1 Introduction

Fix a probability space \((\Omega ,{{\mathcal {A}}},P)\), a complete and separable metric space \((X,\rho )\) with the \( \sigma \)-algebra \( {\mathcal {B}} \) of all its Borel subsets, and a \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable function \( f: X \times \Omega \rightarrow X \).

We continue the research of continuous solutions \(\varphi :X \rightarrow {\mathbb {R}}\) of the equations

$$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$
(1)
$$\begin{aligned} \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ). \end{aligned}$$
(2)

We refer mainly to [2, 5]. Like in these papers we focus on the iteration of random-valued functions:

$$\begin{aligned} f^0 (x, \omega _1, \omega _2, \ldots ) = x, \quad f^n(x, \omega _1, \omega _2, \ldots ) = f\big (f^{n-1} (x, \omega _1, \omega _2, \ldots ), \omega _n\big ) \end{aligned}$$

for \(n \in {\mathbb {N}}\), \(x \in X\) and \( (\omega _1, \omega _2, \ldots ) \) from \( \Omega ^{\infty } \) defined as \( \Omega ^{{\mathbb {N}}} \). Note that for \(n \in {\mathbb {N}}\) the nth iterate \( f^n \) mapping \(X \times \Omega ^{\infty } \) into X is \( {\mathcal {B}} \otimes {\mathcal {A}}_n \)-measurable, where \( {\mathcal {A}}_n \) denotes the \( \sigma \)-algebra of all sets of the form

$$\begin{aligned} \{ (\omega _1, \omega _2, \ldots ) \in \Omega ^{\infty }: (\omega _1, \ldots , \omega _n) \in A \} \end{aligned}$$

with A from the product \( \sigma \)-algebra \( {\mathcal {A}}^n \). (See [13, section 1.4], [10].)

Let \( \pi _n^f (x, \cdot ) \) denote the distribution of \( f^n (x, \cdot ) \), i.e.,

$$\begin{aligned} \pi _n^f (x, B) = P^{\infty } \big (f^n (x, \cdot ) \in B\big ) \quad \text {for } n \in {\mathbb {N}}\cup \{0\}, \ x \in X \text { and } B \in {\mathcal {B}}. \end{aligned}$$

If

$$\begin{aligned} \int _{\Omega } \rho \big (f(x, \omega ), f(z, \omega )\big ) P(d\omega ) \le \lambda \rho (x, z) \quad \text {for } x, z \in X \end{aligned}$$
(3)

with a \( \lambda \in (0,1) \), and

$$\begin{aligned} \int _{\Omega } \rho \big (f(x, \omega ), x\big ) P(d \omega ) <\infty \quad \text {for } x \in X, \end{aligned}$$
(4)

then (see [2, Theorem 3.1]) there exists a probability Borel measure \( \pi ^f\) on X such that for every \(x \in X\) the sequence \((\pi _n^f (x,\cdot ))_{n \in {\mathbb {N}}}\) converges weakly to \(\pi ^f\), and (see [11, Corollary 5.6 and Lemma 3.1], also [3, Lemma 2.2]) for every non-expansive \(u:X \rightarrow {\mathbb {R}}\) the inequality

$$\begin{aligned} \left| \int _X u(z)\pi _n^f(x,dz) - \int _X u(z)\pi ^f(dz) \right| \le \frac{\lambda ^n}{1-\lambda } \int _\Omega \rho \big (f(x, \omega ), x\big ) P(d\omega ) \end{aligned}$$
(5)

holds for \(x \in X\) and \(n \in {\mathbb {N}}\).

This limit distribution \( \pi ^f\) plays an important role in solving (1) and (2), see [5, Theorem 3.1], [2, Corollary 4.1], [3, Theorem 2.1]. In particular:

(I) If \(F:X \rightarrow {\mathbb {R}}\) is continuous and bounded, then any continuous and bounded solution \(\varphi :X \rightarrow {\mathbb {R}}\) of (1) has the form

$$\begin{aligned} \begin{aligned} \varphi (x)&= F(x)-\frac{1}{2}\int _XF(z)\pi ^f(dz)\\&\quad +\sum _{n=1}^{\infty }(-1)^n\left( \int _X F(z)\pi _n^f(x,dz)-\int _XF(z)\pi ^f(dz)\right) \quad \text {for } x\in X; \end{aligned} \end{aligned}$$
(6)

if additionally F is Lipschitz, then (6) defines a Lipschitz solution \(\varphi :X \rightarrow {\mathbb {R}}\) of (1).

