Multiple positive solutions for a nonlocal boundary-value problem with nonconvex vector-valued response
type: Item
instance of: scholarly article
title: Multiple positive solutions for a nonlocal boundary-value problem with nonconvex vector-valued response
zbMATH Open document ID: 1026.34029
DOI: 10.1016/S0022-247X(02)00574-7
author: Aleksandra Orpel
author: Andrzej F. Nowakowski
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-04-09T00:00:00Z
review text: In their recent paper (not yet published), the authors consider the nonlinear problem \[ {d\over dt}\bigl( k(t)x'(t)\bigr) +V_x\bigl(t,x(t) \bigr) =0,\text{ a.e. in }[0,T],\;x(0)=0, \] where \(T>0\) is arbitrary, \(V:\mathbb{R} \times\mathbb{R}^n \to\mathbb{R}\) is Gateaux differentiable in the second variable and measurable in \(t\), and \(k:[0,T] \to\mathbb{R}^+\) satisfies certain conditions. They prove that if the function \(V(t,\cdot)\) is convex then the problem admits a countable family of positive solutions. In the present paper, they show that the conclusion remains true if one requires only the convexity of the function \(x\to \int^T_0 V(t,x(t))dt\) in the set \(\{x\in L^2:u \gamma\leq x(t),\;t\in [t_0T]\), \(x(t)\leq \beta v\), \(x(t)\in P\), \(t\in [0,t]\}\) where \(P\) is a positive cone in \(\mathbb{R}^n\).
Mathematics Subject Classification ID: 34B18
zbMATH DE Number: 1895772
zbMATH Keywords: nonlocal boundary-value problems
zbMATH Keywords: positive solutions
zbMATH Keywords: duality method
zbMATH Keywords: variational method
Wikidata QID: Q104413757
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
cites work: Q3712561
cites work: Q3820938
cites work: Q4528802
cites work: Multiple positive solutions for a nonlocal boundary-value problem with response function quiet at zero
cites work: Q3787011
cites work: A new variational principle and duality for periodic solutions of Hamilton's equations
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description: scientific article; zbMATH DE number 1895772