almost periodic function.In this paper, in the value of |p|≠1,we construct the almost periodic solutions of the series form of two kinds of different equations different solutions, and prove the existence and uniqueness of the solutions. This article is divided to three chapters.The first chapter is the introduction, which mainly introduces the research back-ground and research status of almost periodic function, the main research works and some basic symbols, lemmas.The second chapter main research the almost periodic solutions of the first kind of differential equations. This chapter first get the corresponding difference equations by integral from n to t (t ∈[n, n+1)) on either side of the first kind of equation, then to limit the module of characteristic roots of the difference equations not equal to 1. Through the reduction we get the value range that p, q should satisfy. Then change the wtitten forms of the difference equations, we can get its equivalent form. Under its equivalent form We obtain the existence and uniqueness of almost periodic solutions of the difference equations. Finally, through the solutions of difference equations,we construct convergent series as the almost periodic solutions for the first category of differential equation.The third chapter main research the almost periodic solutions of the second kind of differential equations. In this chapter, we use the method is similar to the second chapter that can prove the almost periodic solutions. first, we also to obtain the corresponding difference equations of the second differential equations by integrating from 2n to t (t E [2n-1,2n+1)), as the time delaied is different between the first kind of differential equations and the second kind of differential equation,the integrating of the two class of two order delaied differential equations may be different. Then we get the value range that p, q should satisfy, through the solutions of difference equations to construct the convergent series as the almost periodic solutions the first category of differential equations.