An application of medial limits to iterative functional equations

Assume that $(\Omega,\mathcal A,P)$ is a probability space, $f\colon[0,1] \times \Omega\to[0,1]$ is a function such that $f(0,\omega)=0$, $f(1,\omega)=1$ for every $\omega\in\Omega$, $g\colon[0,1]\to\mathbb R$ is a bounded function such that $g(0)=g(1)=0$, and $a,b\in\mathbb R$. Applying medial limits we describe bounded solutions $\varphi\colon[0,1] \to \mathbb R$ of the equation \begin{equation*} \varphi(x) = \int_\Omega \varphi(f(x,\omega)) dP(\omega)+g(x) \end{equation*} satisfying the boundary conditions $\varphi(0)=a$ and $\varphi(1)=b$.

We say that a function ϕ : [0, 1] → R is a solution of equation (E g ) if for every x ∈ [0, 1] the function ϕ • f (x, ·) is measurable and (E g ) holds.
The main purpose of this paper is to describe all solutions of equation (E g ) in some classes of bounded functions h : [0, 1] → R such that h(0) = a and h(1) = b. We are also interested under which assumptions any bounded solution ϕ : [0, 1] → R of equation (E g ) with a certain property can be expressed in the form ϕ = Φ + ϕ * , where Φ is a solution of the equation having the same property as ϕ and ϕ * is a specific solution of equation (E g ). This problem seems to be easy to answer, but the difficulty is that the classes considered in this paper are not linear spaces. It is even not clear when the existence of a solution with a certain property of one of the equations (E g ) and (E 0 ) implies the existence of a solution with the same property of the other of these equations. Such a problem is quite natural in the theory of functional equations and it has been studied several times by many authors for different functional equations in various classes of functions; mainly in cases where the class of considered functions forms a vector space. Functional equations (E 0 ) and (E g ), as well as their 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 [18,Chapter XIII], [19,Chapters 6,7] and [4,Section 4]). As emphasized in [19, section 0.3] iteration is the fundamental technique for solving functional equations in a single variable, and iterates usually appear in the formulae for solutions. In most cases such formulas are obtained by taking the limit of sequences in which iterates are involved. In this paper we make use of this fundamental technique, but the goal is to 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, in contrast to the case when we want to calculate the limit of such a sequence.
This paper is organized as follows. Section 2 contains the notation and basic tools required for our considerations. In sections 3 and 4 we describe bounded solutions ϕ : [0, 1] → R with ϕ(0) = a and ϕ(1) = b of equations (E 0 ) and (E g ), respectively. Finally, in section 5, we formulate some consequences of the main results obtained and we present a few examples of the possible applications of those results.

Preliminaries
Denote by B( Note that T is linear and continuous with T = 1. Moreover, equation (E g ) can be written now in the form To the end of this paper we fix a subspace B(   To describe solutions of equation (E g ) in the case where A = 2 Ω we need the concept of Banach limits, established in [1]. However in the general case, when integration is required, we need the concept of medial limits, established in [25] (cf. [24]) as a very special class of Banach limits.
Denote by l ∞ (N) the space of all bounded real sequences equipped with the supremum norm and by B the family of all Banach limits defined on l ∞ (N). Recall that B ∈ B if B : l ∞ (N) → R is a linear, positive, shift invariant and normalized operator. It is easy to see that any B ∈ B is continuous with B = 1. It is known that the cardinality of B is equal to 2 c (see [14]), and even that the cardinality of the set of all extreme points of B is equal to 2 c (see [11], cf. [29]); here c is the cardinality of the continuum.
As it was mentioned above, in the general case we need to integrate the pointwise Banach limit of a bounded sequence of measurable functions. However, the problem is that there is no guarantee that the pointwise Banach limit of a bounded sequence of measurable functions is a measurable function (see [31, page 288]). Fortunately, it is known that there are Banach limits possessing exactly the required property. More precisely, a Banach limit B is called a medial limit if Ω B((h m (ω)) m∈N )dP (ω) is defined and equal to B(( Ω h m (ω)dP (ω)) m∈N ) whenever (h m ) m∈N is a bounded sequence of measurable real-valued functions on Ω. It is also 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 [15,Chapter 53] and in [20].
It is also clear that if we determine all solutions of equation (E g ) in the class B b a , then we can easily describe all solutions of this equation in the class B([0, 1], R). Now we are in a position to begin describing solutions of equation (E g ) in the class B b a . Our first lemma is a simple consequence of (1) and (3).
The functions B h plays a crucial role in this section as well as in this paper. So, we need some fact about them.
a . Applying properties of medial limits we obtain for every x ∈ [0, 1].

We now want to find conditions under which
This observation suggests the following definition.
We say that the class To prove the conclusion let us put B([0, 1], R) = C x0 ([0, 1], R); this is possibly according to Example 2.5, because (H 3 ) yields the continuity of f (·, ω) at x 0 for every ω ∈ Ω. Let B ∈ B and let x ∈ A, and by an easy induction, we obtain sup Finally, (H 1 ) jointly with properties of Banach limits implies that both the functions B h1 and B h2 are increasing.
We are now in a position to formulate the main results of this section. To simplify their statements, let us denote by sol b a (E 0 ) the family of all functions from B b a satisfying equation (E 0 ).
The opposite inclusion follows from Lemma 3.1.
Solutions of equations (E 0 ) was investigated in [26,27], basically in almost the same classes of bounded functions. However, Theorem 3.2 is incomparable with the results obtained in the papers mentioned, in which the existence and the uniqueness problems have been considered as well as properties of the unique solution have been studied.

