Weak and Strong Convergence Theorems for Common Attractive Points of Widely More Generalized Hybrid Mappings in Hilbert Spaces

Panadda Thongpaen,Warunun Inthakon,Suthep Suantai,Attapol Kaewkhao,Narawadee Phudolsitthiphat

doi:10.3390/math9192491

Panadda Thongpaen, Warunun Inthakon + Show 3 more

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https://doi.org/10.3390/math9192491

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Journal: Mathematics	Publication Date: Oct 5, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Chiang Mai University

Abstract

In this work, we study iterative methods for the approximation of common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and obtain weak and strong convergence theorems without assuming the closedness for the domain. A numerical example supporting our main result is also presented. As a consequence, our main results can be applied to solving a common fixed point problem.

Highlights

Let H be a real Hilbert space and C be a nonempty subset of H
In 2010, Kocourek et al [1] introduced the notion of a generalized hybrid mapping T : C → H by the condition that there exist α, β ∈ R such that α Tx − Ty 2 + (1 − α) Ty − x 2 ≤ β Tx − y 2 + (1 − β) x − y 2 for all x, y ∈ C and proved a fixed point theorem of this kind of mapping; see [2,3] for more results for fixed point theorems of generalized hybrid mappings
In 2020, Thongpaen and Inthakon [12] used the iteration (3) to prove a weak convergence theorem for common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and applied the main result to some common fixed point problems

Summary

Introduction

Let H be a real Hilbert space and C be a nonempty subset of H. In 2020, Thongpaen and Inthakon [12] used the iteration (3) to prove a weak convergence theorem for common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and applied the main result to some common fixed point problems. In 2019, Ali and Ali [17] proved some weak and strong convergence theorems for common fixed points of two generalized nonexpansive mappings using the iteration presented in (4) in uniformly Banach spaces. Motivated by [17] and abovementioned works, our goal in this paper is to employ the iteration (4) for finding common attractive points of two widely more generalized hybrid mappings without assuming the closedness of the domain and prove weak and strong convergence theorems of (4). We complete this paper by applying our main results to some common fixed point theorems together with some numerical experiments

Preliminaries

Main Results

Weak Convergence Theorems

Conclusions