A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem

Luckraz, Shravan

Abstract

In a recent paper Xie et al. (Fixed Point Theory Appl. 2013:192, 2013) gave several extensions and some applications of the Abian-Brown (AB) fixed point theorem. While the AB fixed point theorem and its extensions (as well as other related fixed point theorems) assume that the mapping is isotone, this note shows that for single-valued finite maps this condition relates to the acyclicity of the map, which in turn relates to Abian’s (Nieuw Arch. Wiskd. XVI:184-185, 1968) most basic fixed point theorem for finite sets.

Citation

Luckraz, S. (2014). A note on the relationship between the isotone assumption of the Abian-Brown fixed point theorem and Abian’s most basic fixed point theorem. Fixed Point Theory and Applications, 2014(1), https://doi.org/10.1186/1687-1812-2014-129

• Journal Article Type: Article
• Acceptance Date: May 22, 2014
• Publication Date: May 29, 2014
• Deposit Date: Oct 17, 2017
• Publicly Available Date: Oct 17, 2017
• Journal: Fixed Point Theory and Applications
• Electronic ISSN: 1687-1812
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 2014
• Issue: 1
• DOI: https://doi.org/10.1186/1687-1812-2014-129
• Keywords: Isotone assumption; Fixed point theorem
• Public URL: https://nottingham-repository.worktribe.com/output/728078
• Publisher URL: https://doi.org/10.1186/1687-1812-2014-129