A Large Class of Nonweakly Compact Closed Bounded and Convex Sets with Fixed Point Property for Affine Nonexpansive Mappings in c0 When It Is Renormed.

In 2011, Lennard and Nezir showed that very large class of closed bounded convex sets in c₀ fails the fixed point property for affine nonexpansive mappings respect to c₀'s usual norm since they proved that closed convex hull of any asymptotically isometric (ai) c₀-summing basis fails the fixed point property for nonexpansive mappings and in fact their class is one of these. Then, Nezir recently worked on these sets and constructed several equivalent norms. In one of his works, he defined the equivalent norm ΙΙΙ·ΙΙΙ on c₀ by ... for all x ∈ c₀. Then, he studied a subclass of the class S below introduced by Lennard and Nezir and showed that it has the fixed point property for affine ΙΙΙ·ΙΙΙ-nonexpansive mappings for some α > 1 when Q₁ > 1-γ1+ΙαΙ/1+2ΙαΙ ... In this paper, we will show that the below larger class G given by Lennard and Nezir that contains S has the fixed point property for affine ΙΙΙ·ΙΙΙ-nonexpansive mappings for all α > 1 when Q₁ > 1-γ1+ΙαΙ/1+2ΙαΙ ... Moreover and most importantly, we generalize our results for the closed convex hull of the sequence η_η = γ_η (b₁e₁ + b₂e₂ + ... + bnen) when 0 < b_n and 0 γn are arbitrarily choosen. [ABSTRACT FROM AUTHOR]