(II) If \(F:X \rightarrow {\mathbb {R}}\) is continuous and bounded and (2) has a continuous and bounded solution \(\varphi :X \rightarrow {\mathbb {R}}\), then

$$\begin{aligned} \int _XF(x)\pi ^f(dx)=0, \end{aligned}$$
(7)

and any such solution has the form

$$\begin{aligned} \varphi (x)= c+F(x)+\sum _{n=1}^{\infty }\int _XF(z)\pi _n^f(x,dz) \quad \text {for } x\in X \end{aligned}$$

with a real constant c.

(III) If \(F:X \rightarrow {\mathbb {R}}\) is Lipschitz, then it is integrable for \( \pi ^f\) and (2) has a Lipschitz solution \(\varphi :X \rightarrow {\mathbb {R}}\) if and only if (7) holds.

The limit distribution \( \pi ^f\) and facts cited above will be used in the main part of the paper. A characterization of this limit for some special random-valued functions in Hilbert spaces have been given by [3, Theorem 3.1] and, in Banach spaces, by [4, Theorem 2.1].

Actually we do not have a sufficiently satisfactory theorem to guarantee the existence of continuous solutions to the equations considered. An explanation of this situation is given in the paper [9] by Witold Jarczyk (see also [13, Note 3.8.4]). Namely, in the case where \(\Omega \) is a singleton and X is a compact real interval, for the appropriate f the set of continuous \(F:X \rightarrow {\mathbb {R}}\) such that the equation has a continuous solution is small in the sense of Baire category. It is also small from the measure point of view (see [1]). We will go also in this direction but, above all, we are looking for conditions under which Eqs. (1) and (2) have continuous and Hölder continuous solutions \(\varphi :X \rightarrow {\mathbb {R}}\). In the case where \(\Omega \) is a singleton, see [12, Chapter II, §7].

2 Results

We will consider Eqs. (1) and (2) assuming the following hypothesis (H).

(H) \((\Omega ,{{\mathcal {A}}},P)\) is a probability space, \((X,\rho )\) is a complete and separable metric space, \( f: X \times \Omega \rightarrow X \) is \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable, (3) holds with a \( \lambda \in (0,1) \) and (4) is satisfied.

We regard \(\lambda \) as fixed in (0, 1), and for any metric space X we define \({\mathcal {F}}(X)\) as the set of all continuous functions \(F:X \rightarrow {\mathbb {R}}\) such that there are a sequence \((F_n)_{n \in {\mathbb {N}}}\) of real functions on X and constants \(\vartheta \in (0,1)\), \(L \in (0,\frac{1}{\lambda } )\) and \(\alpha , \beta \in (0, \infty )\) such that

$$\begin{aligned} |F(x)-F_n(x)|\le \alpha \vartheta ^n \quad \text {for } x \in X, \ n \in {\mathbb {N}}, \end{aligned}$$

and

$$\begin{aligned} |F_n(x)-F_n(z)|\le \beta L^n\rho (x,z) \quad \text {for } x,z \in X, \ n \in {\mathbb {N}}. \end{aligned}$$

Clearly any real Lipschitz function defined on X belongs to \({\mathcal {F}}(X)\).

Theorem 2.1

Assume (H). If \(F \in {\mathcal {F}}(X)\), then formula (6) defines a continuous solution \(\varphi :X \rightarrow {\mathbb {R}}\) of (1), and if additionally (7) holds, then the formula

$$\begin{aligned} \varphi _0(x)=F(x)+\sum _{n=1}^{\infty }\int _XF(z)\pi _n^f(x,dz) \quad \text {for } x\in X \end{aligned}$$
(8)

defines a continuous solution \(\varphi _0 :X \rightarrow {\mathbb {R}}\) of (2).

The proof will be based on three lemmas. In each of them we assume (H).

