scholarly journals Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 750
Author(s):  
Natthaphon Artsawang ◽  
Kasamsuk Ungchittrakool
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Image Restoration ◽  
Zero Point ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Type Method ◽  
Type Algorithm ◽  
Inertial Terms

In this article, we establish a new Mann-type method combining both inertial terms and errors to find a fixed point of a nonexpansive mapping in a Hilbert space. We show strong convergence of the iterate under some appropriate assumptions in order to find a solution to an investigative fixed point problem. For the virtue of the main theorem, it can be applied to an approximately zero point of the sum of three monotone operators. We compare the convergent performance of our proposed method, the Mann-type algorithm without both inertial terms and errors, and the Halpern-type algorithm in convex minimization problem with the constraint of a non-zero asymmetric linear transformation. Finally, we illustrate the functionality of the algorithm through numerical experiments addressing image restoration problems.

scholarly journals Split equality monotone variational inclusions and fixed point problem of set-valued operator

2017 ◽  
Vol 9 (1) ◽  
pp. 94-121 ◽  
Author(s):  
Mohammad Eslamian ◽  
Ashkan Fakhri
Keyword(s):  
Fixed Point ◽  
Suitable Condition ◽  
Zero Point ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Infinite Dimensional ◽  
Equality Problem ◽  
Point Set

AbstractIn this paper, we are concerned with the split equality problem of finding an element in the zero point set of the sum of two monotone operators and in the common fixed point set of a finite family of quasi-nonexpansive set-valued mappings. Strong convergence theorems are established under suitable condition in an infinite dimensional Hilbert spaces. Some applications of the main results are also provided.

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 A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Young-Ye Huang ◽  
Chung-Chien Hong
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Special Cases ◽  
Split Feasibility

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

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 Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yeong-Cheng Liou ◽  
Yonghong Yao ◽  
Chun-Wei Tseng ◽  
Hui-To Lin ◽  
Pei-Xia Yang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Nonlinear Operator ◽  
Iterative Algorithms ◽  
Fixed Point Problem ◽  
Solution Set ◽  
Point Problem ◽  
General Variational Inequality

We consider a general variational inequality and fixed point problem, which is to find a pointx*with the property that (GVF):x*∈GVI(C,A)andg(x*)∈Fix(S)whereGVI(C,A)is the solution set of some variational inequalityFix(S)is the fixed points set of nonexpansive mappingS, andgis a nonlinear operator. Assume the solution setΩof (GVF) is nonempty. For solving (GVF), we suggest the following methodg(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0. It is shown that the sequence{xn}converges strongly tox*∈Ωwhich is the unique solution of the variational inequality〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for allx∈Ω.

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