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On probability distribution solutions of a functional equation

Morawiec J., Reich L.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

389-399

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation [WZÓR] and its solutions in two classes of functions, namely Z ={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 Z and we show the exact connection between solutions in both classes.