Topological analysis of doubly nonlocal boundary value problems
type: Item
instance of: scholarly article
title: Topological analysis of doubly nonlocal boundary value problems
zbMATH Open document ID: 1473.45016
author: Christopher Goodrich
published in: Journal of Fixed Point Theory and Applications
publication date: 2021-06-09T00:00:00Z
review text: This is an interesting paper dealing with the perturbed Hammerstein integral equation of the form \[u(t)=\gamma(t)H(\varphi(u))+\lambda\int_0^1\left(A\left(\int_0^1|u(\xi)|^qd\xi\right)\right)^{-1}G(t,s)f(s,u(s))ds,\] with \(t\in(0,1)\), where \(q\geq1\) and the functions \(A:\mathbb{R}^+\to\mathbb{R},\) \(f:[0,1\times\to\mathbb{R}^+,\) \(G:[0,1]\times[0,1]\to\mathbb{R}^+,\) \(H:\mathbb{R}^+\to\mathbb{R}^+\) \(\gamma:[0,1]\to[0,1]\) are all continuous. The function \(\varphi\) has the linear integral form \[\int_0^1u(s)d\alpha(s),\] where \(\alpha\) is a monotone increasing function of bounded variation on the interval \([0,1].\) This equation unifies the integral forms produced by various nonlocal boundary value problems of second order, such as \[-M\left(\int_0^1(u(s))^q\right)u''(t)=\lambda f(t,u(t)),\] and \[-M\left(\int_0^1(u'(s))^q\right)u''(t)=\lambda f(t,u(t)),\] with \(t\in(0,1).\) In order to apply a standard fixed point method lying on the index on the boundary of some suitable sets, the author considers a specific cone that is the intersection of two obvious cones, to be used separately to discuss the two specific problems above. An application of these results is given explicitly.
Mathematics Subject Classification ID: 45M20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 7356819
zbMATH Keywords: nonlocal differential equation
zbMATH Keywords: nonstandard cone
zbMATH Keywords: Hammerstein integral equation
zbMATH Keywords: positive solution
reviewed by: George Karakostas
MaRDI profile type: Publication
full work available at URL: s11784-021-00865-1
OpenAlex ID: W3156988578
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Topological analysis of doubly nonlocal boundary value problems (DOI)
DOI: 10.1007/S11784-021-00865-1