Positive periodic solution for Brillouin electron beam focusing system
type: Item
instance of: scholarly article
title: Positive periodic solution for Brillouin electron beam focusing system
zbMATH Open document ID: 1242.34072
author: Jingli Ren
author: Zhibo Cheng
author: S. Siegmund
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2011-07-27T00:00:00Z
review text: The authors consider the problem of existence of positive solutions of the equations of the form \[ x^{''}+a(1+\cos {2t})x=\frac{1}{x}, \tag{1} \] where \(a>0\) is a constant. The approach is first to consider the following singular Liénard equation \[ x''(t)+f(x(t))x^{'}(t)+g(t,x(t))=0, \tag{2} \] where \(f : \mathbb R\rightarrow \mathbb R\) is continuous, \(g : [0,T]\times (0, \infty)\rightarrow \mathbb R\) is an \(L^2\)-Carathéodory function and \(g(t,x)\) is unbounded as \(x\rightarrow 0^{+}\). The following theorem (Theorem 3.1) on equation (2) is proved. Assume that (\(H_1\)) There exist constants \(0<D_1<D_2\) such that if \(x\) is a positive continuous \(T\)-periodic function satisfying \[ \int_{0}^{T}g(t,x(t)) dt=0, \] then \[ D_1\leq x(\tau)\leq D_2 \] for some \(\tau \in [0,T]\); (\(H_2\)) \(\bar {g}(x)<0\) for all \(x \in (0,D_1)\), and \(\bar {g}(x)>0\) for all \(x>D_2\), where \(\bar {g}(x):= \frac{1}{T} \int_{0}^{T}g(t,x)dt\); (\(H_3\)) \(g(t,x)=g_0(x)+g_1(t,x)\), where \(g_0 \in C((0,\infty);\mathbb R)\) and \(g_1\) is an \(L^2\)-Carathéodory function above; (\(H_4\)) \[ \int_{0}^{1} g_0(x) dt=-\infty; \] then equation (2) has at least one positive \(T\)-periodic solution if \[ a(1+\cos {2t})<\frac{2\pi^2}{T^2}. \] Mawhin's topological degree theorem and some inequalities are used to prove the theorem (Theorem 3.1). Theorem 3.1 is later applied to the specialized case of Brillouin electron beam focusing equation given by the equation (1), i.e., if in equation (1), \(0<a<1\), then it has at least one positive \(\pi\)-periodic solution.
reviewed by: Olufemi Adeyinka Adesina
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 5933123
zbMATH Keywords: Brillouin electron beam focusing system
zbMATH Keywords: Lienard equation
zbMATH Keywords: singularity
MaRDI profile type: Publication
full work available at URL: dcdsb.2011.16.385
OpenAlex ID: W1981580750
DOI: 10.3934/DCDSB.2011.16.385
description: scientific article; zbMATH DE number 5933123