Multiple solutions for a super-linear three-point boundary value problem
type: Item
instance of: scholarly article
title: Multiple solutions for a super-linear three-point boundary value problem
zbMATH Open document ID: 1037.34010
DOI: 10.1016/S0362-546X(01)00738-6
author: Chaitan P. Gupta
author: Bruce D. Calvert
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The authors study the three-point BVP \[ x''(t)+g(x(t))=p(t,x(t),x'(t)),\quad x(0)=0, x(\eta)=\beta x(1), \tag{1} \] where \(\eta\in (0,1)\), \(\beta>0, \beta\not=1\), \(g: \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function with \(g(0)=0\), \(g(x)x>0\) for \(x\not= 0\). Moreover \(g(x)/x\) is supposed to be increasing and \(\lim_{| x| \rightarrow\infty}(g(x)/x)= \infty\). Further, \(p:[0,1]\times \mathbb{R}^2 \rightarrow \mathbb{R}\) is supposed to satisfy the Carathéodory conditions and \[ | p(t,x,y)| \leq M_1(t,\max(| x| ,| y| )),\quad (t,x,y)\in [0,1]\times \mathbb{R}^2, \] where \(M_1(t,s)\) is integrable in \(t\) on \([0,1]\) and increasing in \(s\) on \([0,\infty)\) with \[ \lim_{s\rightarrow \infty} {1\over s} \int_0^1 M_1(t,s)\,dt =0. \] This problem with \(\beta =1\) was studied earlier by \textit{M. Henrard} [Adv. Differ. Equ. 2, 753--778 (1997; Zbl 1023.34501)] where a variant of the Leray-Schauder continuation theorem was used. Here, the authors use some of the ideas and results of Hernard and prove the existence of at least two solutions to (1).
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1787998
zbMATH Keywords: multiple solutions
zbMATH Keywords: super-linear three-point boundary value problem
zbMATH Keywords: Leray-Schauder degree
MaRDI profile type: Publication
cites work: Q3712561
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description: scientific article; zbMATH DE number 1787998