Positive periodic solutions of functional differential equations

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131 Scopus citations

Abstract

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t) g(x)x(t)-λb(t)f(x(t-τ(t))), where a,b∈C(ℝ, [0,∞)) are ω-periodic, ∫0ω a(t) dt>0, ∫0ωb(t) 0, f,g∈ C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define f0 =limu→0+ f(u)/u, f =limu→∞ f(u)/u, i0=number of zeros in the set {f0,f} and i=number of infinities in the set {f0, f}. We show that the equation has i0 or i positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.

Original languageEnglish (US)
Pages (from-to)354-366
Number of pages13
JournalJournal of Differential Equations
Volume202
Issue number2
DOIs
StatePublished - Aug 1 2004

Keywords

  • Existence
  • Fixed index theorem
  • Multiplicity
  • Nonexistence
  • Positive periodic solution

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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