A new iteration scheme for approximating fixed points of nonexpansive mappings

Abstract

In this paper, we introduce a new three-step iteration scheme and establish convergence results for approximation of fixed points of nonexpansive mappings in the framework of Banach space. Further, we show that the new iteration process is faster than a number of existing iteration processes. To support the claim, we consider a numerical example and approximated the fixed point numerically by computer using Matlab.

Convergence Analysis for a Three-Step Thakur Iteration for Suzuki-Type Nonexpansive Mappings with Visualization

“…x n+1 = (1 − α n )Ty n + α n Tz n y n = (1 − β n )Tx n + β n Tz n z n = (1 − γ n )x n + γ n Tx n , n ≥ 0, where {α n }, {β n }, {γ n } are real number sequences in (0, 1). In the sequel, we will consider the following iterative process defined by Thakur et al in [7] for numerical reckoning fixed points of nonexpansive mappings; see, also [8]: for an arbitrary chosen element x 0 ∈ C, the sequence {x n } is generated by…”

CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

“…This was in part due to the fact that Picard's iterative sequence for nonexpansive mappings does not necessarily converge. For more recently introduced iterative schemes, one can see Noor [8], Agrawal et al [9], Abbas and Nazir, [10], Sintunavarat and Pitea [11], Thakur et al [12][13][14], etc.…”

Convergence analysis of modified iterative approaches in geodesic spaces with curvature bounded above

“…In 2016, Thakur et al introduced the iterative method for nonlinear mapping in which sequence { x n } is generated by …”

“…x nþ1 ¼ ð1 − α n ÞTx n þ α n Ty n y n ¼ ð1 − β n Þx n þ β n Tx n (6) ∀n ≥ 0, where the sequences {α n } and {β n } are in (0,1). They proved some convergence theorems and show that the independent and rate of convergence of method (6) is better than methods (3) and (4), respectively.…”

“…∀n ≥ 0, where sequences {α n }, {β n }, and {γ n } are in (0,1). In 2016, Thakur et al 6 introduced the iterative method for nonlinear mapping in which sequence {x n } is generated by…”