Lemma 2.2

If \(F \in {\mathcal {F}}(X)\), then the integrals

$$\begin{aligned} \int _{\Omega }\big |F\big (f(x,\omega )\big )\big |P(d\omega ) \quad \text {for }x \in X, \quad \int _X|F(z)|\pi ^f(dz) \end{aligned}$$

are finite, and the function

$$\begin{aligned} x \mapsto \int _{\Omega }F\big (f(x,\omega )\big )P(d\omega ), \quad x \in X, \end{aligned}$$
(9)

is continuous.

Proof

Corresponding to F choose a sequence \((F_n)_{n \in {\mathbb {N}}}\) of real functions on X and constants \(\vartheta \in (0,1)\), \(L \in (0,\frac{1}{\lambda } )\) and \(\alpha , \beta \in (0, \infty )\) as in the definition of \({\mathcal {F}}(X)\). Then

$$\begin{aligned} \int _{\Omega }\big |F\big (f(x,\omega )\big )\big |P(d\omega )\le \alpha \vartheta +\beta L\int _{\Omega } \rho \big (f(x, \omega ), x\big ) P(d \omega )+|F_1(x)| \end{aligned}$$

for \(x \in X\), and

$$\begin{aligned} \int _X|F(z)|\pi ^f(dz)\le \alpha \vartheta +\int _X|F_1(z)|\pi ^f(dz), \end{aligned}$$

see also (III). Moreover, for every \(n \in {\mathbb {N}}\) the function

$$\begin{aligned} x \mapsto \int _{\Omega }F_n\big (f(x,\omega )\big )P(d\omega ), \quad x \in X, \end{aligned}$$

is Lipschitz:

$$\begin{aligned} \left| \int _{\Omega }F_n\big (f(x,\omega )\big )P(d\omega )-\int _{\Omega }F_n\big (f(z,\omega )\big )P(d\omega )\right| \le \beta L^n\lambda \rho (x,z)\quad \end{aligned}$$

for \(x,z \in X\), and therefore function (9), as their uniform limit, is continuous.

\(\square \)

Lemma 2.3

If \(F \in {\mathcal {F}}(X)\), then

$$\begin{aligned} \int _X|F(z)|\pi _n^f(x,dz)<\infty \quad \text {for }x \in X \ \text {and } n \in {\mathbb {N}}, \end{aligned}$$

and for every \(n \in {\mathbb {N}}\) the function

$$\begin{aligned} x \mapsto \int _XF(z)\pi _n^f(x,dz), \quad x \in X, \end{aligned}$$

is continuous.

Proof

By induction, (3) and (4),

$$\begin{aligned} \int _{\Omega ^\infty }\rho \big (f^n(x, \omega ),f^n(z,\omega )\big ) P^{\infty }(d \omega ) \le \lambda ^n \rho (x, z) \quad \text {for } x, z \in X \ \text {and } n \in {\mathbb {N}}\nonumber \\ \end{aligned}$$
(10)

and

$$\begin{aligned} \int _{\Omega ^\infty }\rho \big (f^n(x, \omega ),x\big ) P^{\infty }(d \omega )< \infty \quad \text {for } x \in X \ \text {and } n \in {\mathbb {N}}. \end{aligned}$$

Since

$$\begin{aligned} \int _XF(z)\pi _n^f(x,dz)=\int _{\Omega ^\infty }F\big (f^n(x, \omega )\big )P^\infty (d \omega ) \quad \text {for } x \in X \ \text {and } n \in {\mathbb {N}}, \end{aligned}$$
(11)

an application of Lemma 2.2 with f replaced by \(f^n, \ n \in {\mathbb {N}},\) finishes the proof.

\(\square \)

Lemma 2.4

If \(F \in {\mathcal {F}}(X)\), then there are constants \(\theta \in (0,1)\) and \(M \in (0,\infty )\) such that

$$\begin{aligned} \begin{aligned}&\left| \int _XF(z)\pi _n^f(x,dz)- \int _XF(z)\pi ^f(dz) \right| \\&\quad \le M\theta ^n\big (1+\rho (x,x_0)+ \int _{\Omega } \rho \big (f(x_0, \omega ), x_0\big ) P(d \omega )\big ) \end{aligned} \end{aligned}$$

for \(x, x_0 \in X\) and \(n \in {\mathbb {N}}\).

