An iterative method and its application to stable inversion

Abstract

In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely, normal-S iterative method, performs better and so the latter is implemented in the stable inversion of nonlinear discrete time dynamical systems to yield convergence results when Picard iterative method diverges. This is also illustrated with a numerical example. Our work extends and improves upon many results existing in the literature.

On Berinde’s method for comparing iterative processes

“…Suppose that for two fixed point iteration procedures {u n } ∞ n=0 and {v n } ∞ n=0 , both converging to the same fixed point p, the error estimates This concept turned out to be a very useful and versatile tool in studying the fixed point iterative schemes and hence various authors have used it, see [1]- [5], [18], [22], [23], [28], [32]- [34], [37]- [41], [40], [43]- [46], [55]- [57], [66], [68]- [72], [74], [78]- [81], to cite just an incomplete list. b…”

“…b Throughout this paper the references mentioned in the quoted texts are those in the works from where the texts are taken. c Among the 35 papers from our bibliography published in the period 2017-2020, our reference [18] is mentioned only in [28,37,38], and [40]. However, [15,Definition 2.7] is used in [28,37], and [40] without any mention that the obtained estimates are the best possible.…”

“…c Among the 35 papers from our bibliography published in the period 2017-2020, our reference [18] is mentioned only in [28,37,38], and [40]. However, [15,Definition 2.7] is used in [28,37], and [40] without any mention that the obtained estimates are the best possible. d Of course, when (X, • ) is (R, | • |), Definition 3.5 of Popescu [75] reduces to Definition 2.5(a) of Berinde [15] when a = b. e This manner of comparing the rate of convergence for sequences of real numbers is attributed to Knopp [56]…”

Some observations on generalized non-expansive mappings with an application

“…The existence and convergence theorems for non-expansive and generalized nonexpansive mappings have been studied by notable authors, e.g. see Bogin (1976), Wong (1974), Goebel et al (1973), Gürsoy et al (2018), Gürsoy et al (2019).…”

On Berinde's method for comparing iterative processes

“…Among the 19 papers from our bibliography published in 2017 and 2018, our reference[15] is mentioned only in[24] and[31].…”