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

Abstract

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 ||| · ||| on c0 by |||x|||:=1γ1limp→∞supk∈ℕγk(∑j=k∞|ξj|pj)1p+γ1supj∈ℕ∑k=1∞Qk|ξ*k−αξ*j|where γ2=γ1,γk↑k1,γk+1<γk+2,∀k∈ℕ, x*:=(ξ*j)j∈ℕisthedecreasingrearrangementofx,∑k=1∞Qk=1,Qk↓k0andQk>Qk+1,∀k∈ℕ 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 ||| · |||-nonexpansive mappings for some α > 1 when Q1>1−γ1+2|α|1+2|α|. S := { Eb⊂c0:Eb={ ∑n=1∞αn∑k=1nfk:∀αn≥0 and ∑n=1∞αn=1 } where b∈(0,1), f1=be1,f2=be2,fn=en for all n≥3 }. 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 ||| · |||-nonexpansive mappings for all α > 1 when 1−γ1+2|α|1+2|α|. G := { E⊂c0:E={ ∑n=1∞αn∑k=1nbkek:∀αn≥0 and ∑n=1∞αn=1 } where (bn)n∈ℕ∈ℝ with 0<m:=infn∈ℕbn and M:=supn∈ℕbn<∞ }. Moreover and most importantly, we generalize our results for the closed convex hull of the sequence ηn:=γn(b1e1+b2e2+…+bnen) when 0 < bn and 0 < γn are arbitrarily choosen.