Some properties of the mapping Tμ introduced by a representation in Banach and locally convex spaces

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Let ={Tₛ:s∈ S} be a representation of a semigroup S. First, we prove that the mapping Tμ introduced by a mean on a subspace of l∞(S) has many properties of the mappings in the representation, 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 Tμ is a Q-G-nonexpansive mapping if Tₛ is a Q-G-nonexpansive mapping for each s∈ S. Then we define Q-G-attractive point of and show if a point a is a Q-G-attractive point of then a is a Q-G-attractive point of Tμ.