scholarly journals On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Hong-Kun Xu ◽  
Luigi Muglia
Keyword(s):  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Multivalued Mappings ◽  
Multivalued Nonexpansive Mapping ◽  
Weakly Continuous ◽  
Duality Map ◽  
Implicit And Explicit ◽  
Essential Assumption

AbstractWe are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points.

scholarly journals A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces

2016 ◽  
Vol 32 (2) ◽  
pp. 241-250
Author(s):  
T. M. M. SOW ◽  
◽  
N. DJITTE ◽  
C.E. CHIDUME ◽  
◽  
...  
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Nonexpansive Maps ◽  
Weakly Continuous ◽  
Duality Map

In this paper, we introduce a new iterative process to approximate fixed points of nonexpansive maps in real Banach spaces having weakly continuous duality map and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption on K or on T. Our results improve important recent results.

scholarly journals Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 347-360
Author(s):  
Mujahid Abbas ◽  
Hira Iqbal ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Approximation Method ◽  
Multivalued Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Multivalued Nonexpansive Mapping ◽  
Convergence Results

AbstractWe prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete ℝ-trees without the endpoint condition.

scholarly journals An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 61 ◽  
Author(s):  
Yonghong Yao ◽  
Mihai Postolache ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonlinear Transformation ◽  
Generalized Variational Inequalities ◽  
Generalized Variational Inequality ◽  
Suggested Algorithm

In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated.

scholarly journals Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 270 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Ching-Feng Wen ◽  
Yonghong Yao
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Variational Inclusions ◽  
Minimization Problems ◽  
Convergence Results ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Implicit And Explicit ◽  
Mixed Equilibria

Multistep composite implicit and explicit extragradient-like schemes are presented for solving the minimization problem with the constraints of variational inclusions and generalized mixed equilibrium problems. Strong convergence results of introduced schemes are given under suitable control conditions.

scholarly journals Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Li-Jun Zhu
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Linear Operators ◽  
Nonexpansive Semigroup ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nonexpansive Semigroups

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a pointx*such thatx*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0,  ∀x∈Ω, whereΩis the set of the solutions of the following variational inequality:x*∈Ϝ,〈(A-S)x*,x-x*〉≥0,  ∀x∈Ϝ, whereA,Bare two strongly positive bounded linear operators,fis aρ-contraction,Sis a nonexpansive mapping, andϜis the fixed points set of a nonexpansive semigroup{T(s)}s≥0. We present a double-net convergence hierarchical to some elements inϜwhich solves the above hierarchical constrained variational inequalities.

scholarly journals Fixed Points Approximation and Solutions of Some Equilibrium and Variational Inequalities Problems

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Bashir Ali
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem ◽  
Points Approximation

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.

scholarly journals Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6267-6281
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
General System ◽  
Inequality Constraint ◽  
Implicit And Explicit

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

scholarly journals Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Jian-Wen Peng ◽  
Yan Wang
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

scholarly journals A strong convergence theorem for contraction semigroups in Banach spaces

2005 ◽  
Vol 72 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Contraction Semigroup ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Weakly Continuous ◽  
Duality Map ◽  
Image Position

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined byfor all n ≥ 1 converges strongly to a member of Fix(F).

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