利用最小作用原理研究一类非自治二阶系统周期解

On Periodic Solutions for a Class of Non-autonomous Second Order Systems by the Least Action Principle

主要研究以下二阶系统{u(t)-A(t)u(t)=▽F(t,u(t)),u(0)-u(T)=u(0)-u(T)=0,a.e.t∈[0,T]的周期解的存在性。当F(t,x)=F1(t,x)+F2(x)满足条件A且具有局部有界性T1lim inf x→+∞x 2α∫F(t,x)dt>0T2T∫(r1(t)dt)2/0T12-T∫k(t)dt及A(t)满足条件(A(t)x,x)≥h(t)|x|β+w(t)时,通过使用最小作用原理得到了一个新的周期解的存在性定理,改进了已有结果。

The purpose of this paper is to study the existence of periodic solutions of the following second order systems {u（t）-A（t）u（t）=△F（t,u（t）,u（0）-u（T）=u（0）-u（T）=0,a,e,t∈[0,T].When F（t, x） = F1 （t, x） ＋ F2 （x） satisfies assumption A and the potential function satisfies the locally boundary condition .lim inf 1/{x}→＋∞｜x｜2a∫T0F（t,x）dt〉2T（∫T0r1（t）dt）^2/12-T∫T0k（t）dtand A（t） satisfies the condition （A（t）x,x）≥h（t）｜x｜β＋w（t）.One new existence theorem of the periodic solution is obtained by using the least action principle. Our result im-proves previously known result.