scholarly journals Weak and strong convergence of an iterative method for nonexpansive mappings in Hilbert spaces

2008 ◽  
Vol 2 (2) ◽  
pp. 197-204 ◽  
Author(s):  
Yu Miao ◽  
Li Junfen
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Iterative Process ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Initial Point ◽  
Lipschitzian Mapping ◽  
Weak And Strong Convergence ◽  
Convergence Results ◽  
Arbitrary Initial Point

In a real Hilbert space H, starting from an arbitrary initial point x0 H, an iterative process is defined as follows: xn+1 = anxn +(1-an)T?n+1 f yn, yn = bnxn + (1 - bn)T?n g xn, n ? 0, where T ?n+1 f x = Tx - ?n+1?f f(Tx), T?n g x = Tx - ?n?gg(Tx), (8 x 2 H), T : H ? H a nonexpansive mapping with F(T) 6= ; and f (resp. g) : H ? H an ?f (resp. ?g)-strongly monotone and kf (resp. kg)-Lipschitzian mapping, {an} _ (0, 1), {bn} _ (0, 1) and {?n} _ [0, 1), {?n} _ [0, 1). Under some suitable conditions, several convergence results of the sequence {xn} are shown.

Some convergence results of M iterative process in Banach spaces

2019 ◽  
pp. 2150017 ◽  
Author(s):  
Kifayat Ullah ◽  
Faiza Ayaz ◽  
Junaid Ahmad
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Weak And Strong Convergence ◽  
Convergence Results

In this paper, we prove some weak and strong convergence results for generalized [Formula: see text]-nonexpansive mappings using [Formula: see text] iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

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 Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Lin Wang ◽  
Yi-Juan Chen ◽  
Rong-Chuan Du
Keyword(s):  
Nonexpansive Mappings ◽  
Initial Point ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Lipschitzian Mapping ◽  
Strong And Weak Convergence ◽  
Arbitrary Initial Point

LetEbe a real uniformly convex Banach space, and let{Ti:i∈I}beNnonexpansive mappings fromEinto itself withF={x∈E:Tix=x, i∈I}≠ϕ, whereI={1,2,…,N}. From an arbitrary initial pointx1∈E, hybrid iteration scheme{xn}is defined as follows:xn+1=αnxn+(1−αn)(Tnxn−λn+1μA(Tnxn)),n≥1, whereA:E→Eis anL-Lipschitzian mapping,Tn=Ti,i=n(mod N),1≤i≤N,μ>0,{λn}⊂[0,1), and{αn}⊂[a,b]for somea,b∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of{xn}to a common fixed point of the mappings{Ti:i∈I}are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

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.

Modified CQ-Algorithms for G-Nonexpansive Mappings in Hilbert Spaces Involving Graphs

2020 ◽  
Vol 16 (01) ◽  
pp. 89-103
Author(s):  
W. Cholamjiak ◽  
D. Yambangwai ◽  
H. Dutta ◽  
H. A. Hammad
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Rate Of Convergence ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Iterative Schemes ◽  
Cq Method

In this paper, we introduce four new iterative schemes by modifying the CQ-method with Ishikawa and [Formula: see text]-iterations. The strong convergence theorems are given by the CQ-projection method with our modified iterations for obtaining a common fixed point of two [Formula: see text]-nonexpansive mappings in a Hilbert space with a directed graph. Finally, to compare the rate of convergence and support our main theorems, we give some numerical experiments.

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.

scholarly journals Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Nonlinear Operator ◽  
Real Hilbert Space ◽  
Initial Point ◽  
Nonempty Closed Convex Subset ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Hybrid Steepest Descent Method ◽  
Arbitrary Initial Point ◽  
Mann’S Iteration ◽  
Fixed Point Sets

Assume thatFis a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subsetCof a real Hilbert spaceH. Assume also thatΩis the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings onC. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbationF, which generates two sequences from an arbitrary initial pointx0∈H. These two sequences are shown to converge in norm to the same pointPΩx0under very mild assumptions.

scholarly journals Iterative methods for zero points of accretive operators in Banach spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 329-344
Author(s):  
Sheng Hua Wang ◽  
Sun Young Cho ◽  
Xiao Long Qin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Real Banach Space ◽  
Real Hilbert Space ◽  
Lower Semicontinuous ◽  
Iterative Algorithms ◽  
Accretive Operators ◽  
Weak And Strong Convergence ◽  
Zero Points

Abstract The purpose of this paper is to consider the problem of approximating zero points of accretive operators. We introduce and analysis Mann-type iterative algorithm with errors and Halpern-type iterative algorithms with errors. Weak and strong convergence theorems are established in a real Banach space. As applications, we consider the problem of approximating a minimizer of a proper lower semicontinuous convex function in a real Hilbert space

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.

Sign in / Sign up

Export Citation Format

Share Document