Proof

Corresponding to F choose a sequence \((F_n)_{n \in {\mathbb {N}}}\) of real functions on X and constants \(\vartheta \in (0,1)\), \(L \in (0,\frac{1}{\lambda } )\) and \(\alpha , \beta \in (0, \infty )\) as in the definition of \({\mathcal {F}}(X)\), and put

$$\begin{aligned} \theta =\mathrm{max}\{\vartheta ,\lambda L\}, \quad M =2\mathrm{max}\left\{ \alpha ,\frac{\beta }{1-\lambda }\right\} . \end{aligned}$$

Then \(\theta \in (0,1)\), and by Lemmas 2.3 and 2.2, (5) with \(u=\frac{F_n}{\beta L^n}\) and (3) for every \(x, x_0 \in X\) and \(n \in {\mathbb {N}}\) we have

$$\begin{aligned} \begin{aligned}&\left| \int _XF(z)\pi _n^f(x,dz)- \int _XF(z)\pi ^f(dz) \right| \\&\quad \le \left| \int _XF(z)\pi _n^f(x,dz)- \int _XF_n(z)\pi _n^f (x,dz) \right| \\&\qquad +\left| \int _XF_n(z)\pi _n^f(x,dz)-\int _XF_n(z)\pi ^f(dz) \right| \\&\qquad +\left| \int _XF_n(z)\pi ^f(dz)-\int _XF(z)\pi ^f(dz) \right| \\&\quad \le \int _X|F(z)-F_n(z)|\pi _n^f (x,dz) +\beta L^n\frac{\lambda ^n}{1-\lambda }\int _{\Omega } \rho \big (f(x, \omega ), x\big ) P(d \omega ) \\&\qquad +\int _X|F_n(z)-F(z)|\pi ^f(dz) \le 2\alpha \vartheta ^n +\beta L^n\frac{\lambda ^n}{1-\lambda }\int _{\Omega } \rho \big (f(x, \omega ), x\big ) P(d \omega ) \\&\quad \le 2\alpha \theta ^n+\frac{\beta }{1-\lambda }\theta ^n\big (\lambda \rho (x,x_0)+\int _{\Omega } \rho \big (f(x_0, \omega ), x_0\big ) P(d \omega )+\rho (x,x_0)\big ) \\&\quad \le M\theta ^n\big (1+\rho (x,x_0)+ \int _{\Omega } \rho \big (f(x_0, \omega ), x_0\big ) P(d \omega )\big ). \end{aligned} \end{aligned}$$

\(\square \)

Proof of Theorem 2.1

It follows from Lemmas 2.2–2.4 that formula (6) defines a continuous function \(\varphi :X \rightarrow {\mathbb {R}}\) and arguing like in the proof of Theorem 3.1(ii) of [5] (see also the calculations below) we show that it solves (1).

Assume now that also (7) holds. Then it follows from Lemmas 2.3 and 2.4 that formula (8) defines a continuous function \(\varphi _0 :X \rightarrow {\mathbb {R}}\). Applying (11), Lemma 2.4, the Lebesgue dominated convergence theorem and the Fubini theorem we observe that for every \(x\in X\) the function \(\varphi _0 \circ f(x,\cdot )\) is integrable for P and

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\varphi _0\big (f(x,\omega )\big )P(d\omega ) =\int _{\Omega }F\big (f(x,\omega )\big )P(d\omega ) \\&\quad +\int _{\Omega }\sum _{n=1}^\infty \left( \int _X F(z)\pi _n^f\big (f(x,\omega ),dz\big )\right) P(d\omega ) =\int _X F(z)\pi _1^f(x,dz) \\&\quad +\sum _{n=1}^\infty \int _{\Omega }\left( \int _{\Omega ^\infty }F\big (f^n\big (f(x,\omega _1),\omega _2,\omega _3,\ldots \big )\big )P^\infty \big (d(\omega _2,\omega _3,\ldots )\big )\right) P(d\omega _1) \\&=\int _X F(z)\pi _1^f(x,dz)+\sum _{n=1}^\infty \int _{\Omega ^\infty }F\big (f^{n+1}(x,\omega _1,\omega _2,\ldots )\big )P^\infty \big (d(\omega _1,\omega _2,\ldots )\big ) \\&=\int _X F(z)\pi _1^f(x,dz)+\sum _{n=1}^\infty \int _XF(z)\pi _{n+1}^f(x,dz)=\varphi _0(x)-F(x). \end{aligned} \end{aligned}$$

