scholarly journals Modified Iterative Algorithms for Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Hilbert Space ◽  
Approximate Solution ◽  
Real Number ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Contractive Mapping ◽  
Linear Bounded Operator ◽  
Special Cases

Let H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)∩F(T)≠∅, let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1−βn)Syn, yn=αnγf(xn)+(I−αnA)Txn, n≥0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following control conditions: (C1) lim⁡n→∞ αn=0, (C3) 0<lim⁡inf⁡n→∞ βn≤lim⁡sup⁡n→∞ βn<1, then ‖xn+1−xn‖→0. We also discuss several special cases of this iterative algorithm.

scholarly journals Iterative methods for nonlinear quasi complementarity problems

1987 ◽  
Vol 10 (2) ◽  
pp. 339-344 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Hilbert Space ◽  
Approximate Solution ◽  
Complementarity Problems ◽  
Real Hilbert Space ◽  
Continuous Mapping ◽  
Convergence Criteria ◽  
Polar Cone ◽  
Point Mapping ◽  
Special Cases ◽  
Point To Point

In this paper, we consider and study an iterative algorithm for finding the approximate solution of the nonlinear quasi complementarity problem of findingu ϵ k(u)such thatTu ϵ k*(u)  and  (u−m(u),Tu)=0wheremis a point-to-point mapping,Tis a (nonlinear) continuous mapping from a real Hilbert spaceHinto itself andk*(u)is the polar cone of the convex conek(u)inH. We also discuss the convergence criteria and several special cases, which can be obtained from our main results.

scholarly journals New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 466
Author(s):  
Li Wei ◽  
Yingzi Shang ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Special Cases ◽  
Accretive Mappings ◽  
Number Sequences ◽  
Zero Points

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

scholarly journals General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nopparat Wairojjana ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Linear Bounded Operator ◽  
New Methods

This paper deals with new methods for approximating a solution to the fixed point problem; findx̃∈F(T), whereHis a Hilbert space,Cis a closed convex subset ofH,fis aρ-contraction fromCintoH,0<ρ<1,Ais a strongly positive linear-bounded operator with coefficientγ̅>0,0<γ<γ̅/ρ,Tis a nonexpansive mapping onC,andPF(T)denotes the metric projection on the set of fixed point ofT. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality〈(A-γf)x̃+τ(I-S)x̃,x-x̃〉≥0forx∈F(T), whereτ∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

scholarly journals Two iterative algorithms for solving the split common fixed point problems

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4375-4386
Author(s):  
Dingping Wu ◽  
Mihai Postolache
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Fixed Point Problems ◽  
Split Common Fixed Point

The purpose of this paper is to study the split common fixed point problems (SCFP) involved in nonexpansive mappings in real Hilbert space. We introduce two iterative algorithms for finding a solution of the SCFP involved in nonexpansive mappings, where one is a Mann-type iterative algorithm and another is a Halpern-type iterative algorithm.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 480
Author(s):  
Manatchanok Khonchaliew ◽  
Ali Farajzadeh ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Constraint Qualifications ◽  
Solution Sets ◽  
New Algorithms

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm.

scholarly journals Iterative methods for zero points of accretive operators in Banach spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 329-344
Author(s):  
Sheng Hua Wang ◽  
Sun Young Cho ◽  
Xiao Long Qin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Real Banach Space ◽  
Real Hilbert Space ◽  
Lower Semicontinuous ◽  
Iterative Algorithms ◽  
Accretive Operators ◽  
Weak And Strong Convergence ◽  
Zero Points

Abstract The purpose of this paper is to consider the problem of approximating zero points of accretive operators. We introduce and analysis Mann-type iterative algorithm with errors and Halpern-type iterative algorithms with errors. Weak and strong convergence theorems are established in a real Banach space. As applications, we consider the problem of approximating a minimizer of a proper lower semicontinuous convex function in a real Hilbert space

scholarly journals Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Baogui Xin
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings

Let{T}i=1NbeNquasi-nonexpansive mappings defined on a closed convex subsetCof a real Hilbert spaceH. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.

scholarly journals Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Songnian He ◽  
Caiping Yang
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Functions ◽  
Level Sets ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Original Domain

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0, for allx∈C, whereCis the intersection of finite level sets of convex functions defined on a real Hilbert spaceHandF:H→His anL-Lipschitzian andη-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution ofVI(C,F). Since our algorithm avoids calculating the projectionPC(calculatingPCby computing several sequences of projections onto half-spaces containing the original domainC) directly and has no need to know any information of the constantsLandη, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

scholarly journals Convergence Theorems of Iterative Schemes For Nonexpansive Mappings

2017 ◽  
Vol 12 (12) ◽  
pp. 6845-6851
Author(s):  
Inaam Mohammed Ali Hadi ◽  
Dr. salwa Salman Abd
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Common Fixed Point ◽  
Convex Analysis ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Iterative Schemes

In this paper, we give a type of iterative scheme for sequence of nonexpansive mappings and we study the strongly convergence of these schemes in real Hilbert space to common fixed point which is also a solution of a variational inequality. Also there are some consequent of this results in convex analysis

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