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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
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