scholarly journals On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
F. O. Isiogugu ◽  
P. Pillay ◽  
P. U. Nwokoro
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Contemporary Literature ◽  
Iteration Scheme ◽  
Common Elements ◽  
Type Mapping ◽  
The Common ◽  
Selection Of

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multivalued (or single-valued) k-strictly pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence {Kn}n=1∞ of closed convex subsets of H from an arbitrary x0∈H and a sequence {xn}n=1∞ of the metric projections of x0 into Kn. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

scholarly journals Strong convergence of a selection of ishikawa-reich-sabach-type algorithm

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1505-1516
Author(s):  
Obiageli Isiogugu ◽  
Paranjothi Pillay ◽  
Osuo-Siseken Uzoma-Oguguo
Keyword(s):  
Strong Convergence ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Contemporary Literature ◽  
Iteration Scheme ◽  
Common Elements ◽  
Type Mapping ◽  
Type Algorithm ◽  
The Common ◽  
Selection Of

We establish the strong convergence of a selection of an Ishikawa-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multi-valued (or singlevalued) pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a contribution to the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequence {Kn}?n=1 of closed convex subsets of H from an arbitrary x0 ? H and the sequence {xn}?n=1 of the metric projections of x0 into Kn. The results obtained are contributions to the resolution of the controversy over the computability and applicability of such algorithms in the contemporary literature.

scholarly journals Strong Convergence Theorems for Equilibrium Problems andk-Strict Pseudocontractions in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Dao-Jun Wen
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Common Element ◽  
Convergence Theorems

We introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family ofk-strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.

scholarly journals Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

2021 ◽  
Vol 54 (1) ◽  
pp. 280-298
Author(s):  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Type Condition ◽  
Step Size ◽  
Extragradient Methods ◽  
Iterative Schemes ◽  
Mild Conditions

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

scholarly journals A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Solution

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

scholarly journals Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 76
Author(s):  
Chainarong Khanpanuk ◽  
Nuttapol Pakkaranang ◽  
Nopparat Wairojjana ◽  
Nattawut Pholasa
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Inertial Term ◽  
Mild Conditions ◽  
Lipschitz Constants

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

scholarly journals Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Dan Zhang ◽  
Xiaolong Qin ◽  
Feng Gu
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Common Fixed Point ◽  
Hilbert Spaces ◽  
Real Hilbert Space ◽  
Adjoint Operator ◽  
Common Fixed Points ◽  
Nonexpansive Semigroup ◽  
The Common

LetHbe a real Hilbert space. Consider onHa nonexpansive semigroupS={T(s):0≤s<∞}with a common fixed point, a contractionfwith the coefficient0<α<1, and a strongly positive linear bounded self-adjoint operatorAwith the coefficientγ¯>  0. Let0<γ<γ¯/α. It is proved that the sequence{xn}generated by the iterative methodx0∈H, xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)(1/sn)∫0snT(s)xnds, n≥0converges strongly to a common fixed pointx*∈F(S), whereF(S)denotes the common fixed point of the nonexpansive semigroup. The pointx*solves the variational inequality〈(γf-A)x*,x-x*〉≤0for allx∈F(S).

scholarly journals Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
C. E. Chidume ◽  
C. O. Chidume ◽  
N. Djitté ◽  
M. S. Minjibir
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Iteration Sequence ◽  
Strictly Pseudocontractive Mappings

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

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.

Sign in / Sign up

Export Citation Format

Share Document