Positive solutions of \(n\)th-order \(m\)-point impulsive boundary value problems
type: Item
instance of: scholarly article
title: Positive solutions of \(n\)th-order \(m\)-point impulsive boundary value problems
zbMATH Open document ID: 1329.34054
author: Ilkay Yaslan Karaca
author: Fatma Tokmak
published in: Georgian Mathematical Journal
publication date: 2015-09-08T00:00:00Z
review text: The authors study the following nonlinear m-point impulsive boundary value problem \[ \begin{aligned} & u^{(n)}(t)+q(t)f(t,u(t),u'(t),\dots,u^{(n-2)}(t))=0,\;t\in[0,1],t\neq t_{k}\\ & u^{(n-2)}(t_{k}^+)-u^{(n-2)}(t_{k}^-)=I_{k}(u^{(n-2)}(t_{k})),\\ & u^{(n-1)}(t_{k}^+)-u^{(n-1)}(t_{k}^-)=-J_{k}(u^{(n-2)}(t_{k})),\\ & au^{(n-2)}(0)-bu^{(n-1)}(0)=\sum_{i=1}^{m-2}\alpha_{i}u^{(n-2)}(\xi_{i}),\\ & cu^{(n-2)}(1)+du^{(n-1)}(1)=\sum_{i=1}^{m-2}\beta_{i}u^{(n-2)}(\xi_{i}),\\ & u^{(j)}(0)=0,\;0\leq j\leq n-3\end{aligned} \] where \(t_{k}\in(0,1)\), \(k=1,\dots,l\), \(0<t_1<\dots<t_{l}<1\), \(a,b,c,d\in\mathbb R_+:=[0,\infty)\), \(ac+ad+bc>0\), \(\alpha_{i},\beta_{i}\in\mathbb R_+\), \(\xi_{i}\in(0,1)\), \(i=1,\dots,m-2\), \(f\in C([0,1]\times\mathbb R_+^{n-1},\mathbb R_+)\), \(q\in C([0,1],\mathbb R_+)\), \(I_{k}\in C(\mathbb R_+,\mathbb R_+)\) and \(J_{k}\in C(\mathbb R_+,\mathbb R_+)\) are bounded functions such that \[ (c(1-t_{k})+d)J_{k}(u(t_{k}))>cI_{k}(u(t_{k})). \] Using Avery-Peterson's fixed point theorem the authors prove that there exist at least three positive solutions of the problem.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B37
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 6479290
zbMATH Keywords: Green's function
zbMATH Keywords: impulsive boundary value problems
zbMATH Keywords: fixed-point theorem
zbMATH Keywords: positive solution
reviewed by: Ekin Ugurlu
MaRDI profile type: Publication
full work available at URL: gmj-2015-0020
OpenAlex ID: W2318430188
DOI: 10.1515/GMJ-2015-0020
description: scientific article; zbMATH DE number 6479290