On Schauder's fixed point theorem and forced second-order nonlinear oscillations
On Schauder's fixed point theorem and forced second-order nonlinear oscillations
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DOI：
10.1016/0022-247x(68)90225-4
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发表时间：
1968-02
影响因子：
1.3
通讯作者：
A. Lazer
中科院分区：
文献类型：
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作者：
A. Lazer
令e (t) 是连续的、周期性的，周期为T 3-s 0，并且均值为零。在这篇笔记中，我们将展示对于任何数字 c 的微分方程。 i:-+ ct+ g (x)= e (t)(1) 具有 T 周期解，前提是 g 是连续的。 vg (. v) 2 0 1 s 足够大，并且 g (x)/. y-0 为 1 s/--f ‘x。我们的证明利用了一种应用 Schauder 不动点定理的新方法来确定非线性微分方程周期解的存在性。在未来的论文中，我们希望能够在一般情况下制定该方法，从而建立更一般的非线性微分方程的周期解的存在性。为了简洁起见，我们引入一些符号。 P 表示周期为 T 的实值连续函数集。Q 表示抵消 E P ，其中
Let e (t) be continuous, periodic with period T 3-s 0, and have mean zero. In this note we will show that for any number c the differential equation. i:-+ ct+ g (x)= e (t)(1) has a T-periodic solution provided that g is continuous,. vg (. v) 2 0 for 1 s sufficiently large, and g (x)/. y-0 as 1 s/--f ‘x. Our proof makes use of what appears to be a new method of applying the Schauder fixed point theorem to establish the existence of periodic solutions of nonlinear differential equations. In a future paper we hope to be able to formulate this method in a general setting and thereby establish the existence of periodic solutions of more general nonlinear differential equations. For brevity we introduce some notation. P will denote the set of realvalued continuous functions with period T. Q will denote the set off E P with