Results in Mathematics ( IF 2.2 ) Pub Date : 2021-02-22 , DOI: 10.1007/s00025-021-01354-0 Janusz Morawiec
Applying medial limits we describe bounded solutions \(\varphi :S\rightarrow {\mathbb {R}}\) of the functional equation
$$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi (f(x,\omega ))d\mu (\omega )+G(x), \end{aligned}$$where \((\Omega ,{\mathcal {A}},\mu )\) is a measure space, \(S\subset \mathbb R\), \(f:S\times \Omega \rightarrow S\), \(g:\Omega \rightarrow {\mathbb {R}}\) is integrable and \(G:S\rightarrow {\mathbb {R}}\) is bounded. The main purpose of this paper is to extend results obtained in Morawiec (Results Math 75(3):102, 2020) to the above general functional equation in wider classes of functions and under weaker assumptions.
中文翻译：
中间极限在迭代函数方程中的应用，II
应用中间极限，我们描述了函数方程的有界解\（\ varphi：S \ rightarrow {\ mathbb {R}} \）
$$ \ begin {aligned} \ varphi（x）= \ int _ {\ Omega} g（\ omega）\ varphi（f（x，\ omega））d \ mu（\ omega）+ G（x），\结束{aligned} $$其中\（（\ Omega，{\ mathcal {A}}，\ mu）\）是一个度量空间，\（S \ subset \ mathbb R \），\（f：S \ times \ Omega \ rightarrow S \），\（g：\ Omega \ rightarrow {\ mathbb {R}} \）是可积的，并且\（G：S \ rightarrow {\ mathbb {R}} \）是有界的。本文的主要目的是将Morawiec（Results Math 75（3）：102，2020）中获得的结果扩展到上述广义泛函方程中，并用较宽泛的函数类别和较弱的假设得出。
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