Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications

Pongsakorn Sunthrayuth,Nuttapol Pakkaranang,Phatiphat Thounthong,Prasit Cholamjiak,Poom Kumam

doi:10.3390/math7020161

Abstract

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge to a fixed point of the given mapping and a solution of the variational inequality. As applications, we apply our main results to show the existence of fixed points of strict pseudo-contractions and periodic solutions of nonlinear evolution equations and Fredholm integral equations. Finally, we give some numerical examples to illustrate the efficiency and implementation of our method.

Highlights

In the real world, many engineering and science problems can be reformulated as ordinary differential equations
The main objective of this paper is to introduce a generalized viscosity explicit rule (9) for nonexpansive mappings in Banach spaces
The results presented in the paper extend and improve the main results of Ke and Ma [21], Marino et al [25], and previously known results in the earlier and recent literature to Banach spaces

Summary

Introduction

Many engineering and science problems can be reformulated as ordinary differential equations. Ke and Ma [21] improved the VIMRby replacing the midpoint by any point of the interval [ xn , xn+1 ] They constructed the so-called method generalized viscosity implicit rules for a nonexpansive mapping as follows: xn+1 = αn f ( xn ) + (1 − αn ) T (sn xn + (1 − sn ) xn+1 ), ∀n ≥ 1. They showed that { xn } defined by (7) converges strongly to x ∗ ∈ F ( T ), which solves the variational inequality problem (6). The results presented in the paper extend and improve the main results of Ke and Ma [21], Marino et al [25], and previously known results in the earlier and recent literature to Banach spaces

Preliminaries

The Main Results

Convergence Theorems for a Strict Pseudo-Contraction Mapping

Periodic Solution of a Nonlinear Evolution Equation

Nonlinear Fredholm Integral Equation

Numerical Examples