Periodic points of positive linear operators and Perron-Frobenius operators

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Abstract

Let C(S) denote the Banach space of continuous, real-valued maps f : S → IR and let A denote a positive linear map of C(S) into itself. We give necessary conditions that the operator A have a strictly positive periodic point of minimal period m. Under mild compactness conditions on the operator A, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal period m. We study a class of Perron-Frobenius operators defined by (Ax)(t) = ∑i=1 bi(t)x(wi(t)), and we show how to verify the necessary compactness conditions to apply our theorems concerning existence of positive periodic points.

Original languageEnglish (US)
Pages (from-to)41-97
Number of pages57
JournalIntegral Equations and Operator Theory
Volume39
Issue number1
DOIs
StatePublished - Dec 1 2001

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory

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