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On Probability Distribution Solutions of a Functional Equation

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Morawiec J., Reich L.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom 53

nr 4

389-399

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.

The set of probability distribution solutions of a linear functional equation

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Morawiec J., Reich L.

Annales Polonici Mathematici

2008

tom 93

nr 3

253-261

Let (Ω,𝓐,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation $F(x) = ∫_{Ω} F(τ(x,ω))dP(ω)$ we determine the set of all its probability distribution solutions.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Bulletin of the Polish Academy of Sciences. Mathematics

2 Morawiec J.

2 Reich L.

1 2008

1 2005

Wyniki wyszukiwania

On Probability Distribution Solutions of a Functional Equation

The set of probability distribution solutions of a linear functional equation