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## On Probability Distribution Solutions of a Functional Equation

### Volume 53 / 2005

Bulletin Polish Acad. Sci. Math. 53 (2005), 389-399 MSC: Primary 39B22, 39B12; Secondary 60E05, 26A30. DOI: 10.4064/ba53-4-4

#### Abstract

Let $0<\beta<\alpha<1$ and let $p\in (0,1)$. We consider the functional equation $$\varphi(x)=p\varphi \biggl(\frac{x-\beta}{1-\beta}\biggr) +(1-p)\varphi \biggl(\!\min\biggl\{\frac{x}{\alpha}, \frac{x(\alpha-\beta)+\beta(1-\alpha)}{\alpha(1-\beta)}\biggr\}\biggr)$$ and its solutions in two classes of functions, namely \eqalign{ {\cal I}&=\{\varphi\colon\mathbb R\to\mathbb R\mid \varphi\hbox{ is increasing, } \varphi|_{(-\infty,0]}=0,\,\varphi|_{[1,\infty)}=1\},\cr {\cal C}&=\{\varphi\colon\mathbb R\to\mathbb R\mid \varphi\hbox{ is continuous, } \varphi|_{(-\infty,0]}=0,\,\varphi|_{[1,\infty)}=1\}.} We prove that the above equation has at most one solution in $\mathcal C$ and that for some parameters $\alpha,\beta$ and $p$ such a solution exists, and for some it does not. We also determine all solutions of the equation in $\mathcal I$ and we show the exact connection between solutions in both classes.

#### Authors

• Janusz MorawiecInstitute of Mathematics
Silesian University
Bankowa 14
PL-40-007 Katowice, Poland
e-mail
• Ludwig ReichInstitut für Mathematik
Karl Franzens Universität
Heinrichstrasse 36
A-8010 Graz, Austria
e-mail

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