Solutions of equation (E g )
In this section we describe all functions belonging to the class B b a which are solutions of equation (E g ). We also give the formula for these solutions showing that each of them can be written in the form Φ + ϕ * , where Φ ∈ B b a is a solution of equation (E 0 ) and ϕ * ∈ B 0 0 is a particular solution of equation (E g ). To find ϕ * we need define a certain family of functions generated by g ∈ B 0 0 ; recall that g ∈ B 0 0 is a necessary condition for equation (E g ) to have a solution in the class B b a by Lemma 2.1(ii). If g ∈ B 0 0 , then Lemma 2.1(i) yields {T l g : l ∈ N} ⊂ B 0 0 . Therefore, given g ∈ B 0 0 and k ∈ N we can define a function g k : [0, 1] → R by putting As in the previous section, denote by sol b a (E g ) the family of all functions from B b a satisfying equation (E g ). Proof. Fix ϕ ∈ sol b a (E g ). Then Lemma 2.1 implies that G ⊂ B 0 0 . Applying (2) we obtain The above lemma shows that boundedness of the family G is a necessary condition for equation (E g ) to have a solution in the class B b a . This also demonstrate, that B g k is well defined for all k ∈ N and B ∈ B whenever equation for every x ∈ [0, 1]. Now, it only remains to see that B g k = kB g .
If g ∈ B 0 0 and G is bounded, then for every B ∈ B we define a function B * : [0, 1] → R by putting for every x ∈ [0, 1].
We now want to find conditions under which B * ∈ B 0 0 . The situation is similar to that for B h ∈ B b a . Namely, to prove that B * ∈ B 0 0 , we would have to show that T B * ∈ B 0 0 , but by Lemma 4.3 we have T B * = B * − g. This leads us to the following definition.
We say that a function g ∈ B 0 0 is admissible for B ∈ B, if the family G is bounded and B * ∈ B 0 0 . Note that the assumption on boundedness of G in the admissibility definition is not restrictive, because if the family G is unbounded, then B * can not be a solution of equation (E g ) by Lemma 4.1.
Before we give examples of conditions guaranteeing admissibility of a given function under a Banach limit, let us recall the definition of almost convergence of sequences. Namely, a bounded sequence (x m ) m∈N of real numbers is said to be almost convergent to a real number x if B((x m ) m∈N ) = x for any B ∈ B. The sequence (0, 1, 0, 1, 0, 1, . . .) is a simple example of a non-convergent sequence which is almost convergent. However almost none of the sequences consisting of 0's and 1's are almost convergent (see [12]). It is proved in [22] that a sequence (x k ) k∈N is almost convergent to x if and only if lim n→∞    Let us note that condition (5) is not very far from a necessary condition for g derived in Lemma 4.2, which says that B (T m g(x)) m∈N = 0 for all x ∈ [0, 1] and B ∈ B.
We now formulate the main result of this paper.

Theorem 4.4.
(i) Assume that a (E g ) and B ∈ B. Obviously, B ϕ is well defined. From Lemma 4.1 we conclude that B * is also well defined. Applying induction to (2) we get (7) ϕ = T k ϕ + g k for every k ∈ N, and hence ϕ( a , and making use of (7) we obtain that sup k∈N g k ≤ 2 ϕ and B * = ϕ − B ϕ ∈ B 0 0 .
a . Then Lemmas 3.1, 4.1 and 4.3 give which means that B h + B * ∈ sol b a (E g ). (ii) It suffices to apply assertion (i). (iii) Fix ϕ ∈ sol b a (E g ). Lemma 4.3 jointly with the admissibility of g implies that a (E 0 ). Then again Lemma 4.3 jointly with the admissibility of g implies that Φ + B * ∈ sol b a (E g ). a (E g ) = ∅, then it may happen that there is no B ∈ B for which B * is well defined; see e.g. the equation ϕ(x) = ϕ(x)+ 1. Therefore, assumption (6) can not be omitted in assertion (i) of Theorem 4.4. The above exemplary equation also shows that the admissibility assumption in assertion (iii) of Theorem 4.4 is necessary.

Consequence of the main results
In this section we formulate some exemplary consequences of the main results, making use of the presented examples and applying some know results on equation (E g ). We begin with the case where A = 2 Ω . p n ϕ(f n (x)) + g(x), which is discussed in more details in [18,Chapter XIII] and in [19,Subsections 6.3 and 6.7]).
Purely bounded solutions of equation (E g ) are considered rather rarely. Usually some additional property is requited, such as monotonicity (see e.g. [16,17,28]), Borel measurability (see e.g. [2,6]), continuity at a point (see e.g. [5]). The next two corollaries concern just such cases. To formulate the first one we need some notion. Namely, following [8] (cf. [13]) we define iterates of a function h : for all x ∈ [0, 1], ω = (ω 1 , ω 2 , . . .) ∈ Ω ∞ and n ∈ N. Note that if h is an rvfunction, then all its iterates are also rv-functions defined on the product space (Ω ∞ , A ∞ , P ∞ ).  The next example is in the spirit of the idea of the manuscript [30] with the use of Corollary 5.3.
Lipschitzian solutions of equation (E g ), in a more general setting than in this paper, were recently examined in [3,7,9,10]. However, the next Corollary gives a general formulae for a wide class of Lipschitzian solutions of equation (E g ), in contrast to the papers mentioned, in which assumptions made force uniqueness or uniqueness up to an additive constant of Lipschitzian solutions of the equation considered. The next corollary gives a formulae for the general solution of equation (E g ) in the space BV ([0, 1], R), and hence, partially solves the problem considered in [23] for a very spacial case of equation (E 0 ). Before we formulate the last corollary of this paper let us to extend the main result of [23] to equation (E 0 ). Proposition 5.6. Assume (H 1 ). If Φ ∈ BV ([0, 1], R) satisfies (E 0 ), then also Φ + and Φ − satisfy (E 0 ).