Application of a lemma on bilinear forms to a problem in nonlinear oscillations
Application of a lemma on bilinear forms to a problem in nonlinear oscillations
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DOI：
10.1090/s0002-9939-1972-0293179-9
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发表时间：
1972
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影响因子：
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通讯作者：
A. Lazer
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文献类型：
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作者：
A. Lazer
本文给出了无限维向量空间上对称双线性型非退化的一个简单条件。利用这个条件和傅里叶级数的基本性质，证明了一类二阶非线性微分系统周期解的唯一性定理。在[5]中，作者和D. A. S 'anchez考虑了微分方程（1）x”+格拉德G（x）= p（t）= p（t +27 r）其中peC（R，Rn），GeC ~ 2（R'，R）.该方程可看作是受保守内力和周期激振力作用的力学系统的牛顿运动方程。在[5]中证明了：如果存在整数N和数[tN和fN+1]使得（2）N2 <，uN < YN+1 <（N + 1）2，且对所有aeRn，（3）-v <8 G（a）1（axiax）< YN+1，其中I是n × n单位矩阵，则（1）至少存在一个27周期解.这个存在性定理的证明是基于对C. L. Dolph [1]和Brouwer不动点定理，并不意味着唯一性。本文的目的是证明比（2）和（3）限制性小得多的条件意味着（1）至多存在一个217周期解。我们的证明将基于两个非常基本的抽象代数引理和傅立叶级数的最基本的性质。LEMMA 1.设V是一个真实的向量空间，H：Vx V-+R是V上的一个真实的值对称双线性型。如果V是子空间X和Z的直和，使得H在X上是正定的，在Z上是负定的，即1971年6月21日由编辑收到，1971年7月21日修订。AMS 1970主题分类。小学34 C25、15 A63;中学70 D 05。
In this note we give a simple condition for nondegeneracy of symmetric bilinear forms on infinite dimensional vector spaces. We apply this condition and elementary properties of Fourier series to prove a uniqueness theorem for periodic solutions of a class of second order nonlinear differential systems. In [5] the present author and D. A. S'anchez considered the differential equation (1) x" + grad G(x) = p(t) = p(t + 27r) where peC(R, Rn), GeC2(R', R). The equation may be considered as the Newtonian equations of motion of a mechanical system subject to conservative internal forces and periodic exciting forces. In [5] it was shown that if there exists an integer N and numbers [tN and fN+1 such that (2) N2 < ,uN < YN+1 < (N + 1)2, and for all aeRn, (3) -v < 8G(a)1axiaxj)< YN+1l, where I is the n x n identity matrix, then there exists at least one 27rperiodic solution of (1). The proof of this existence theorem, which was based on a slight modification of a theorem of C. L. Dolph [1] and the Brouwer fixed point theorem, did not imply uniqueness. The purpose of this note is to show that conditions far less restrictive than (2) and (3) imply that there can exist at most one 217-periodic solution of (1). Our proof will be based on two very elementary abstract algebraic lemmas and the most basic properties of Fourier series. LEMMA 1. Let V be a real vector space and let H: Vx V-+R be a real valued symmetric bilinear form on V. If V is the direct sum of subspaces X and Z such that H is positive definite on X and negative definite on Z, i.e. Received by the editors June 21, 1971 and, in revised form, July 21, 1971. AMS 1970 subject classifications. Primary 34C25, 15A63; Secondary 70D05.