Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equations

Abstract

In this paper, we establish some convergence results for a monotone nonexpansive mapping in a space. We prove the Δ- and strong convergence of the Mann iteration scheme. Further, we provide a numerical example to illustrate the convergence of our iteration scheme, and also, as an application, we discuss the solution of integral equation. Our results extend some of the relevant results.

On Some Convergence Results for Hybrid Maximal Functions in Cat(0) Spaces

“…Mann in 1953(Mann, 1953 has studied for the first time the convergence of a sequence based on an iteration in Banach spaces. Many researchers have studied the convergence of an iterative sequence in various spaces (Ishikawa, 1974), (Ullah et al, 2018), (Uddin et al, 2018). In this paper, there are given some convergence results for modified hybrid maximal functions in a complete CAT(0) space.…”

On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

“…As an application, fixed point theory of nonexpansive mapping and its generalization has many applications in different fields such as applications of nonexpansive mapping to solve an integral equation (see [18]) and to solve a variational inequality problem (see [19]). Also, there are applications of some classes of generalized nonexpansive mappings like quasi-nonexpansive mappings under contraction to find the minimum norm fixed point and generalized α-nonexpansive mappings to solve split feasibility problem (see [20,21]).…”

Results on a faster iterative scheme for a generalized monotone asymptotically

“…Note that, the continuity of monotone nonexpansive mapping may be not achieved, see [33] or [4]. At the beginning of studying the existence of fixed point for the nonexpansive mapping G, Mann formed the following iterative scheme which was later known by his name, Mann' iteration:…”