Fixed Point Theorems for Digital Images

Abstract

Abstract. In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

“…Using a method similar to the AFPP in DTC in Remark 3, we can refer to the AFPP for simple closed curves under Kor Marcus-Wyse topology, as follows: Remark 4. In the category of K-or Marcus-Wyse topological spaces, every simple closed curve with l element does not have the AFPP (see [9,[29][30][31]).…”

Fixed point property for digital spaces

“…It is well known that only a singleton has the FPP [18]. Although the authors of [3] studied the DHFP in terms of a graph-theoretical approach, the work can be simplified as one statement (see Proposition 1.1 below) because it is wellknown that a digital image (X, k) does not have the FPP with |X| 2 (see Theorems 3.3 and 4.1 of [18] and the papers [6,7]). Namely, only a singleton has the FPP and further, the DHFP of (X, k) requires its digital fixed point property (DFP for short) (see Corollary 3.3 of [3]).…”

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

“…Remark 1. It is obvious that SC n,l K [4], SC 2,l M [7] and SC n,l k [3] do not have the AFPP in the categories KTC, MTC and DTC, respectively. For instance, for SC n,l K := (x i ) i∈[0,l−1] Z , consider a self-map of SC n,l K such that f (x i ) = x i+2(mod l) .…”