In the present paper, we introduce two classes \({\mathcal {L}}^+\) and \({\mathcal {L}}^-\) of nonlinear stochastic operators acting on the simplex of \(\ell ^1\)-space. For each operator V from these classes, we study omega limiting sets \(\omega _V\) and \(\omega _V^{(w)}\) with respect to \(\ell ^1\)-norm and pointwise convergence, respectively. As a consequence of the investigation, we establish that every operator from the introduced classes is weak ergodic. However, if V belongs to \({{\mathcal {L}}}^-\), then it is not ergodic (w.r.t \(\ell ^1\)-norm) while V is weak ergodic.

中文翻译：

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无限维非线性随机算子的遍历性

在本文中，我们介绍了作用在\（\ ell ^的单纯形上的非线性随机算子的两个类\（{\ mathcal {L}} ^ + \）和\（{\ mathcal {L}} ^-\）1 \） -空间。对于这些类别中的每个算子V，我们针对\（\ ell ^ 1 \）-范数和逐点收敛性研究omega极限集\（\ omega _V \）和\（\ omega _V ^ {（w）} \）， 分别。作为调查的结果，我们确定所介绍类别中的每个运算符都是弱遍历的。但是，如果V属于\（{{\ mathcal {L}}} ^-\），则在V时它不是遍历的（wrt \（\ ell ^ 1 \）- norm） 弱遍历。