scholarly journals The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Phayap Katchang ◽  
Somyot Plubtieng
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Projection Methods ◽  
Common Element ◽  
Nonexpansive Semigroup ◽  
Iterative Schemes ◽  
Nonlinear Mappings

We modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

scholarly journals A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Watcharaporn Cholamjiak ◽  
Suthep Suantai
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Multivalued Maps

We introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established.

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 A New Iterative Method for Equilibrium Problems and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Abdul Latif ◽  
Mohammad Eslamian
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Common Element ◽  
Continuous Mappings ◽  
New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

INERTIAL RELAXED TSENG METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACE

2021 ◽  
Vol 10 (12) ◽  
pp. 3597-3623
Author(s):  
F. Akusah ◽  
A.A. Mebawondu ◽  
H.A. Abass ◽  
M.O. Aibinu ◽  
O.K. Narain
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extrapolation Method ◽  
Minimum Norm ◽  
Minimum Norm Solution ◽  
Inequality Problem ◽  
Weaker Conditions

The research efforts of this paper is to present a new inertial relaxed Tseng extrapolation method with weaker conditions for approximating the solution of a variational inequality problem, where the underlying operator is only required to be pseudomonotone. The strongly pseudomonotonicity and inverse strongly monotonicity assumptions which the existing literature used are successfully weakened. The strong convergence of the proposed method to a minimum-norm solution of a variational inequality problem are established. Furthermore, we present an application and some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature.

scholarly journals Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjie Liu ◽  
Junqing Wang ◽  
Qiansheng Feng
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
Nonspreading Mappings

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mappingTand the solution sets of zero of a maximal monotone mapping and anα-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

scholarly journals Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
D. R. Sahu ◽  
Shin Min Kang ◽  
Vidya Sagar
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

scholarly journals Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Jing Zhao
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Common Element ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for aγ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

scholarly journals Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 609 ◽  
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Abdulkarim Hassan Ibrahim
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Lipschitz Constant ◽  
Pseudomonotone Operator ◽  
Infinite Dimensional ◽  
Iterative Schemes ◽  
Computational Performance ◽  
Variable Step ◽  
Adaptive Step

Two inertial subgradient extragradient algorithms for solving variational inequality problems involving pseudomonotone operator are proposed in this article. The iterative schemes use self-adaptive step sizes which do not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we show the weak and strong convergence of the sequences generated by the proposed algorithms. The strong convergence in the second algorithm follows from the use of viscosity method. Numerical experiments both in finite- and infinite-dimensional spaces are reported to illustrate the inertial effect and the computational performance of the proposed algorithms in comparison with the existing state of the art algorithms.

scholarly journals An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Shamshad Husain ◽  
Nisha Singh
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Common Fixed Points ◽  
Finite Family ◽  
Common Element ◽  
Point Problem

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

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