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Keywords:
iterative functional equation; invariant curves; monotone solutions
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 a previous paper 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.
References:
[1] J. Dhombres: Applications associatives ou commutatives. C. R. Acad. Sci. Paris 281 (1975), 809–812. MR 0419662 | Zbl 0344.39009
[2] P. Kahlig, J. Smítal: On the solutions of a functional equation of Dhombres. Results Math. 27 (1995), 362–367. DOI 10.1007/BF03322840 | MR 1331109
[3] P. Kahlig, J. Smítal: On a parametric functional equation of Dhombres type. Aequationes Math. 56 (1998), 63–68. DOI 10.1007/s000100050044 | MR 1628303
[4] P. Kahlig, J. Smítal: On a generalized Dhombres functional equation. Aequationes Math. 62 (2001), 18–29. DOI 10.1007/PL00000138 | MR 1849137
[5] M. Kuczma: Functional Equations in a Single Variable. Polish Scientific Publishers, Warsaw, 1968. MR 0228862 | Zbl 0196.16403
[6] M. Kuczma, B. Choczewski, R. Ger: Iterative Functional Equations. Encyclopedia of Mathematics and its Applications Vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720

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