Assume 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 .

中文翻译：

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中极限在迭代函数方程中的应用

假设 $$(\Omega ,{\mathcal {A}},P)$$ ( Ω , A , P ) 是一个概率空间， $$f:[0,1]\times \Omega \rightarrow [0, 1]$$ f : [ 0 , 1 ] × Ω → [ 0 , 1 ] 是一个函数使得 $$f(0,\omega )=0$$ f ( 0 , ω ) = 0 , $$f( 1,\omega )=1$$ f ( 1 , ω ) = 1 对于每个 $$\omega \in \Omega $$ ω ∈ Ω , $$g:[0,1]\rightarrow \mathbb R$$ g : [ 0 , 1 ] → R 是一个有界函数，使得 $$g(0)=g(1)=0$$ g ( 0 ) = g ( 1 ) = 0 ，并且 $$a,b\in \ mathbb R$$ a , b ∈ R 。应用中间极限，我们描述了有界解 $$\varphi :[0,1]\rightarrow \mathbb R$$ φ : [ 0 , 1 ] → R 的方程 $$\begin{aligned} \varphi (x)=\ int _{\Omega }\varphi (f(x,\omega ))dP(\omega )+g(x) \end{aligned}$$ φ ( x ) = ∫ Ω φ ( f ( x , ω ) ) d P ( ω ) + g ( x ) 满足边界条件 $$\varphi (0)=a$$ φ ( 0 ) = a 和 $$\varphi (1)=b$$ φ ( 1 ) = b 。