Abstract
AbstractAssume that $$(\Omega ,{\mathcal {A}},P)$$
(
Ω
,
A
,
P
)
is a probability space, $$f:[0,1]\times \Omega \rightarrow [0,1]$$
f
:
[
0
,
1
]
×
Ω
→
[
0
,
1
]
is a function such that $$f(0,\omega )=0$$
f
(
0
,
ω
)
=
0
, $$f(1,\omega )=1$$
f
(
1
,
ω
)
=
1
for every $$\omega \in \Omega $$
ω
∈
Ω
, $$g:[0,1]\rightarrow \mathbb R$$
g
:
[
0
,
1
]
→
R
is a bounded function such that $$g(0)=g(1)=0$$
g
(
0
)
=
g
(
1
)
=
0
, and $$a,b\in \mathbb R$$
a
,
b
∈
R
. Applying medial limits we describe bounded solutions $$\varphi :[0,1]\rightarrow \mathbb R$$
φ
:
[
0
,
1
]
→
R
of the equation $$\begin{aligned} \varphi (x)=\int _{\Omega }\varphi (f(x,\omega ))dP(\omega )+g(x) \end{aligned}$$
φ
(
x
)
=
∫
Ω
φ
(
f
(
x
,
ω
)
)
d
P
(
ω
)
+
g
(
x
)
satisfying the boundary conditions $$\varphi (0)=a$$
φ
(
0
)
=
a
and $$\varphi (1)=b$$
φ
(
1
)
=
b
.
Funder
University of Silesia in Katowice
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
Reference31 articles.
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5. Baron, Karol, Kuczma, Marek: Iteration of random-valued functions on the unit interval. Colloq. Math. 37(2), 263–269 (1977)
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