Approximating common fixed points by a new faster iteration process

Abstract

In this paper, we propose a three-step iteration process and show that this process converges faster than a number of existing iteration processes. We give a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating common fixed points for two nonexpansive mappings. Again we reconfirm our results by examples and tables. Further, we provide some applications of the our iteration process.

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

“…Recently, Khan et al [18] introduced a new iteration for finding common fixed points of two mappings S, T : X → X, as follows: given…”

A Modified Iterative Approach for Fixed Point Problem in Hadamard Spaces

“…Mann [11], Ishikawa [12], and Halpern [13] are the fundamental iterative algorithms to approximate the fixed points of nonexpansive mappings. Following them, several new iterative algorithms were developed by Noor [14], Agarwal et al [15], Garodia et al [16], Abbas and Nazir [17], and Garodia and Uddin [18]. More specifically, the following iterative algorithm was defined using total asymptotically nonexpansive mappings in [19]:…”

An Analytical Study of the Convergence and Stability of the New Four-Step Iterative Schemes

“…We solve this problem by selecting an initial value, ᶉ 0 , and solving it iteratively. The iterative definition of the sequence {ᶉ 𝑛 } 𝑛=0 ∞ leads to an increasing f-point in the constant 𝑈, or a mapping 𝑔 from 𝑈 on to 𝑈 nonexpansive if the next statement holds for any ᶉ, 𝑠 ∈ 𝑈 𝑑(𝑔ᶉ, 𝑔𝑠) ≤ 𝑑(ᶉ, 𝑠) (1.4) Numerous writers have presented and explored alternative iteration strategies for approximating fixed points for diverse classes of contractive circumstances and studied the convergence, rate of convergence and stability (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]). The function Ƿ 𝑈 : 𝑆 → 𝑈 is called the projective metric function, such that 𝑑(ᶉ, Ƿ 𝑈 (ᶉ)) ≤ 𝑚𝑖𝑛 𝑠∈𝑈 𝑑(ᶉ, 𝑠) for all ᶉ ∈ 𝑆 and 𝑠 ∈ 𝑈.…”

“…Some authors have defined new iterative schemes in different spaces and functions,and demonstrated their acceleration compared to the currently leading iterative schemes, see [18][19][20][21].…”