\(\square \)

Proposition 2.5

If F is a real function on a metric space X and

$$\begin{aligned} |F(x)-F(z)|\le \beta \rho (x,z)^\alpha \quad \text {for } x,z \in X \end{aligned}$$
(12)

with some constants \(\alpha \in (0,1), \ \beta \in [0,\infty )\), then \(F \in {\mathcal {F}}(X)\).

Proof

Fix \(L \in (1,\frac{1}{\lambda })\), put

$$\begin{aligned} \vartheta =L^{-\frac{\alpha }{1-\alpha }}, \quad \theta =\vartheta ^ {\frac{1}{\alpha }}, \end{aligned}$$

and for every \(n \in {\mathbb {N}}\) let \(A_n\) be a maximal for inclusion subset of X such that

$$\begin{aligned} \rho (x,z)\ge \theta ^n \quad \text {for every pair of distinct points } x,z \text { of } A_n. \end{aligned}$$

By the maximality,

$$\begin{aligned} X=\bigcup _{z \in A_n}\{x\in X: \rho (x,z)<\theta ^n\} \quad \text {for } n \in {\mathbb {N}}. \end{aligned}$$

If \(n \in {\mathbb {N}}\) and x, z are distinct points of \(A_n\), then by (12),

$$\begin{aligned} |F(x)-F(z)|\le \beta \rho (x,z)^{\alpha -1}\rho (x,z)\le \beta \theta ^{(\alpha -1)n}\rho (x,z)=\beta L^n\rho (x,z). \end{aligned}$$

It follows from this, using Kirszbraun–McShane extension theorem [7, Theorem 6.1.1], that for every \(n \in {\mathbb {N}}\) there exists an \(F_n:X \rightarrow {\mathbb {R}}\) such that

$$\begin{aligned} F_n\mid _{A_n}=F\mid _{A_n} \quad \text {and } \quad |F_n(x)-F_n(z)|\le \beta L^n\rho (x,z) \quad \text {for }x,z \in X. \end{aligned}$$

If \(n \in {\mathbb {N}}\) and \(x \in X\), then there is a \(z \in A_n\) such that \(\rho (x,z)<\theta ^n\), and

$$\begin{aligned} \begin{aligned}&|F(x)-F_n(x)|\le |F(x)-F(z)|+|F_n(z)-F_n(x)|\\&\quad \le \beta \rho (x,z)^\alpha +\beta L^n\rho (x,z) \\&\quad \le \beta \theta ^{\alpha n}+\beta L^n\theta ^n=2\beta \vartheta ^n. \end{aligned} \end{aligned}$$

\(\square \)

Corollary 2.6

Assume (H). If \(F:X \rightarrow {\mathbb {R}}\) satisfies (12) with some constants \(\alpha \in (0,1), \ \beta \in [0,\infty )\), then formula (6) defines a solution \(\varphi :X \rightarrow {\mathbb {R}}\) of (1) such that

$$\begin{aligned} |\varphi (x)-\varphi (z)|\le \frac{\beta }{1-\lambda ^\alpha }\rho (x,z)^\alpha \quad \text {for } x, z \in X, \end{aligned}$$

and if additionally (7) holds, then formula (8) defines a solution \(\varphi _0 :X \rightarrow {\mathbb {R}}\) of (2) such that

$$\begin{aligned} |\varphi _0 (x)-\varphi _0 (z)|\le \frac{\beta }{1-\lambda ^\alpha }\rho (x,z)^\alpha \quad \text {for } x, z \in X. \end{aligned}$$

