An Application of Medial Limits to Iterative Functional Equations, II

Applying medial limits we describe bounded solutions φ:S→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :S\rightarrow {\mathbb {R}}$$\end{document} of the functional equation φ(x)=∫Ωg(ω)φ(f(x,ω))dμ(ω)+G(x),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi (f(x,\omega ))d\mu (\omega )+G(x), \end{aligned}$$\end{document}where (Ω,A,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Omega ,{\mathcal {A}},\mu )$$\end{document} is a measure space, S⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subset \mathbb R$$\end{document}, f:S×Ω→S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:S\times \Omega \rightarrow S$$\end{document}, g:Ω→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\Omega \rightarrow {\mathbb {R}}$$\end{document} is integrable and G:S→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G:S\rightarrow {\mathbb {R}}$$\end{document} is bounded. The main purpose of this paper is to extend results obtained in Morawiec (Results Math 75(3):102, 2020) to the above general functional equation in wider classes of functions and under weaker assumptions.


Introduction
Throughout this paper we assume that (Ω, A, μ) is a measure space, S ⊂ R is a non-empty set, f : S × Ω → S is a function, G : S → R is a bounded function and g : Ω → R is an integrable function, i.e. Ω |g(ω)|dμ(ω) < ∞. (1) We are interested in solutions ϕ ∈ B(S, R) of the following iterative functional equation (ω).
But, in the general case it is not easy to find a non-trivial solution of (E 0 ) in M(S, R). Moreover, it is unclear whether equation (E G ) has a solution in M(S, R). The problem is that we do not have tools to solve this problem in the full generality. However, some results can be proved with the use of the following observation. Obviously, T is linear and continuous with T ≤ Ω |g(ω)|dμ(ω), by (1). Moreover, equations (E G ) and (E 0 ) can be written now in the following forms and Φ = TΦ, respectively. From now on we fix a non-trivial subspace B(S, R) of M(S, R) that is invariant under T, i.e.
Before we give examples showing how the space B(S, R) looks like in a certain situation, let us introduce symbols, which we will use for basic spaces of functions. Here and subsequently, C A (S, R), Lip(S, R), BV (S, R) and Borel(S, R) denote subspaces of the space B(S, R) consisting of all functions that are continuous at every point of a set A ⊂ S, Lipschitzian, of bounded variation (in general S may not be a compact set, and in such a case by a function with bounded variation we mean a function which can be written as a difference of two increasing functions, which is justified by [   In later sections we will need the medial limits. To introduce them denote by B the family of all Banach limits defined on B(N, R). Recall that M ∈ B if M : B(N, R) → R is a linear, positive, shift invariant and normalized operator. It is easy to see that any M ∈ B is continuous with M = 1. Let us note that the cardinality of B is equal to 2 c (see [5]; cf. [3], where it is proved that the cardinality of the set of all extreme points of B is equal to 2 c ); here c is the cardinality of the continuum. A Banach limit M is called a medial limit, with respect to a probability space (Ω, A, P ), if Ω M ((h m (ω)) m∈N )dP (ω) is defined and equal to M (( Ω h m (ω)dP (ω)) m∈N ) whenever (h m ) m∈N is a bounded sequence of measurable functions from Ω to R; the sentence Ω M ((h m (ω)) m∈N )dP (ω) is defined means, in particular, that the function M ((h m (·)) m∈N ) is A-measurable. It is known that the continuum hypothesis implies the existence of medial limits. More results on the existence and non-existence of medial limits can be found in [6, Chapter 53] and in [9]. Denote by M P the family of all medial limits, with respect to a probability

The Case Where T is Contractive
We begin with two simple results on equation (E G ). The first one concerns the uniqueness of solutions in the space M(S, R), whereas in the second one we get the existence of a solution in the space B(S, R).
Before we formulate the second result let us note that G ∈ B(S, R) is a necessary condition for equation (E G ) to have a solution in B(S, R), by (4).
Proof. According to Proposition 2.1 we conclude that the space B(S, R) is complete. By (5) the operator P : B(S, R) → B(S, R) given by P h = Th + G is a contraction. Thus the Banach fixed point theorem implies that sol(E G ) = {ϕ 0 }. To get the formula for ϕ 0 we first note that (5) yields convergence of the series m∈N0 T m G. From (4) and the fact that In general, it can happen that equation (E G ) has a solution in the space M(S, R), which is not of the form m∈N0 T m G (see [15]).
Combining Proposition 3.2 with Example 2.1 (i) and (iv) we obtain the following corollaries.
has exactly one bounded solution ϕ ∈ B(S, R); it is given by the formula (5) holds, then equation (E G ) has exactly one solution ϕ ∈ Borel(S, R); it is given by the formula

