Fixed point theorems in finite discrete spaces

The continuity and compactness properties do not directly characterize finite adjacency structures, making the problems of fixed point analysis in finite discrete spaces challenging.This paper puts forward an implementable framework for fixed point existence and convergence.An adjacency-preserving mapping, a discrete contraction condition, a distance matrix and an iterative verification method are proposed to develop a finite discrete fixed point verification (FDFPV) method.Experiments were performed on randomly generated admissible mappings on graph based discrete spaces with vertices 25, 50, 100, 250 and 500 vertices.All but 5.4% of the mappings had a fixed point found by FDFPV, and 93.2% of them converged after 10 iterations, with an average of 6.4 iterations.Computational assessments are 28.7% less than exhaustive verification.The novelty is an algorithmically testable fixed-point criterion, based on an amalgamation of metric and adjacency properties.The suggested framework provides a verifiable algorithm that is easily reproducible, and it is shown that it can be applied to the fields of digital topology and discrete dynamical systems.