Proof

By Proposition 2.5 and Theorem 2.1 formula (6) defines a solution \(\varphi :X \rightarrow {\mathbb {R}}\) of (1). Using (6), (11), (12), Jensen’s inequality and (10) for every \(x,z \in X\) we have

$$\begin{aligned} \begin{aligned}&|\varphi (x)-\varphi (z)|\le |F(x)-F(z)| \\&\qquad + \sum _{n=1}^\infty \int _{\Omega ^\infty }\left| F\big (f^n(x,\omega )\big )-F\big (f^n(z,\omega )\big )\right| P^\infty (d\omega ) \\&\quad \le \beta \rho (x,z)^\alpha +\sum _{n=1}^\infty \int _{\Omega ^\infty }\beta \rho \big (f^n(x,\omega ),f^n(z,\omega ) \big )^\alpha P^\infty (d\omega ) \\&\quad \le \beta \rho (x,z)^\alpha +\beta \sum _{n=1}^\infty \left( \int _{\Omega ^\infty } \rho \big (f^n(x,\omega ),f^n(z,\omega ) \big )P^\infty (d\omega )\right) ^\alpha \\&\quad \le \beta \rho (x,z)^\alpha +\beta \sum _{n=1}^\infty \big (\lambda ^n\rho (x,z)\big )^\alpha =\frac{\beta }{1-\lambda ^\alpha }\rho (x,z)^\alpha . \end{aligned} \end{aligned}$$

For the second part we argue similarly. \(\square \)

Regarding the uniqueness of solutions, we have the following theorem.

Theorem 2.7

Assume (H) and let \(F:X \rightarrow {\mathbb {R}}\).

(i) If \(\varphi _1, \varphi _2 \in {\mathcal {F}}(X) \) are solutions of (1), then \(\varphi _1= \varphi _2.\)

(ii) If \(\varphi _1, \varphi _2 \in {\mathcal {F}}(X) \) are solutions of (2), then \(\varphi _1- \varphi _2\) is a constant function.

Proof

Let \(\varphi _1, \varphi _2 \in {\mathcal {F}}(X) \) and put \(\varphi =\varphi _1- \varphi _2\). Then \(\varphi \in {\mathcal {F}}(X) \). Corresponding to \(\varphi \) choose a sequence \((F_n)_{n \in {\mathbb {N}}}\) of real functions on X and constants \(\vartheta \in (0,1)\), \(L \in (0,\frac{1}{\lambda } )\) and \(\alpha , \beta \in (0, \infty )\) as in the definition of \({\mathcal {F}}(X)\).

If \(\varphi _1, \varphi _2\) are solutions of (1), then \(\varphi \) solves (1) with \(F=0\), and, by induction,

$$\begin{aligned} \varphi (x)=(-1)^n\int _{\Omega ^\infty }\varphi \big (f^n(x,\omega )\big )P^\infty (d\omega ) \quad \text {for } x \in X,\ n \in {\mathbb {N}}. \end{aligned}$$

If \(\varphi _1, \varphi _2\) are solutions of (2), then \(\varphi \) solves (2) with \(F=0\), and

$$\begin{aligned} \varphi (x)=\int _{\Omega ^\infty }\varphi \big (f^n(x,\omega )\big )P^\infty (d\omega ) \quad \text {for } x \in X,\ n \in {\mathbb {N}}. \end{aligned}$$

In both cases

$$\begin{aligned} |\varphi (x)-\varphi (z)|\le \int _{\Omega ^\infty }\big |\varphi \big (f^n(x,\omega )\big )-\varphi \big (f^n(z,\omega )\big )\big |P^\infty (d\omega ) \end{aligned}$$

for \(x, z \in X, \ n \in {\mathbb {N}}\). Moreover,

$$\begin{aligned} \begin{aligned} |\varphi (x)-\varphi (z)|&\le |\varphi (x)-F_n(x)|+|F_n(x)-F_n(z)|+|F_n(z)-\varphi (z)| \\&\le 2\alpha \vartheta ^n+|F_n(x)-F_n(z)| \quad \text {for } x, z \in X, \ n \in {\mathbb {N}}. \end{aligned} \end{aligned}$$

