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

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.

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Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

Mathematics ◽

10.3390/math9131504 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1504

Author(s):

Li Wei ◽

Xin-Wang Shen ◽

Ravi P. Agarwal

Keyword(s):

Hilbert Space ◽

Variational Inequalities ◽

Strong Convergence ◽

Real Hilbert Space ◽

Iterative Algorithms ◽

Iterative Design ◽

Common Solution ◽

Monotone Inclusions ◽

New Ideas ◽

The Common

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work.

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Iterative methods for zero points of accretive operators in Banach spaces

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/v10309-012-0022-7 ◽

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

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

Abstract and Applied Analysis ◽

10.1155/2013/942315 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 25

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.

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Modified Iterative Algorithms for Nonexpansive Mappings

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2009/320820 ◽

2009 ◽

Vol 2009 ◽

pp. 1-11 ◽

Cited By ~ 2

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.

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Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems

Axioms ◽

10.3390/axioms9040115 ◽

2020 ◽

Vol 9 (4) ◽

pp. 115 ◽

Cited By ~ 1

Author(s):

Nopparat Wairojjana ◽

Nuttapol Pakkaranang ◽

Habib ur Rehman ◽

Nattawut Pholasa ◽

Tiwabhorn Khanpanuk

Keyword(s):

Variational Inequalities ◽

Strong Convergence ◽

Real Hilbert Space ◽

Lipschitz Constant ◽

Minimax Problems ◽

Iterative Algorithms ◽

Pseudomonotone Operators ◽

Variable Stepsize ◽

Type Method ◽

Penalization Methods

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method.

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Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

Journal of Mathematics ◽

10.1155/2013/254821 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Bin-Chao Deng ◽

Tong Chen

Keyword(s):

Hilbert Space ◽

Strong Convergence ◽

Fixed Points ◽

Real Hilbert Space ◽

Convergence Theorems ◽

Mild Conditions ◽

Strongly Monotone

LetHbe a real Hilbert space. LetT1,T2:H→Hbek1-,k2-strictly pseudononspreading mappings; letαnandβnbe two real sequences in (0,1). For givenx0∈H, the sequencexnis generated iteratively byxn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn,∀n∈N, whereTwi=1−wiI+wiTiwithi=1,2andB:H→His strongly monotone and Lipschitzian. Under some mild conditions on parametersαnandβn, we prove that the sequencexnconverges strongly to the setFixT1∩FixT2of fixed points of a pair of strictly pseudononspreading mappingsT1andT2.

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

Symmetry ◽

10.3390/sym11040480 ◽

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.

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Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

ISRN Applied Mathematics ◽

10.5402/2012/482869 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22

Author(s):

S. Imnang

Keyword(s):

Hilbert Space ◽

Variational Inequalities ◽

Nonexpansive Mapping ◽

Real Hilbert Space ◽

General System ◽

Projection Algorithm ◽

Common Element ◽

System Of Variational Inequalities ◽

Demiclosedness Principle ◽

Hybrid Projection Algorithm

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

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Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

Symmetry ◽

10.3390/sym11121502 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1502

Author(s):

Sun Young Cho

Keyword(s):

Hilbert Space ◽

Fixed Point ◽

Variational Inequality ◽

Variational Inequalities ◽

Variational Inequality Problem ◽

Real Hilbert Space ◽

Fixed Point Problem ◽

Point Problem ◽

Strict Pseudocontraction ◽

Adaptive Step

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

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On a class of multivalued variational inequalities

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953398000070 ◽

1998 ◽

Vol 11 (1) ◽

pp. 79-93 ◽

Cited By ~ 4

Author(s):

Muhammad Aslam Noor

Keyword(s):

Variational Inequalities ◽

Optimization Problems ◽

Iterative Algorithms ◽

Auxiliary Principle ◽

Existence Of A Solution ◽

Auxiliary Principle Technique ◽

New Class ◽

Special Cases ◽

Projection Techniques ◽

Multivalued Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

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