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Let $$(\Omega , \mathcal {F}, \mathbb {P})$$ be a probability space and let $$\alpha , \beta : \mathcal {F} \rightarrow ~\mathbb {R}$$ be random variables. We provide sufficient conditions under… Click to show full abstract
Let $$(\Omega , \mathcal {F}, \mathbb {P})$$ be a probability space and let $$\alpha , \beta : \mathcal {F} \rightarrow ~\mathbb {R}$$ be random variables. We provide sufficient conditions under which every bounded continuous solution $$\varphi : \mathbb {R} \rightarrow \mathbb {R}$$ of the equation $$ \varphi (x) = \int _{ \Omega } \varphi \left( \alpha (\omega ) (x-\beta (\omega ))\right) \mathbb {P}(d\omega )$$ is constant. We also show that any non-constant bounded continuous solution of the above equation has to be oscillating at infinity.
Journal Title: Aequationes Mathematicae
Year Published: 2020
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