Lagrange Stability for Duffing-Type Equations
Lagrange Stability for Duffing-Type Equations
复制标题
DOI：
10.1006/jdeq.1999.3663
复制
发表时间：
2000
影响因子：
2.4
通讯作者：
Xiaoping Yuan
中科院分区：
文献类型：
--
作者：
Xiaoping Yuan
摘要本文研究了pj为1周期的duffing型方程(d2x / dt 2)+ x2n +1 + p2n (t) x2n +…+ p1 (t) x + p0 (t)=0。当p j (t)（0≤j≤n）在[0,1]中有界变化，且p j (t) （n≤j≤2n）的导数为Lipschitzian时，证明了该方程的所有解都是有界的。还证明了pj处处不连续，使得方程的所有解都是有界的。这意味着pj 's的连续性对于方程解的有界性是不必要的。
Abstract In this paper the Duffing-type equation ( d 2 x / dt 2 )+ x 2 n +1 + p 2 n ( t ) x 2 n +…+ p 1 ( t ) x + p 0 ( t )=0 is studied where the p j 's are 1-periodic. It is shown that all solutions of this equation are bounded, provided that the p j ( t ) (0⩽ j ⩽ n ) are of bounded variation in [0, 1] and that the derivatives of p j ( t ) ( n ⩽ j ⩽2 n ) are Lipschitzian. It is also shown that there exist p j 's being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of p j 's is not necessary for the boundedness of solutions of the equation.