#### Document Type

Article

#### Publication Date

2-26-2006

#### Publication Title

Fixed Point Theory and Applications

#### Department

Department of Mathematics

#### Abstract

Let X be an H-space of the homotopy type of a connected, finite CW-complex, f : X→X any map and p_{k }: X→X the k^{th} power map. Duan proved that p_{k}f : X → X has a fixed point if k ≥ 2. We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces X whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a θ-structure μ_{θ} : X → X as defined by Hemmi-Morisugi-Ooshima. The conclusion is that μ_{θ}f and f_{μθ} each has a fixed point.

#### DOI

10.1155/FPTA/2006/17563

#### Original Citation

Arkowitz, M. Duan's fixed point theorem: Proof and generalization. Fixed Point Theory Appl 2006, 17563 (2006). https://doi.org/10.1155/FPTA/2006/17563

#### Dartmouth Digital Commons Citation

Arkowitz, Martin, "Duan's Fixed Point Theorem: Proof and Generalization" (2006). *Dartmouth Scholarship*. 2419.

https://digitalcommons.dartmouth.edu/facoa/2419