The Case Where T is Non-expansive
The aim of this section is to generalize results obtained in [14] without any assumptions on the function f .
Assume that T = 1.  h(x)) m∈N ∈ B(N, R) for every x ∈ S. Therefore, given a function h ∈ B(S, R) and a Banach limit M ∈ B we can associate with them a function The functions M h play a key role in determining sol(E 0 ) and sol(E G ). So, we need some facts about them. We begin with the following simple observation. Before we formulate the next lemma we introduce two probability measures that will be useful in the next proofs.
Vol. 76 (2021) An Application of Medial Limits Page 7 of 17 47 Note that this equality can also be obtained by the Jordan decomposition of the signed measure defined by ν(A) = A g(ω)dμ(ω) (see [4, Section IX.2]), but the difference is that in the case where P is of constant sing, there is formally only one measure in the Jordan distribution.
To the end of this paper we assume that where M P 1 and M P 2 are families of medial limits with respect to (Ω A, P 1 ) and (Ω, A, P 2 ), respectively. In general, there is no guarantee that (8) , ·))) m∈N is A-measurable for every x ∈ S, which means that M h ∈ M(S, R).
Applying (7) jointly with properties of medial limits we get for every x ∈ S. Now, we want to find conditions under which M h ∈ B(S, R) for every h ∈ B(S, R). Unfortunately, the situation is as in [14], i.e. there is no chance to find such conditions in the general case, because to prove that M h ∈ B(S, R), we would have to show, by the definition of B(S, R) (see (4)), that TM h ∈ B(S, R); unfortunately Lemma 4.2 yields TM h = M h . This observation suggests the following definition.  We are now in a position to describe the family sol(E 0 ). Namely, combining Lemmas 4.1 and 4.2 we obtain the following result, which generalizes [14,Theorem 3.2].  R). Thus for all G ∈ B(S, R) and k ∈ N we can define a function G k : S → R by To the end of this section we set We begin with a necessary condition on the family G for equation (E G ) to have a solution in the class B(S, R).

Lemma 4.3. Assume that (6) holds. If
then G is a bounded subset of B(S, R).
Proof. Fix ϕ ∈ sol(E G ). From (4) we see that G ⊂ B(S, R). Applying induction to (2) we obtain ϕ = T k ϕ + G k (10) for every k ∈ N. This jointly with (6) gives sup k∈N G k ≤ 2 ϕ . From Lemma 4.3 we see that M G k are well defined for all k ∈ N and M ∈ B whenever (9) holds. So, we can ask about the formula of these functions. To answer this question fix ϕ ∈ sol(E G ) and M ∈ B. Then (6) implies that M G is well defined and (2) gives If G ∈ B(S, R) and if the family G is bounded, then with any Banach limit M ∈ B we associate a function M * : S → R defined by Proof. According to our convention we can assume that (9) holds. Fix ϕ ∈ sol(E G ). Obviously, ϕ ∈ B(S, R) and B ϕ is well defined. From Lemma 4.3 we conclude that M * is also well defined. Then making use of (10) we get  Hence M * ∈ B(S, R). Since M ∈ M P 1 ∩M P 2 , it follows that M * ∈ M(S, R); cf. the second part of the proof of Lemma 4.2. Applying (7) jointly with properties of medial limits we get for every x ∈ S.
We now want to find conditions under which M * ∈ B(S, R). The situation is similar to that for M h ∈ B(S, R). Namely, to prove that M * ∈ B(S, R), we would have to show that TM * ∈ B(S, R), but Lemma 4.5 yields TM * = M * −G. This leads us to the following definition. Note that the boundedness assumption on the family G in the above definition is not restrictive; indeed if G is unbounded, then M * can not be a solution of equation (E G ) by Lemma 4.3.
Before we give examples of admissible functions for some Banach limits, let us note the following fact.