Consequently, applying among others (10),

$$\begin{aligned} \begin{aligned} |\varphi (x)-\varphi (z)|&\le 2\alpha \vartheta ^n+\int _{\Omega ^\infty }\big |F_n\big (f^n(x,\omega )\big )-F_n\big (f^n(z,\omega )\big )\big |P^\infty (d\omega )\\&\le \beta L^n\lambda ^n\rho (x,z) \quad \text {for } x, z \in X,\ n \in {\mathbb {N}}, \end{aligned} \end{aligned}$$

whence \(\varphi (x)=\varphi (z)\) for \(x, z \in X\), i.e., \(\varphi \) is a constant function. Noting that if a constant \(\varphi \) solves (1) with \(F=0\), then \(\varphi =0\), we end the proof. \(\square \)

We finish with a qualitative result.

Following [6] by Jens Peter Reus Christensen we say that a Borel subset B of an abelian Polish group G is a Haar zero set if there is a probability Borel measure \(\mu \) on G such that \(\mu (B+x)=0\) for every \(x \in G\). See also [8] where measurability in abelian Polish groups related to Christensen’s Haar zero set is studied.

Assume

(H\(_0\)) \((\Omega ,{{\mathcal {A}}},P)\) is a probability space, \((X,\rho )\) is a compact metric space, \( f: X \times \Omega \rightarrow X \) is \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable and (3) holds with a \( \lambda \in (0,1) \).

Assuming (H\(_0\)) we have in particular (4):

$$\begin{aligned} \int _{\Omega } \rho \big (f(x, \omega ), x\big ) P(d \omega ) \le \mathrm{diam}(X) \quad \text {for } x \in X. \end{aligned}$$

Moreover one can consider the Banach space C(X) of all continuous real functions on X with the uniform norm and its subspace \(C_f\),

$$\begin{aligned} C_f = \left\{ F \in C(X): \int _XF(x)\pi ^f(dx)=0\right\} . \end{aligned}$$

Clearly \(C_f\) is a closed linear subspace of C(X) and (see, e.g., [7, Corollary 11.2.5]) C(X) is a separable Banach space. We have also the following lemma.

Lemma 2.8

Assume (H\(_0\)). If \(F :X \rightarrow {\mathbb {R}}\) is continuous, then so is the function

$$\begin{aligned} x \mapsto \int _{\Omega }F\big (f(x,\omega )\big )P(d\omega ), \quad x \in X. \end{aligned}$$

Proof

Fix \(\varepsilon \in (0,\infty )\) and choose \(\delta \in (0,\infty )\) such that

$$\begin{aligned} |F(x)-F(z)|\le \varepsilon \quad \text {for } x,z \in X \ \text {with } \rho (x,z)\le \delta . \end{aligned}$$

Then, by (3), for all \(x,z \in X\),

$$\begin{aligned} \begin{aligned}&\left| \int _{\Omega }F\big (f(x,\omega )\big )P(d\omega )-\int _{\Omega }F\big (f(z,\omega )\big )P(d\omega )\right| \\&\quad \le \int _{\Omega }\big |F\big (f(x,\omega )\big )-F\big (f(z,\omega )\big )\big |P(d\omega )\\&\quad \le \varepsilon +\int _{\{\omega \in \Omega : \rho (f(x,\omega ),f(z,\omega ))>\delta \}}\big |F\big (f(x,\omega )\big )-F\big (f(z,\omega )\big )\big |P(d\omega )\\&\quad \le \varepsilon +2\Vert F\Vert P\big (\{\omega \in \Omega : \rho \big (f(x,\omega ),f(z,\omega )\big )>\delta \} \big ) \\&\quad \le \varepsilon +2\Vert F \Vert \frac{1}{\delta }\int _{\Omega } \rho \big (f(x, \omega ), f(z, \omega )\big ) P(d\omega ) \le \varepsilon + \frac{2\lambda \Vert F \Vert }{\delta }\rho (x,z), \end{aligned} \end{aligned}$$

and therefore the discussed function is continuous. \(\square \)

Let

$$\begin{aligned} {\mathcal {F}}_1= & {} \{F \in C(X): \text { equation }(1) \text { has a continuous solution } \varphi :X \rightarrow {\mathbb {R}} \},\\ {\mathcal {F}}_2= & {} \{F \in C_f: \text { equation }(2) \text { has a continuous solution } \varphi :X \rightarrow {\mathbb {R}} \}. \end{aligned}$$

Theorem 2.9

Under the assumptions (H\(_0\)):

(i) \({\mathcal {F}}_1\) is a Borel and dense subset of C(X), and if \({\mathcal {F}}_1 \not = C(X)\), then \({\mathcal {F}}_1\) is of first category in C(X) and a Haar zero subset of C(X).

