An Application of Medial Limits to Iterative Functional Equations

Abstract

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.

An Application of Medial Limits to Iterative Functional Equations, II

“…where B(S, R) denotes the space of all bounded functions from S to R endowed with the supremum norm. The main theorems of this paper generalize results obtained in [14] in three directions; to more general functional equations in wider classes of functions and under weaker assumptions. Functional equation (E G ), as well as its generalizations and special cases, are investigated in various classes of functions in connection with their appearance in miscellaneous fields of science (for more details see [7, Chapter XIII], [8,Chapters 6,7] and [2,Section 4]).…”

“…The aim of this section is to generalize results obtained in [14] without any assumptions on the function f .…”

“…Finally, Sect. 6 contains extensions of the main results from [14] under a bounded condition on the function f .…”

“…Iteration is the fundamental technique for solving functional equations in a single variable, and in most cases the formulae for solutions is obtained by taking the limit of sequences in which iterates are involved. In this paper we make use of this fundamental technique, but following the ideas of [14] we apply a subclass of Banach limits instead of the limit. The idea of replacing the limit by a Banach limit seems to be clear, because we do not need any additional assumption guaranteeing the existence of a Banach limit of a bounded sequence.…”

“…4 we proceed with the detailed analysis of the operator in the non-expansive case. This section, the main part of this paper, contains generalizations of all results from [14] without any assumptions on the function f . In Sect.…”