Existence of positive solutions for multi-point boundary value problem with strong singularity
type: Item
instance of: scholarly article
title: Existence of positive solutions for multi-point boundary value problem with strong singularity
zbMATH Open document ID: 1206.34034
author: Chan-Gyun Kim
published in: Acta Applicandae Mathematicae
publication date: 2010-10-14T00:00:00Z
review text: This paper is devoted to the study of the Sturm-Liouville boundary value problem \[ (\varphi_p(u'(t)))'+h(t)f(t,u(t))=0\quad \text{a.e. }\,0<t<1 \] with boundary conditions \[ u(0)=\sum_{i=1}^{m-2}a_iu(\xi_i),\;u(1)=\sum_{i=1}^{m-2}b_iu(\xi_i), \] where \(\varphi_p(s)=|s|^{p-2}s\,(p>1)\) is the \(p\)-Laplacian, \(0<\xi_1<\xi_2<\dots<\xi_{m-2}\) are points and \(\sum_{i=1}^{m-2}a_i<1,\;\sum_{i=1}^{m-2}b_i<1\). \(h\) is a nonnegative measurable function which may be singular at \(t=0\) and/or \(t=1\). The nonlinearity \(f\in C([0,1]\times[0,\infty),[0,\infty))\) does not necessarily satisfy the Carathéodory but some assumptions on the ratio \(\frac{f(t,u)}{\varphi_p(u)}\) are assumed. Existence of single and multiple solutions is then obtained via computation of the fixed point index on cones of Banach spaces. The paper ends with an example.
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5800315
zbMATH Keywords: \(m\)-point boundary value problem
zbMATH Keywords: positive solutions
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: multiplicity
reviewed by: Smail Djebali
MaRDI profile type: Publication
full work available at URL: s10440-009-9554-x
OpenAlex ID: W2085920314
cites work: Existence of multiple positive solutions for nonlinear \(m\)-point boundary value problems
cites work: Multiple positive solutions for \(m\)-point boundary-value problems with a one-dimensional \(p\)-Laplacian
cites work: A \(p\)-Laplacian problem with a multi-point boundary condition
cites work: Q3994493
cites work: Positive solutions for a three-point boundary value problem at resonance
cites work: Q3820938
cites work: Positive solutions of multi-point boundary value problems at resonance
cites work: Multiple positive solutions to a three-point boundary value problem with \(p\)-Laplacian
cites work: Nontrivial solutions of \(m\)-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term
cites work: Global behavior of positive solutions of nonlinear three-point boundary value problems
cites work: Positive solutions of singular three-point boundary value problems for second-order differential equations
cites work: Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional \(p\)-Laplacian
cites work: Existence of multiple positive solutions for one-dimensional \(p\)-Laplacian
cites work: Existence of positive solutions for various boundary value problems
cites work: Positive solutions for singular \(m\)-point boundary value problems with positive parameter.
DOI: 10.1007/S10440-009-9554-X
description: scientific article; zbMATH DE number 5800315