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

Shravan Luckraz

doi:10.1186/1687-1812-2014-129

Shravan Luckraz

Open Access

https://doi.org/10.1186/1687-1812-2014-129

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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.

Highlights

Fixed point theorems play an important part in equilibrium analysis in mathematical sciences
This note looks at one crucial assumption, namely that the mapping is isotone and shows that this assumption is related to another fixed point theorem of Abian ( ) [ ] for finite sets
4 Conclusion We have shown in Theorem that Abian’s theorem relates to the idea of cycles and in Lemma we have demonstrated that for finite maps acyclicity is implied by the isotone assumption as long each element relates to its image via the binary relation

Summary

Introduction

Fixed point theorems play an important part in equilibrium analysis in mathematical sciences. This note looks at one crucial assumption, namely that the mapping is isotone and shows that this assumption is related to another fixed point theorem of Abian ( ) [ ] for finite sets. Lemma Let D be a finite set that is partially ordered by and let f : D → D be an isotone mapping, that is, for any x, y ∈ D, x y implies f (x) f (y).

Results

Conclusion