Solvability of the periodic problem for higher-order linear functional differential equations
type: Item
instance of: scholarly article
title: Solvability of the periodic problem for higher-order linear functional differential equations
zbMATH Open document ID: 1345.34119
DOI: 10.1134/S0012266115050018
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Differential Equations
publication date: 2015-09-28T00:00:00Z
review text: The author studies the solvability of the periodic functional differential equation \[ x^{(n)} = T^+ x - T^- x + f, \] where solutions are real-valued functions \(t \in \mathbb{R} \mapsto x(t)\) with a locally integrable weak derivative of order \(n\), of the same fixed period \(\tau\) than the locally integrable function \(t \in \mathbb{R} \mapsto f(t)\). The operators \(T^+\) and \(T^-\) are linear and positive, e.g. defined by \((Tx)(t) = k(t) x(t- \sigma(t))\) for essentially bounded and nonnegative \(\tau\)-periodic functions \(k\) and \(\sigma\). The question of the unique solvability of such systems has been fully characterized in the past for the sets of operators \(T^+\) and \(T^-\) of fixed integral norms \[ \| T\|_1 = \int_0^{\tau} (T \mathbf{1})(t)\,dt. \] The author extends the analysis to operators with essentially bounded values, subject to pointwise constraints, i.e. based on \[ \| T\|_{\infty} = \sup_t T \mathbf{1}(t). \] Necessary and sufficient conditions for the unique solvability are derived. As in the integral case, these conditions depend on values which change with the order \(n\); these values are explicited in terms of the Favard and Stechkin constants.
reviewed by: Sébastien Boisgérault
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K10
zbMATH DE Number: 6485712
zbMATH Keywords: functional differential equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: higher-order equations
zbMATH Keywords: solvability
MaRDI profile type: Publication
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full work available at URL: s0012266115050018
OpenAlex ID: W2009789170
description: scientific article; zbMATH DE number 6485712