Fixed points of asymptotically nonexpansive type mappings

In this paper, we prove that if K is a nonempty weakly compact convex subset of a UREk Banach space X having the asymptotic (P) and T : K -> K is an asymptotically nonexpansive type mapping, then T has a fixed point in K. This result improves a result by Kim and Xu (Nonlinear Anal Ser A: Theory Methods 41(3-4):405-415, 2000). We also prove the existence of a fixed point for pointwise eventually nonexpansive self-mappings defined on a nonempty weakly compact convex subset K of a UREk Banach space X having the property (P). This result answers a question raised by Dominguez Benavides and Lorenzo Ramirez (Fixed points for mappings of asymptotically nonexpansive type. Preprint. arXiv:2201.02802 [v1]).