(ii) \({\mathcal {F}}_2\) is a Borel and dense subset of \(C_f\), and if \({\mathcal {F}}_2 \not = C_f\), then \({\mathcal {F}}_2\) is of first category in \(C_f\) and a Haar zero subset of \(C_f\).

Proof

By Lemma 2.8 the formulas

$$\begin{aligned} T_1(\varphi )(x)= & {} \varphi (x) +\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ T_2(\varphi )(x)= & {} \varphi (x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ) \quad \text {for } \varphi \in C(X) \text { and } x \in X, \end{aligned}$$

define self-mappings \(T_1, T_2\) of C(X). Clearly, these operators are linear and continuous. Moreover,

$$\begin{aligned} T_1(C(X))={\mathcal {F}}_1. \end{aligned}$$

Furthermore, for every \(F \in T_2(C(X))\) Eq. (2) has a continuous solution \(\varphi :X \rightarrow {\mathbb {R}}\). Hence (II) gives \(T_2(C(X)) \subset C_f\), and

$$\begin{aligned} T_2(C(X))= {\mathcal {F}}_2. \end{aligned}$$

Applying now [1, Lemma] we see that \({\mathcal {F}}_1\) is a Borel subset of C(X), and if \({\mathcal {F}}_1 \not = C(X)\), then \({\mathcal {F}}_1\) is of first category in C(X) and a Haar zero subset of C(X), and \({\mathcal {F}}_2\) is a Borel subset of \(C_f\), and if \({\mathcal {F}}_2 \not = C_f\), then \({\mathcal {F}}_2\) is of first category in \(C_f\) and a Haar zero subset of \(C_f\).

Since by (I) the set

$$\begin{aligned} \{F \in C(X): F \text { is Lipschitz} \} \end{aligned}$$

is contained in \({\mathcal {F}}_1\) and (see [7, Theorem 11.2.4]) dense in C(X), the set \({\mathcal {F}}_1\) is dense in C(X).

To show that \({\mathcal {F}}_2\) is dense in \(C_f\), fix \(F \in C_f\) and \(\varepsilon \in (0,\infty )\). Choose a Lipschitz \(F_1: X \rightarrow {\mathbb {R}}\) so that \(\Vert F-F_1\Vert <\frac{\varepsilon }{2}\). According to (III), \(F_1-\int _XF_1d\pi ^f \in {\mathcal {F}}_2\). Moreover,

$$\begin{aligned} \left\| F-\big (F_1-\int _XF_1d\pi ^f\big )\right\| \le \big \Vert F-F_1\big \Vert +\left| \int _XF_1d\pi ^f-\int _XFd\pi ^f\right| <\varepsilon . \end{aligned}$$

\(\square \)

Remark 2.10

It is possible that (H\(_0\)) holds and \({\mathcal {F}}_1 = C(X), \ {\mathcal {F}}_2 = C_f\).

To see it consider an \({\mathcal {A}} \)-measurable \(\xi :\Omega \rightarrow X\) and let \(f(x,\omega )=\xi (\omega )\) for \((x,\omega ) \in X \times \Omega \). Then \(f^n(x,\omega )=\xi (\omega _n)\) for \((x,\omega ) \in X \times \Omega ^{\infty } \), so \(\pi _n^f(x,B)=P(\xi \in B)=\pi ^f(B)\) for \(n \in {\mathbb {N}}, \ x \in X, \ B \in {\mathcal {B}} \), and \(\int _XFd\pi ^f=\int _\Omega F \circ \xi dP\) for \(F \in C(X)\). Consequently for every \(F \in C(X)\) the function \(F-\frac{1}{2}\int _XFd\pi ^f\) solves (1), and every \(F \in C_f\) solves (2).