Estimation of the complexity of a digital image from the viewpoint of fixed point theory

摘要

The present paper introduces and estimates the complexity of the fixed point property of a digital image (X, k) for any k−DC-self-map f of (X, k), where a k−DC-self-map f of (X, k) means a digitally k-continuous self-map of (X, k) with a digital version of the Banach contraction principle. To do this work, we need to study some properties of iterations of a k−DC-self-map f of (X, k) and to establish the notion of complexity of (X, k) denote by C♯(X, k) (see Definition 7 in the present paper). According to C♯(X, k), we can estimate complexity of the fixed point property of (X, k) for any k−DC-self-map f of (X, k). Based on this approach, the present paper investigates some relationships between the k-adjacency of (X, k) and C♯(X, k). Furthermore, we prove that C♯(X, k) is not a digital topological invariant. Besides, we develop the notions of uniform k-connectedness and strict k-connectivity to calculate C♯(X, k) for some digital images (X, k). In the paper each (X, k) is assumed to be a k-connected and non-empty set and 2 ≤ | X |≨∞, where | X | means the cardinal number of the given set X.