Abstract
AbstractLet $$(\Omega , \mathcal {F}, \mathbb {P})$$
(
Ω
,
F
,
P
)
be a probability space and let $$\alpha , \beta : \mathcal {F} \rightarrow ~\mathbb {R}$$
α
,
β
:
F
→
R
be random variables. We provide sufficient conditions under which every bounded continuous solution $$\varphi : \mathbb {R} \rightarrow \mathbb {R}$$
φ
:
R
→
R
of the equation $$ \varphi (x) = \int _{ \Omega } \varphi \left( \alpha (\omega ) (x-\beta (\omega ))\right) \mathbb {P}(d\omega )$$
φ
(
x
)
=
∫
Ω
φ
α
(
ω
)
(
x
-
β
(
ω
)
)
P
(
d
ω
)
is constant. We also show that any non-constant bounded continuous solution of the above equation has to be oscillating at infinity.
Funder
University of Zielona Gora
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
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