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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.

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 |