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

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. MSC: 46B42; 47H10; 58J20; 91A06; 91A10

On a Fixed Point Theorem for General Multivalued Mappings on Finite Sets with Applications in Game Theory

“…It was showed by [33] that when the periodicity of f is even, then two sets suffice in the statement of Abian's theorem. We now consider the above statement when the mapping is generalized to a set-valued map.…”

“…eorem 2 shows that the closed graph condition relates the acyclicity of the mapping which in turn relates to Abian's fixed point theorem ( [33]). Since from the standpoint of directed graphs the acyclicity condition may be easier to verify, eorem 2 can be useful in the development of graph theoretic fixed point search algorithms.…”

“…[2] also observes that Abian's partitioning method does not work for multivalued maps. In this paper, we solve this problem by using an equivalent xed point result due to the author [33] who relates the idea of cycle to Abian's fixed point theorem. By representing the fixed point problem on a directed graph, we make use of [33]'s idea to generalize Abian's theorem by first defining a set of all infinite walks of the mapping and second, by providing the necessary and sufficient conditions that these walks need to satisfy so that the mapping has a fixed point.…”

“…In this paper, we solve this problem by using an equivalent xed point result due to the author [33] who relates the idea of cycle to Abian's fixed point theorem. By representing the fixed point problem on a directed graph, we make use of [33]'s idea to generalize Abian's theorem by first defining a set of all infinite walks of the mapping and second, by providing the necessary and sufficient conditions that these walks need to satisfy so that the mapping has a fixed point. In its most general form, the fixed point theorem we propose shows that a fixed point exists if and only if at least one of these walks converges in the Cauchy sense.…”