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• # Artykuł - szczegóły

## Bulletin of the Polish Academy of Sciences. Mathematics

2005 | 53 | 4 | 389-399

## On Probability Distribution Solutions of a Functional Equation

EN

### Abstrakty

EN
Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation
φ(x) = pφ (x-β)/(1-β) + (1-p)φ(min{x/α, (x(α-β)+β(1-α))/α(1-β)})
and its solutions in two classes of functions, namely
ℐ = {φ: ℝ → ℝ|φ is increasing, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$},
𝒞 = {φ: ℝ → ℝ|φ is continuous, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$}.
We prove that the above equation has at most one solution in 𝒞 and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection between solutions in both classes.

389-399

wydano
2005

### Twórcy

autor
• Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland
autor
• Institut für Mathematik, Karl Franzens Universität, Heinrichstrasse 36, A-8010 Graz, Austria