scholarly journals Weak and strong convergence of inertial algorithms for solving split common fixed point problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hong-Yi Chen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Linear Operators ◽  
Fixed Point Problems ◽  
Inclusion Problem ◽  
Mann Iteration ◽  
Contractive Mappings ◽  
Variational Inclusion Problem ◽  
Split Common Fixed Point

AbstractIn this paper, we propose two iterative schemes for approximating solutions of split common fixed point problems in multiple linear operators case. The first algorithm implements the Krasnosel’skiĭ–Mann iteration with an inertial effect for which the weak convergence is established under mild assumptions. With the tool of nearly contractive mappings, we introduce a viscosity-type iteration which ensures strong convergence. We apply our results to solve a multiple split monotone variational inclusion problem. A numerical example is given to demonstrate the efficiency of the proposed algorithms.

scholarly journals Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jing Zhao ◽  
Hang Zhang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Fixed Point Problem ◽  
Linear Operators ◽  
Point Problem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Common Fixed Point Problem ◽  
Split Common Fixed Point

Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

scholarly journals The split common fixed point problem of two infinite families of demicontractive mappings and the split common null point problem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3859-3874 ◽  
Author(s):  
Ali Abkar ◽  
Elahe Shahrosvand
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Problem ◽  
Null Point ◽  
Inclusion Problem ◽  
Point Problem ◽  
Variational Inclusion Problem ◽  
Common Fixed Point Problem ◽  
Demicontractive Mappings ◽  
Split Common Fixed Point

In this paper we introduce a new algorithm based on the viscosity iteration method for solving the split common fixed point problem of two infinite families of k-demicontractive mappings. We shall also study the split common null point problem, and the split equilibrium problem for this class of mappings. As an application, we obtain strong convergence theorems for the split monotone variational inclusion problem and the split variational inequality problem. Our results improve and extend the recent results of Cui and Wang [9], Takahashi [21], Tang and Lui [22], Moudafi [15], Eslamian and Vahidi [17], and many others.

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