New results for the Liebau phenomenon via fixed point index
type: Item
instance of: scholarly article
title: New results for the Liebau phenomenon via fixed point index
zbMATH Open document ID: 1370.34043
author: Gennaro Infante
author: Mirosława Zima
author: Milan Tvrdý
author: J. Ángel Cid
published in: Nonlinear Analysis. Real World Applications
publication date: 2017-01-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1606.03602
review text: The authors are concerned with the second-order nonlinear differential equation: \[ x''+ax'+m^2x=f_m(t,x),\quad 0<t<T, \] together with periodic conditions \(x(0)=x(T)\) and \(x'(0)=x'(T)\). Here, \(f_m=rx^\alpha-sx^\beta+m^2x\) with \(0<\alpha<\beta<1\), \(m\in\mathbb{R}\), and \(r, s\) are continuous functions on \([0,T]\). This problem arises in the circulation of a fluid in some mechanical system. The authors formulate the problem with a fixed point Green operator then use Krasonesl'skii's fixed point index in a positive cone of a Banach space to prove the existence of at least one solution. Some conditions on the nonlinearity \(f_m\) are assumed. Examples of applications with numerical results are provided at the end of the paper.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 6676302
zbMATH Keywords: valveless pumping
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: cone
zbMATH Keywords: fixed point index
zbMATH Keywords: Green's function
reviewed by: Smail Djebali
MaRDI profile type: Publication
OpenAlex ID: W2442779389
arXiv ID: 1606.03602
cites work: Numerical investigation of the Liebau phenomenon
cites work: An experimental investigation and a simple model of a valveless pump
cites work: Pumping effects in models of periodically forced flow configurations
cites work: Mathematical models with singularities. A zoo of singular creatures
cites work: On the pumping effect in a pipe/tank flow configuration with friction
cites work: A topological approach to periodic oscillations related to the Liebau phenomenon
cites work: Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory
cites work: Multiple Positive Solutions of Semilinear Differential Equations with Singularities
cites work: Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems
cites work: Existence of positive solutions for the cantilever beam equations with fully nonlinear terms
cites work: Existence of positive solutions for a thermostat model
cites work: Positive solutions of nonlinear equations via comparison with linear operators
cites work: Remarks on the lower and upper solutions method for second- and third- order periodic boundary value problems
cites work: Q3336252
cites work: Q3994493
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: Q5185568
cites work: Q4271322
DOI: 10.1016/J.NONRWA.2016.11.009
description: scientific article