scholarly journals Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 881 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Xiaolong Qin ◽  
Yekini Shehu ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Common Solution ◽  
Asymptotically Nonexpansive

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

scholarly journals Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 860 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petruşel ◽  
Ching-Feng Wen ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Pseudomonotone Operator ◽  
Strictly Pseudocontractive Mapping ◽  
Common Solution ◽  
Convergence Results ◽  
Asymptotically Nonexpansive

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

scholarly journals On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 925 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petruşel ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Line Search ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Iteration Algorithm ◽  
Point Problem ◽  
Search Process ◽  
Pseudomonotone Operator ◽  
Common Solution

In a real Hilbert space, we denote CFPP and VIP as common fixed point problem of finitely many strict pseudocontractions and a variational inequality problem for Lipschitzian, pseudomonotone operator, respectively. This paper is devoted to explore how to find a common solution of the CFPP and VIP. To this end, we propose Mann viscosity algorithms with line-search process by virtue of subgradient extragradient techniques. The designed algorithms fully assimilate Mann approximation approach, viscosity iteration algorithm and inertial subgradient extragradient technique with line-search process. Under suitable assumptions, it is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the CFPP and VIP, which is the unique solution to a hierarchical variational inequality (HVI).

scholarly journals On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3322
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yekini Shehu
Keyword(s):  
Line Search ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Search Process ◽  
Convergence Results ◽  
Common Fixed Point Problem ◽  
Asymptotically Nonexpansive

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

scholarly journals Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Line Search ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Continuous Mapping ◽  
Solution Set ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Search Process ◽  
Common Solution

Abstract In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

scholarly journals Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Long He ◽  
Yun-Ling Cui ◽  
Lu-Chuan Ceng ◽  
Tu-Yan Zhao ◽  
Dan-Qiong Wang ◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Implicit Rule

AbstractIn a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

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.

scholarly journals Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 67
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao
Keyword(s):  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Common Fixed Points ◽  
Point Problem ◽  
Weak Continuity ◽  
Common Solution ◽  
Pseudomonotone Mapping ◽  
Common Fixed Point Problem ◽  
The Common

Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples.

scholarly journals Inertial KM-type extragradient scheme for solving a variational inequality and a hierarchical fixed point problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ghada AlNemer ◽  
Rehan Ali ◽  
K. R. Kazmi
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Iterative Schemes ◽  
Common Solution ◽  
Hierarchical Fixed Point ◽  
Hierarchical Fixed Point Problems

AbstractWe propose an inertial KM-type extragradient scheme to approximate a common solution of a variational inequality problem and a hierarchical fixed point problem for nonexpansive mappings. This scheme generalizes and unifies a number of known iterative schemes. Furthermore, we discuss the weak convergence for the proposed scheme. We also discuss an example to illustrate the main theorem.

scholarly journals The iterative solutions of split common fixed point problem for asymptotically nonexpansive mappings in Banach spaces

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yuanheng Wang ◽  
Xiuping Wu ◽  
Chanjuan Pan
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Optimization Problems ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Iteration Algorithm ◽  
Point Problem ◽  
Asymptotically Nonexpansive ◽  
Split Common Fixed Point

AbstractIn this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors.

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