Solvability of a nonlinear two point boundary value problem at resonance
type: Item
instance of: scholarly article
title: Solvability of a nonlinear two point boundary value problem at resonance
zbMATH Open document ID: 0887.34016
author: Chung-Cheng Kuo
published in: Journal of Differential Equations
publication date: 1998-05-14T00:00:00Z
review text: The paper deals with the solvability of the boundary value problem at resonance: \[ u''+k^2u +g(x,u)=h \quad \text{on} \quad (0,\pi),\;u(0)= u(\pi)=0. \] The main result is the following theorem: Let \(g:(0,\pi) \times\mathbb{R}\to\mathbb{R}\) be a Carathéodory function such that \(|g(x,u) |\leq p(x) |u|+b(x)\) a.e. \(x\in (0,\pi)\), \(|u|\geq r_0>0\), where \(p\), \(b\in L^1(0, \pi;\mathbb{R}_+)\) and \(|p|_1< 2k(k+1) \tan \pi/(2(k+1))\). If there exist \(c\), \(d\in L^1(0,\pi)\) such that for a.e. \(x\in(0,\pi)\) and \(u\geq r_0\), \(c(x)\leq g(x,u)\), and for a.e. \(x\in (0,\pi)\) and \(u\leq-r_0\), \(g(x,u)\leq d(x)\), then the problem is solvable for any \(h\in L^1(0,\pi)\) provided that \(\int^\pi_0 hv dx <\int_{v>0} g_+(x)v(x) dx+\int_{v<0} g_-(x) v(x)dx\), where \(g_+ (x) =\liminf_{u\to\infty} g(x,u)\), \(g_-(x)= \limsup_{u\to-\infty} g(x,u)\) and \(v(x)= \alpha\sin kx\) for \(\alpha\in\mathbb{R}\setminus \{0\}\).
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1080195
zbMATH Keywords: boundary value problem
zbMATH Keywords: resonance
zbMATH Keywords: Carathéodory function
reviewed by: Radu Precup
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3312
OpenAlex ID: W2011980435
DOI: 10.1006/JDEQ.1997.3312
description: scientific article; zbMATH DE number 1080195