Global attractivity in monotone and subhomogeneous almost periodic systems
type: Item
instance of: scholarly article
title: Global attractivity in monotone and subhomogeneous almost periodic systems
zbMATH Open document ID: 1023.37043
DOI: 10.1016/S0022-0396(02)00054-2
author: Xiao-Qiang Zhao
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: The paper is devoted to the study of global dynamics in monotone and subhomogeneous almost periodic systems. By using the theory of skew-product semiflows and the concept of part metric, the global attractivity theorem is proved for a class of skew-product semiflows \(\Pi _t(x,y)=(u(x,y,t),\sigma (t)y)\) on \(P\times Y\), where \(P\) is a the positive cone of a Banach space \(X\) with nonempty interior \(\text{int}(P)\), and \(\sigma (t):Y\to Y\) is a compact minimal and distal flow. It is shown that the skew-product semiflow is contracting with respect to the part metric \(\rho\) on \(\text{int}(P)\). It should be noted that the monotonicity is not assumed for the nonautonomous system. The global attractivity theorem proved in the paper is applied to certain types of monotone and subhomogeneous almost periodic systems. The following results are obtained: 1. A threshold dynamics is established for Kolmogorov reaction-diffusion equations under general boundary conditions. 2. The global attractivity of a unique positive almost periodic solution for either irreducible cooperative and strictly subhomogeneous, or cooperative and strongly subhomogeneous ordinary differential equations is shown. 3. The global attractivity of a unique positive almost periodic solution for cooperative and strictly subhomogeneous delay differential equations is shown.
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 35B15
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1884233
zbMATH Keywords: skew-product semiflows
zbMATH Keywords: part metric
zbMATH Keywords: global attractivity
zbMATH Keywords: almost periodic systems
zbMATH Keywords: principal spectrum point
reviewed by: Eugene Ershov
MaRDI profile type: Publication
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full work available at URL: s0022-0396(02)00054-2
OpenAlex ID: W2058611356
description: scientific article; zbMATH DE number 1884233