We study the problem of the existence of increasing and continuous solutions ϕ : [0, 1] → [0, 1] such that ϕ(0) = 0 and ϕ(1) = 1 of the functional equationwhere N ∈ N and f0, . . . , fN : [0, 1] → [0, 1] are strictly increasing contractions satisfying the following condition 0 = f0(0) < f0(1) = f1(0) < · · · < fN−1(1) = fN (0) < fN (1) = 1. In particular, we give an answer to the problem posed in [9] by Janusz Matkowski concerning a very special case of that equation.Mathematics Subject Classification (2010). Primary 39B12; Secondary 26A30, 26A46, 28A80.

Abstract. We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a generalized problem of Zoltán Boros posed during the 50th ISFE.Mathematics Subject Classification. 39B12, 39A06.

Assume that (Ω, A, P) is a probability space, f : [0, 1] × Ω → [0, 1] is a function such that f (0, ω) = 0, f (1, ω) = 1 for every ω ∈ Ω, g : [0, 1] → R is a bounded function such that g(0) = g(1) = 0, and a, b ∈ R. Applying medial limits we describe bounded solutions ϕ : [0, 1] → R of the equation ϕ(x) = Ω ϕ(f (x, ω))dP (ω) + g(x) satisfying the boundary conditions ϕ(0) = a and ϕ(1) = b.

No abstract

Given a sequence (ξ n , η n ) of independent identically distributed vectors of random variables we consider the Grincevičjus series ∞ n=1 η n n−1 k=1 ξ k and a functional-integral equation connected with it. We prove that the equation characterizes all probability distribution functions of the Grincevičjus series. Moreover, some application of this characterization to a continuous refinement equation is presented.