Bi-commutative digital contraction mapping and fixed point theorem on digital image and metric spaces

Abstract

The contraction mapping theorem is a powerful and most useful method in constructing the solution of linear and nonlinear systems. The principle of Banach fixed point guarantees that a self-contraction mapping of a complete metric space has a unique fixed point which can be obtain using the limit of iteration technique defined by repeated images under mapping of an arbitrary starting point in space. In this paper, we propose a Bi-commutative (II-commuting) digital maps for Kannan and Reich contraction mapping theorem and analyze the existence and uniqueness of the fixed point in the framework of digital metric spaces. We applied the contraction and commutative maps methods in proving our results. The results obtained satisfies the existence and uniqueness condition of the Banach contraction principle for a digital contraction mapping in digital metric space.