A case of boundedness in Littlewood's problem on oscillatory differential equations
A case of boundedness in Littlewood's problem on oscillatory differential equations
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DOI：
10.1017/s0004972700024862
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发表时间：
1976-02
影响因子：
0.7
通讯作者：
G. R. Morris
中科院分区：
文献类型：
--
作者：
G. R. Morris
结果表明，ẍ + 2x3 = p(t) 的所有解都是有界的，该符号表明 p 是周期性的。不需要乘以 p 的小参数。关键的一步是利用Moser定理证明，在方程对应的（初值平面）映射下，存在不变的简单闭合曲线。这还意味着存在不可数无穷个近周期解，并且对于每个正整数 m，存在无穷多个最小周期 2mπ 的周期解（2π 被视为 p 的最小周期）。建议对于一大类方程，相同的攻击将显示 ẍ + g(x) = p(t) 有界的所有解。然而，为了清楚地展示该方法，这里不尝试进行概括。
It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p. The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ). It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.