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.

令\（（\ Omega，\ mathcal {F}，\ mathbb {P}）\）为一个概率空间，并令\（\ alpha，\ beta：\ mathcal {F} \ rightarrow〜\ mathbb {R} \）是随机变量。我们提供了足够的条件，在该条件下方程\（\ varphi（x）= \ int _ {\ Omega} \ varphi \的每个有界连续解\（\ varphi：\ mathbb {R} \ rightarrow \ mathbb {R} \）left（\ alpha（\ omega）（x- \ beta（\ omega））\ right）\ mathbb {P}（d \ omega）\）是恒定的。我们还表明，上述方程式的任何非常数有界连续解都必须在无穷大处振动。