Abstract
AbstractBased on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP.
Highlights
It is well known that Schauder’s fixed point theorem [ ] implies that a nonempty compact convex subset X of a Banach space has a fixed point for any continuous self-map of X
Unlike the difference between contractibility and local contractibility in classical mathematics, the present paper proves that their digital versions have their own features
Developing the notion of K -homotopy in the category of Khalimsky topological spaces, we have developed the notions of contractibility and local contractibility induced by the
Summary
It is well known that Schauder’s fixed point theorem [ ] implies that a nonempty compact convex subset X of a Banach space has a fixed point for any continuous self-map of X. ) as follows: let X be a K -topological space with K -contractibility It has the FPP for K-continuous mappings. ), the present paper proves that K -contractibility of a finite K -topological space need not imply the existence of fixed points of K -continuous maps To classify K -topological spaces in terms of a certain homotopy equivalence in KTC, we use the following: Definition In KTC, for two spaces (X, κXn ) := X and (Y , κYn ) := Y , if there are K -continuous maps h : X → Y and l : Y → X such that l ◦ h is K -homotopic to X and h ◦ l is K -homotopic to Y , the map h : X → Y is called a K -homotopy equivalence, denoted X K·h·e Y. Proof The process presented in Figures (a) and (b) explains the following K -contractibility of
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