A novel scheme of $ k $-step iterations in digital metric spaces

Abstract

<abstract><p>In computational mathematics, the comparison of convergence rate in different iterative methods is an important concept from theoretical point of view. The importance of this comparison is relevant for researchers who want to discover which one of these iterations converges to the fixed point more rapidly. In this article, we study the different numerical methods to calculate fixed point in digital metric spaces, introduce a new k-step iterative process and conduct an analysis on the strong convergence, stability and data dependence of the mentioned scheme. Some illustrative examples are given to show that this iteration process converges faster.</p></abstract>

Remarks on fixed point assertions in digital topology, 5

“…Perhaps appropriate rewriting can yield valid results out of the assertions of [1]. However, as written, none of the "Main Results" of this paper can be regarded as all of well defined, proven, and valid.…”

“…The paper [1] is concerned with comparing the rates of convergence of convergent sequences in a digital image, perhaps especially of sequences converging to a fixed point of a given function. This is a problem of greater theoretical than practical interest, as in the "real world," a digital image is finite and, usually, of small to moderate size.…”

“…• In Definition 2.5 of [1], a digital metric space (E, µ), a function T : E → E, and a sequence {α n } ∞ n=0 , where 0 ≤ α n ≤ 1 are hypothesized. Statement (2.2) of this definition calls for…”

“…• The assertion of [1] labeled as Theorem 4.1 uses "F (T )" without definition in its hypothesis. Its usage suggests this is intended to be "F T ", defined earlier as the fixed point set of the function T .…”