Existence and uniqueness of periodic solution for a class of differential systems
type: Item
instance of: scholarly article
title: Existence and uniqueness of periodic solution for a class of differential systems
zbMATH Open document ID: 1116.34037
author: Xiaojing Yang
author: Kueiming Lo
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: The authors use a global inverse function theorem to prove the existence and uniqueness of periodic solutions for differential systems of the type \[ x_k' = f_k (t,x_1,x_2,\ldots,x_n) \] where \(f_k\) is continuous and \(2\pi\)-periodic in \(t\) and \(\partial f_k / \partial x_j\) is continuous and bounded, for each \(1 \leq k,j \leq n.\) Moreover, the following assumption is supposed: for any continuous vector function \(x(t) = (x_1(t),\ldots,x_n(t)), \;t \in [0,2\pi],\) define the matrix \(A(x) = (a_{ij}(x)),\) where \[ a_{ij}(x) = \int_0^{2\pi} \frac{\partial f_i}{\partial x_j} \;(t,x_1 (t),\ldots,x_n (t)) \;dt \] Then \(A(x)\) is nonsingular and the eigenvalues \(\lambda_k = \lambda_k (x)\) of \(A(x)\) satisfy: For those \(\{ \lambda_j \} \), where \(R(\lambda_j) \neq 0,\) there exists a positive constant \(\delta\) such that \(| R(\lambda_j) | \geq \delta\) (\(R(\lambda_j)\) is the real part of \(\lambda_j\)), and for those \(\{ \lambda_k \} \) with \(R(\lambda_k)=0,\) there exist nonnegative integers \(\{ m_k \}\) and positive constants \(\{a_k\}, \{b_k\}\) such that \[ 2\pi m_k < a_k \leq | \lambda_k (x) | \leq b_k < 2\pi (m_k+1). \] Some applications to even-order differential scalar equations are shown.
reviewed by: Antonio Cañada
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5083256
zbMATH Keywords: periodic solutions
zbMATH Keywords: differential systems
zbMATH Keywords: global inverse function theorem
zbMATH Keywords: existence and uniqueness
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.077
OpenAlex ID: W1991995219
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DOI: 10.1016/J.JMAA.2006.03.077
description: scientific article; zbMATH DE number 5083256