Banach fixed point theorem from the viewpoint of digital topology

Abstract

The present paper studies the Banach contraction principle for digital metric spaces such as digital intervals, simple closed k-curves, simple closed 18-surfaces and so forth. Furthermore, we prove that a digital metric space is complete, which can strongly contribute to the study of Banach fixed point theorem for digital metric spaces. Although Ege, et al. [O. Ege, I. Karaca, J. Nonlinear Sci. Appl., 8 (2015), 237-245] studied "Banach fixed point theorem for digital images", the present paper makes many notions and assertions of the above mentioned paper refined and improved.

Banach and Edelstein Fixed Point Theorems for Digital Images

“…Definition 1.1: We say that two distinct points , ∈ ℤ are -(or ( , )) adjacent for digital images if they satisfy the followings [13]: Figure 1. Configuration of the digital -connectivity of ℤ , ∈ {1,2} [12,17] For a natural number , 1 ≤ ≤ , two distinct points Using above fact, we can obtain the -adjacency relations of ℤ as follows [13]:…”

“…Concretely, for ⊂ ℤ , we obtain Since the sequence { } is defined in the digital metric space ( , , ), Han [12] observed that the Euclidean distance between any two distinct points , ∈ is greater than or equal to 1 as follows: Proposition 3.1 [12]: In a digital metric space ( , , ), consider two points , in a sequence { } of such that they are -adjacent . .…”

“…Thus by using the above Proposition 3.1, we obtain the following: Proposition 3.2 [12]: A sequence { } of points of a digital metric space ( , , ) is a Cauchy sequence if and only if there is ∈ ℕ such that for all , > , we have ( )…”

“…It is an important tool for solution of some problems in mathematics and engineering. Up to now, there are many generalizations of Banach fixed point theorem have been established [12]. In 1961, Michael Edelstein generalized that theorem on -chainable metric spaces [10,12].…”

“…In 2015, they studied the Banach fixed point theorem for digital images [5]. Han [12] refined and improved several notions of that paper such as digital versions of both Cauchy sequence and limit of a sequence in a digital metric spaces. Recently, approximate fixed points and the approximate fixed point property (AFPP) of digitally continuous functions are introduced [4].…”