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.

Full text (pdf)

• 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
• Revised: 2008-02-10.
• Accepted: 2008-01-20.