×

zbMATH — the first resource for mathematics

On a generalized Dhombres functional equation. II. (English) Zbl 1007.39016
Summary: We consider the functional equation \(f(xf(x))=\varphi (f(x))\) where \(\varphi: J\rightarrow J\) is a given increasing homeomorphism of an open interval \(J\subset (0,\infty)\) and \(f\:(0,\infty)\rightarrow J\) is an unknown continuous function. In part I [Aequationes Math. 62, No. 1-2, 18-29 (2001; Zbl 0994.39013)] we proved that no continuous solution can cross the line \(y=p\) where \(p\) is a fixed point of \(\varphi \), with a possible exception for \(p=1\). The range of any non-constant continuous solution is an interval whose end-points are fixed by \(\varphi \) and which contains in its interior no fixed point except for \(1\). We also gave a characterization of the class of continuous monotone solutions and proved a sufficient condition for any continuous function to be monotone.
In the present paper we give a characterization of the equations (or equivalently, of the functions \(\varphi \)) which have all continuous solutions monotone. In particular, all continuous solutions are monotone if either (i) 1 is an end-point of \(J\) and \(J\) contains no fixed point of \(\varphi \), or (ii) \(1\in J\) and \(J\) contains no fixed points different from 1.

MSC:
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
39B22 Functional equations for real functions
PDF BibTeX XML Cite
Full Text: EuDML