A Modified Krasnosel’skiǐ–Mann Iterative Algorithm for Approximating Fixed Points of Enriched Nonexpansive Mappings

Abstract

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple Krasnosel’skiǐ fixed point algorithm. Based on the numerical experiments reported in the paper we conclude that, for the class of enriched nonexpansive mappings, it is more convenient to work with the simple Krasnosel’skiǐ fixed point algorithm than with the modified Krasnosel’skiǐ–Mann algorithm.

A modified Ishikawa iteration scheme for b‐enriched nonexpansive mapping to solve split variational inclusion problem and fixed point problem in Hilbert spaces

“…, and 𝛿 n = 0 for each n ≥ 0, then Algorithm 1 reduces to the algorithm (6) of Berinde [12,Theorem 2] for enriched nonexpansive mapping.…”

“…In this section, we present numerical experiments to demonstrate the behaviors of the proposed algorithms. Furthermore, we also compare it with some existing methods, namely, the Krasnosel'ski ǐ iteration (3) [6], the modified Krasnosel'ski ǐ-Mann iteration (6) in Berinde [12], and algorithm (9) in Shehu and Ogbuisi [23]. For more convenient, we shall denote by ALG 1, ALG 2, KRAS (3), MKMB (6), and SHOG (9), Algorithm 1, Algorithm 2, (6), and (9), respectively.…”

“…">The sequences $$$ \left\{{x}_n\right\},\left\{{v}_n\right\},\left\{{y}_n\right\} $$$, and $$$ \left\{{z}_n\right\} $$$ are bounded.Let $$$ {x}^{\ast}\in {S}_{F\left({T}_{\lambda}\right)\cap \Omega} $$$. Following the line proof in Berinde [12], we have $$$$ {\left\Vert {T}_{\lambda }x-x\right\Vert}^2\le 2\left\langle x-{x}^{\ast },x-{T}_{\lambda }x\right\rangle . $$$$ By the definition of $$$ \left\{{y}_n\right\},\left\{{\gamma}_n\right\} $$$, Definition 2.2 (2) and Lemma 2.3, we have …”

“…For decades, many authors have studied the Mann and Ishikawa approaches to estimate the fixed point of nonexpansive mapping, for instance, previous studies [8][9][10][11]. Very recently, Berinde [12] introduced a new class of contractive mapping which covered the nonexpansive mappings. Such a mapping is called an enriched nonexpansive mapping: There exists b ∈ [0, ∞) such that ||b(x − 𝑦) + Tx − T𝑦|| ≤ (b + 1)||x − 𝑦||, ∀x, 𝑦 ∈ X.…”

“…For decades, many authors have studied the Mann and Ishikawa approaches to estimate the fixed point of nonexpansive mapping, for instance, previous studies [8–11]. Very recently, Berinde [12] introduced a new class of contractive mapping which covered the nonexpansive mappings. Such a mapping is called an enriched nonexpansive mapping : There exists $$$ b\in \left[0,\infty \right) $$$ such that $$$$ \left\Vert b\left(x-y\right)+ Tx- Ty\right\Vert \le \left(b+1\right)\left\Vert x-y\right\Vert, \kern0.30em \forall x,y\in X.$$…”