Fixed point property for digital spaces

The paper compares the fixed point property (FPP for short) of a compact Euclidean plane with its digital versions associated with Khalimsky and Marcus-Wyse topology. More precisely, by using a Khalimsky and a Marcus-Wyse topological digitization, the paper studies digital versions of the FPP for Euclidean topological spaces. Besides, motivated by the digital homotopy fixed point property (DHFP for brevity) [O. Ege, I. Karaca, C. R. Math. Acad. Sci. Paris, 353 (2015), 1029–1033], the present paper establishes the digital homotopy almost fixed point property (DHAFP for short) which is more generalized than the DHFP. Moreover, the present paper corrects some errors in [O. Ege, I. Karaca, C. R. Math. Acad. Sci. Paris, 353 (2015), 1029–1033] and improves it.