### A class of ${I}_{0}$-sets

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Let $G$ be a locally compact group. We continue our work [A. Ghaffari: $\Gamma $-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of $\Gamma $-amenability of a locally compact group $G$ defined with respect to a closed subgroup $\Gamma $ of $G\times G$. In this paper, among other things, we introduce and study a closed subspace ${A}_{\Gamma}^{p}\left(G\right)$ of ${L}^{\infty}\left(\Gamma \right)$ and then characterize the $\Gamma $-amenability of $G$ using ${A}_{\Gamma}^{p}\left(G\right)$. Various necessary and sufficient conditions are found for a locally compact group to possess...

In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case...

Let $N$ and $K$ be groups and let $G$ be an extension of $N$ by $K$. Given a property $\mathcal{P}$ of group compactifications, one can ask whether there exist compactifications ${N}^{\text{'}}$ and ${K}^{\text{'}}$ of $N$ and $K$ such that the universal $\mathcal{P}$-compactification of $G$ is canonically isomorphic to an extension of ${N}^{\text{'}}$ by ${K}^{\text{'}}$. We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties $\mathcal{P}$ and then apply this result to the almost periodic and weakly almost periodic compactifications of $G$.

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set $X$ in a metric space, it is proved the existence of an almost periodic sequence ${\left\{{\psi}_{k}\right\}}_{k\in \mathbb{Z}}$ such that $\{{\psi}_{k};\phantom{\rule{0.166667em}{0ex}}k\in \mathbb{Z}\}=X$ and ${\psi}_{k}={\psi}_{k+lq\left(k\right)}$, $l\in \mathbb{Z}$ for all $k$ and some $q\left(k\right)\in \mathbb{N}$ which depends on $k$.

This paper generalizes earlier author's results where the linear and quasilinear equations with constant coefficients were treated. Here the method of limit passages and a fixed-point theorem is used for the linear and quasilinear equations with almost periodic coefficients.