On Probability Distribution Solutions of a Functional Equation

• Autorzy: Janusz Morawiec , Ludwig Reich
• Języki publikacji: 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.

Kategorie tematyczne

• 39B12: Iteration theory, iterative and composite equations
• 39B22: Equations for real functions
• 60E05: Distributions: general theory
• 26A30: Singular functions, Cantor functions, functions with other special properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: 53
• Numer: 4
• Strony: 389-399

Daty

wydano

2005

Twórcy

autor

Janusz Morawiec

• Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland

autor

Ludwig Reich

• Institut für Mathematik, Karl Franzens Universität, Heinrichstrasse 36, A-8010 Graz, Austria

Identyfikatory

DOI

10.4064/ba53-4-4