On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec ; Ludwig Reich

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005), p. 389-399 / Harvested from The Polish Digital Mathematics Library

Résumé

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(minx/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, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$, = φ: ℝ → ℝ|φ is continuous, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$. We prove that the above equation has at most one solution in 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.

Détail

Publié le : 2005-01-01

Zbl 1107.39025

EUDML-ID : urn:eudml:doc:281092