Mann iteration process for monotone nonexpansive mappings

Abstract

Let (X, · ) be a Banach space. Let C be a nonempty, bounded, closed, and convex subset of X and T : C → C be a monotone nonexpansive mapping. In this paper, it is shown that a technique of Mann which is defined byis fruitful in finding a fixed point of monotone nonexpansive mappings.MSC: Primary 06F30; 46B20; 47E10

Convergence Analysis of an Accelerated Iteration for Monotone Generalizedα-Nonexpansive Mappings with a Partial Order

“…Furthermore, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration. The method and results presented in this paper extend and improve the corresponding results of [2,17,19,20,25,26] and some others previously.…”

“…In 2015, Bin Dehaish-Khamsi [25] applied the Mann iteration (6) to the case of a monotone nonexpansive mapping in a Banach space endowed with a partial order. Moreover, they proved that { } generated by (6) weakly converges to * ∈ ( ) and * and 1 are comparable.…”

“…In particular, the continuity property probably is not valid, which not only reduces the efficiency of numerical approach but also increases the difficulty of convergence analysis. This is also the main reason why Mann iteration has become popular in approximating the fixed point of monotone-type mappings [2,25,26]. Therefore, it is important and interesting to construct an iterative accelerator method for finding the fixed points problem of such class of monotone-type mappings.…”

“…Our purpose is not only to extend Mann iteration of Bin Dehaish-Khamsi [25] and Song et al [26] to an accelerated iteration for monotone generalized -nonexpansive mappings, but also to establish some weak and strong convergence theorems of fixed point for monotone generalized -nonexpansive mapping in a uniformly convex Banach space with a partial order. Furthermore, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration.…”

Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces

“…Recently, many authors studied the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces (see for example [10][11][12][13][14][15]). Recall that a self mapping T on X is said to be monotone nonexpansive if T is monotone and Tx − Ty ≤ x − y , for every comparable elements x and y.…”