Opuscula Math. 28, no. 4 (2008), 529-539

Opuscula Mathematica

Iteration groups, commuting functions and simultaneous systems of linear functional equations

Janusz Matkowski

Abstract. Let \(( f^t )_{t \in \mathbb{R}}\) be a measurable iteration group on an open interval \(I\). Under some conditions, we prove that the inequalies \(g\circ f^a \leq f^a \circ g\) and \(g\circ f^b \leq f^b\circ g\) for some \(a,b \in \mathbb{R}\) imply that \(g\) must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function \(g\) commutes in a neighbourhood of \(f\) with two suitably chosen elements \(f^a\) and \(f^b\) of an iteration group of \(f\) then, in this neighbourhood, \(g\) coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.

Keywords: iteration group, commuting functions, functional equation, functional inequalities.

Mathematics Subject Classification: 39B12, 39B72, 26A18.

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  • Janusz Matkowski
  • University of Zielona Góra, Institute of Mathematics, Computer Science and Econometry, ul. Podgórna 50, 65-246 Zielona Góra, Poland
  • Silesian University, Institute of Mathematics, 40-007 Katowice, Poland
  • Received: 2007-11-06.
  • Revised: 2008-02-10.
  • Accepted: 2008-01-20.
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Cite this article as:
Janusz Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28, no. 4 (2008), 529-539

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