Convergence for essentially strongly increasing discrete time semi-flows
type: Item
instance of: scholarly article
title: Convergence for essentially strongly increasing discrete time semi-flows
zbMATH Open document ID: 1200.37018
DOI: 10.1216/RMJ-2009-39-3-1013
author: Taishan Yi
author: Bingwen Liu
author: Qingguo Li
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: The article deals with iterations of a monotone and \(\alpha\)-condensing mapping \(T: \;[p,q] \to [p,q]\), where \([p,q]\), \(p \ll q\), is an order interval \(\{x: a\leq x \leq b\}\) in a strongly ordered Banach space \(V\); it is assumed that a cone \(V_+ = \{x \in V: x\geq 0\}\) is solid and, probably, normal (the authors state that order bounded subsets of \(V\) are bounded). About \(T\) it is assumed that it is essentially strongly increasing, i.e. \(x \leq y\) implies that either \(Tx = Ty\) or \(Tx \ll Ty\), and \(x \ll y\) always implies \(Tx \ll Ty\). The main results are the following: if \(F(T) = \{p,q\}\) then \(\{x \in [p,q]:p \ll x \ll q\}\) is attracted by either \(p\) or \(q\); if \(F(T)\) is a singleton, then \(\omega(x) = F(T)\) for all \(x \in [p,q]\); if \(F(T)\) is not a singleton and each element \(F(T)\) is Lyapunov stable then \(F(T) = J([0,1])\) where \(J:[0,1] \to [p,q]\) is a strictly incresing continuous mapping and some others (in these statements \(\omega(x)\) is the set of limit points for \(\{T^nx: n\in {\mathbb N}\}\)). As applications the periodic solutions of delay differential equations are considered. In particular, the systems of type \[ \frac{dx_i(t)}{dt} = F_i(t,x_t), \qquad i = 1,2,\dots,n \] and \[ \begin{cases} dx^1 / dt = f(t,x^n_t) - \alpha_1(t)x^1(t), \\ dx^i / dt = x^{i-1}(t - r_{i-1}) - \alpha_i(t)x^i(t),\end{cases} \qquad 2 \leq i \leq n, \] are studied.
Mathematics Subject Classification ID: 37C65
Mathematics Subject Classification ID: 34C12
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 5566214
zbMATH Keywords: essentially strongly increasing discrete time semi-flow
zbMATH Keywords: periodic quasi-monotone delay differential equations
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
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full work available at URL: rmj-2009-39-3-1013
OpenAlex ID: W2024135317
description: scientific article; zbMATH DE number 5566214