scholarly journals Strong Convergence of a Modified Extragradient Method to the Minimum-Norm Solution of Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Yeong-Cheng Liou
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Minimum Norm ◽  
Minimum Norm Solution ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

We suggest and analyze a modified extragradient method for solving variational inequalities, which is convergent strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.

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 A New Approximation Method for Finding Zeros of Maximal Monotone Operators

2021 ◽  
Vol 31 (2) ◽  
pp. 117-124
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Point ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

scholarly journals A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

2006 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Hilbert Module ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Product Systems ◽  
Dimensional Hilbert Space ◽  
The One

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

scholarly journals Tomography in Abstract Hilbert Spaces

2006 ◽  
Vol 13 (03) ◽  
pp. 239-253 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
A. Simoni ◽  
F. Ventriglia
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Rank One

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

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