scholarly journals On strong convergence theorems for a viscosity-type extragradient method

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1033-1043
Author(s):  
L.C. Ceng ◽  
C.S. Fong
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Asymptotically Nonexpansive Mapping ◽  
Inclusion Problem ◽  
Point Problem ◽  
Asymptotically Nonexpansive

In this paper, we introduce a general viscosity-type extragradient method for solving the fixed point problem of an asymptotically nonexpansive mapping and the variational inclusion problem with two accretive operators. We obtain a strong convergence theorem in the setting of Banach spaces. In terms of this theorem, we establish the strong convergence result for solving the fixed point problem (FPP) of an asymptotically nonexpansive mapping and the variational inequality problem (VIP) for an inverse-strongly monotone mapping in the framework of Hilbert spaces. Finally, this result is applied to deal with the VIP and FPP in an illustrating example.

scholarly journals Approximation of common solutions for a fixed point problem of asymptotically nonexpansive mapping and a generalized equilibrium problem in Hilbert space

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Richard Osward ◽  
Santosh Kumar ◽  
Mengistu Goa Sangago
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Equilibrium Problem ◽  
Fixed Point Problem ◽  
Asymptotically Nonexpansive Mapping ◽  
Point Problem ◽  
Generalized Equilibrium Problem ◽  
Common Solution ◽  
Asymptotically Nonexpansive ◽  
Generalized Equilibrium

Abstract In this paper, we introduce an iterative algorithm to approximate a common solution of a generalized equilibrium problem and a fixed point problem for an asymptotically nonexpansive mapping in a real Hilbert space. We prove that the sequences generated by the iterative algorithm converge strongly to a common solution of the generalized equilibrium problem and the fixed point problem for an asymptotically nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area. Some applications of main results are also provided.

scholarly journals A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-ϕ-Asymptotically Nonexpansive Mapping in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Preedaporn Kanjanasamranwong ◽  
Poom Kumam ◽  
Siwaporn Saewan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Asymptotically Nonexpansive Mapping ◽  
Point Problem ◽  
Common Solution ◽  
Asymptotically Nonexpansive ◽  
System Of Equilibrium Problems

The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalizedf-projection operator for finding a common solution of fixed-point problem of a totally quasi-ϕ-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.

scholarly journals Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

2019 ◽  
pp. 35-50
Author(s):  
Yusuf Ibrahim
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
Asymptotically Nonexpansive ◽  
Split Common Fixed Point

In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition

scholarly journals STRONG CONVERGENCE OF A NEW HYBRID ALGORITHM FOR FIXED POINT PROBLEMS AND EQUILIBRIUM PROBLEMS

2018 ◽  
Vol 24 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Dang Van Hieu
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hybrid Algorithm ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Mann Iteration ◽  
Common Solution ◽  
Mild Conditions

The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fixed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal-like method) and a modified version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods.

scholarly journals Strong Convergence for the Split Common Fixed-Point Problem for Total Quasi-Asymptotically Nonexpansive Mappings in Hilbert Space

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Lawan Bulama Mohammed ◽  
A. Kılıçman
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
Asymptotically Nonexpansive ◽  
Split Common Fixed Point

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.

scholarly journals Weak and Strong Convergence Theorems for the Inclusion Problem and the Fixed-Point Problem of Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 167 ◽  
Author(s):  
Prasit Cholamjiak ◽  
Suparat Kesornprom ◽  
Nattawut Pholasa
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Weak And Strong Convergence ◽  
Convergence Theorems

In this work, we study the inclusion problem of the sum of two monotone operators and the fixed-point problem of nonexpansive mappings in Hilbert spaces. We prove the weak and strong convergence theorems under some weakened conditions. Some numerical experiments are also given to support our main theorem.

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