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Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Korea
Mathematics 2019, 7(12), 1244; https://doi.org/10.3390/math7121244
Received: 12 November 2019 / Revised: 4 December 2019 / Accepted: 8 December 2019 / Published: 16 December 2019
(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)
The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al. View Full-Text
Keywords: fixed point property; almost (approximate) fixed point property; digital surface; digital connected sum; geometric realization; Euler characteristic; minimal (3n − 1)-neighborhood; digital topology fixed point property; almost (approximate) fixed point property; digital surface; digital connected sum; geometric realization; Euler characteristic; minimal (3n − 1)-neighborhood; digital topology
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Han, S.-E. Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces. Mathematics 2019, 7, 1244.

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