Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

John Mallet-Paret, Roger Nussbaum

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We consider a class of compact positive operators L: X→ X given by (Lx)(t)=∫η(t)tx(s)ds, acting on the space X of continuous 2 π-periodic functions x. Here η is continuous with η(t) ≤ t and η(t+ 2 π) = η(t) + 2 π for all t∈ R. We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem κx= Lx exists for some κ> 0 (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally η is analytic, we study the set A⊆ R of points t at which x is analytic; in general A is a proper subset of R, although x is C everywhere. Among other results, we obtain conditions under which the complement N= R\ A of A is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map t→ η(t).

Original languageEnglish (US)
Pages (from-to)1045-1077
Number of pages33
JournalJournal of Dynamics and Differential Equations
Volume31
Issue number3
DOIs
StatePublished - Sep 1 2019

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Analytic solution
  • Delay-differential equation
  • Generalized Cantor set
  • Integral operator
  • Spectral radius
  • Variable delay

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