Some properties of the mapping $T_{\mu}$ introduced by a representation in Banach and locally convex spaces

TL;DR

This paper explores properties of a specific mapping $T_{\mu}$ derived from semigroup representations in Banach and locally convex spaces, establishing its nonexpansiveness and attractive points under certain conditions.

Contribution

It introduces and analyzes the properties of the mapping $T_{\mu}$ in Banach and locally convex spaces, linking it to semigroup representations and nonexpansive mappings.

Findings

$T_{\mu}$ shares properties with the representation $\mathcal{T}$ in Banach spaces.

$T_{\mu}$ is shown to be $Q$-$G$-nonexpansive in locally convex spaces.

Existence of $Q$-$G$-attractive points for $T_{\mu}$ under certain conditions.

Abstract

Let $ \sc=\{T_{s}:s\in S\} $ be a representation of a semigroup $S$. First, we prove that the mapping $T_{\mu}$ introduced by a mean on a subspace of $l^{\infty}(S)$ has many properties of the mappings in the representation $ \sc$, in Banach spaces. Then we consider a directed graph and then we define a $Q$-$G$-nonexpansive mapping in locally convex spaces and show that $T_{\mu}$ is a $Q$-$G$-nonexpansive mapping if $T_{s}$ is a $Q$-$G$-nonexpansive mapping for each $s\in S$. Then we define $Q$-$G$-attractive point of $ \sc$ and show if a point $a$ is a $Q$-$G$-attractive point of $ \sc$ then $a$ is a $Q$-$G$-attractive point of $T_{\mu}$.