Modified Inertial-Type Multi-Choice CQ-Algorithm for Countable Weakly Relatively Non-Expansive Mappings in a Banach Space, Applications and Numerical Experiments

Li Wei,Yibin Xin,Ravi P Agarwal,Ruilan Zhang

doi:10.3390/math8040613

Abstract

In this paper, some new modified inertial-type multi-choice CQ-algorithms for approximating common fixed point of countable weakly relatively non-expansive mappings are presented in a real uniformly convex and uniformly smooth Banach space. New proof techniques are used to prove strong convergence theorems, which extend some previous work. The connection and application to maximal monotone operators are demonstrated. Numerical experiments are conducted to illustrate that the rate of convergence is accelerated compared to some previous ones for some special cases.

Highlights

A Banach space E is said to be uniformly smooth [1] if limt→0 Et = 0
The normalized duality mapping JE : E → 2E is defined by JE (u) = { g ∈ E∗ : hu, gi = kuk2 = k gk2 }, u ∈ E, where hu, gi denotes the value of g ∈ E∗ at u ∈ E
From Theorems 11–14, we can see that the main results of Theorems 3–6 in our paper can be further extended to the topic of designing iterative algorithms to approximate common zero points of two kinds of countable maximal monotone operators

Summary

Introduction and Preliminaries

Suppose E is a real Banach space and E∗ is its dual space. ”un → u” and ”un * u”. In 2005, Matsushita and Takahashi [7] extended the topic of non-expansive mappings to that of strongly relatively non-expansive mappings They presented the following CQ iterative algorithm to approximate the fixed points of a strongly relatively non-expansive mapping T in a real uniformly convex and uniformly smooth Banach space E: vn = J −1 [αn JE un + (1 − αn ) JE Tun ], E. In 2009, Wei et al [14] presented the following hybrid iterative algorithm to approximate the common fixed point of two strongly relatively non-expansive mappings T1 and T2 in a real uniformly convex and uniformly smooth Banach space E: vn = J −1 [αn JE un + (1 − αn ) JE T1 un ],. A−1 0 is closed and convex in E; if xn → x and yn ∈ Axn with yn * y, or xn * x and yn ∈ Axn with yn → y, x ∈ D ( A) and y ∈ Ax

Improvement and New Proof Techniques for Chidume’s Results

Inertial-Type Iterative Algorithms with New Set Cn

Under the following assumptions

Applications

Numerical Experiments