Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications

Godwin Amechi Okeke,Manuel De La Sen,Mujahid Abbas

doi:10.1155/2020/8634050

Abstract

In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. We prove analytically and with series of numerical experiments that the Picard–Ishikawa hybrid iterative process has the same rate of convergence as the CR iterative process.

Highlights

If the existence of the solution of a fixed-point equation involving an operator T is guaranteed, but an exact solution is not possible, the requirement of approximating the solution becomes very pertinent. is gives rise to the need of different iterative processes [1,2,3]
It is worth mentioning that the scheme proposed in [16] is a self-correcting, unlike the variational or weighted residual methods of approximation which depend on the selection of suitable coordinate or basis functions (e.g., [16, 17]). ey established that the proposed fixed-point iteration method is more suitable to approximate the exact solution than other existing methods
Okeke [3] introduced the Picard–Ishikawa hybrid iterative process. e author proved that this iterative process converges faster than all of Picard [15], Krasnosel’skii [10], Mann [11], Ishikawa [9], Noor [12], Picard-Mann [27], and Picard–Krasnosel’skii [2] iterative processes in the sense of Motivated by the investigation of iterative scheme in [3], we introduce the following multivalued fixed-point iterative process

Summary

Introduction

If the existence of the solution of a fixed-point equation involving an operator T is guaranteed, but an exact solution is not possible, the requirement of approximating the solution becomes very pertinent. is gives rise to the need of different iterative processes [1,2,3]. In view of theoretical and practical significance of fixed-point iterative schemes, several authors have constructed and applied different fixedpoint iteration schemes in approximating the solution of equations which model certain physical problems (e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15]). In 2018, Bello et al [16] developed a Mann-type fixedpoint iteration scheme for approximating the solution of two-point boundary value problems. Ey established that the proposed fixed-point iteration method is more suitable to approximate the exact solution than other existing methods. A study of a faster fixed-point iterative method for finding the solution of two-point boundary value problems is carried out. Discrete Dynamics in Nature and Society improve, extend, and generalize several known results in the literature [18,19,20]

Preliminaries

Applications to Solution of Two-point Boundary Value Problems

Numerical Examples