scholarly journals The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Rabian Wangkeeree
Keyword(s):  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Linear Operators ◽  
Duality Mapping ◽  
Approximation Methods ◽  
Iterative Approximation ◽  
Bounded Linear ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Hybrid Approximation

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

scholarly journals A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Duality Mapping ◽  
Strong Convergence Theorem ◽  
Iterative Approximation ◽  
Nonexpansive Semigroups ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Iterative Approximation Method

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

scholarly journals The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Methods ◽  
Reflexive Banach Space ◽  
Iterative Scheme ◽  
Duality Mapping ◽  
Nonexpansive Semigroups ◽  
Weakly Continuous ◽  
Asymptotically Nonexpansive ◽  
Weakly Continuous Duality Mapping

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

scholarly journals Convergence Theorems for Modified Implicit Iterative Methods with Perturbation for Pseudocontractive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 72
Author(s):  
Jong Soo Jung
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Reflexive Banach Space ◽  
Convex Combination ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Pseudocontractive Mappings ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping

In this paper, first, we introduce a path for a convex combination of a pseudocontractive type of mappings with a perturbed mapping and prove strong convergence of the proposed path in a real reflexive Banach space having a weakly continuous duality mapping. Second, we propose two modified implicit iterative methods with a perturbed mapping for a continuous pseudocontractive mapping in the same Banach space. Strong convergence theorems for the proposed iterative methods are established. The results in this paper substantially develop and complement the previous well-known results in this area.

scholarly journals Convergence of iterative algorithms for continuous pseudocontractive mappings

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1767-1777 ◽  
Author(s):  
Jong Jung
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Convex Combination ◽  
Iterative Algorithms ◽  
Duality Mapping ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Implicit Iterative Algorithm

In this paper, we prove strong convergence of a path for a convex combination of a pseudocontractive type of operators in a real reflexive Banach space having a weakly continuous duality mapping J? with gauge function ?. Using path convergency, we establish strong convergence of an implicit iterative algorithm for a pseudocontractive mapping combined with a strongly pseudocontractive mapping in the same Banach space.

scholarly journals Strong Convergence of New Two-Step Viscosity Iterative Approximation Methods for Set-Valued Nonexpansive Mappings in CAT(0) Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Ting-jian Xiong ◽  
Heng-you Lan
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Schwarz Inequality ◽  
Approximation Methods ◽  
Iterative Approximation ◽  
Convergence Theorems

This paper is for the purpose of introducing and studying a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings in CAT(0) spaces. By means of some properties and characteristic to CAT(0) space and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete CAT(0) spaces are provided. The results of this paper improve and extend the corresponding main theorems in the literature.

Solving the general split common fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operator norms

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 559-573 ◽  
Author(s):  
Jing Zhao ◽  
Songnian He
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Linear Operators ◽  
Point Problem ◽  
Bounded Linear ◽  
Common Fixed Point Problem ◽  
Operator Norms ◽  
Split Common Fixed Point

Let H1, H2, H3 be real Hilbert spaces, let A : H1 ? H3, B : H2 ? H3 be two bounded linear operators. The general multiple-set split common fixed-point problem under consideration in this paper is to find x ??p,i=1F(Ui), y ??r,j=1 F(Tj) such that Ax = Bym, (1) where p, r ? 1 are integers, Ui : H1 ? H1 (1 ? i ? p) and Tj : H2 ? H2 (1 ? j ? r) are quasi-nonexpansive mappings with nonempty common fixed-point sets ?p,i=1 F(Ui) = ?p,i=1 {x ? H1 : Uix = x} and ?r,j=1F(Tj) = ?r,j=1 {x ? H2 : Tjx = x}. Note that, the above problem (1) allows asymmetric and partial relations between the variables x and y. If H2 = H3 and B = I, then the general multiple-set split common fixed-point problem (1) reduces to the multiple-set split common fixed-point problem proposed by Censor and Segal [J. Convex Anal. 16(2009), 587-600]. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

scholarly journals The Choquet–Deny Equation in a Banach Space

2007 ◽  
Vol 59 (4) ◽  
pp. 795-827 ◽  
Author(s):  
Wojciech Jaworski ◽  
Matthias Neufang
Keyword(s):  
Banach Space ◽  
Harmonic Functions ◽  
Linear Operators ◽  
Crossed Product ◽  
Poisson Formula ◽  
Bounded Linear ◽  
Weakly Continuous ◽  
Bounded Harmonic Functions ◽  
Dual Banach Space ◽  
Special Case

AbstractLet G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure μ on G, we study the Choquet–Deny equation π(μ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law μ. The relation between the space of solutions of the Choquet–Deny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling theW*-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space E = B(L2(G)) of bounded linear operators on L2(G). Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

scholarly journals On totally global solvability of evolutionary equation with unbounded operator

2021 ◽  
Vol 31 (2) ◽  
pp. 331-349
Author(s):  
A.V. Chernov
Keyword(s):  
Cauchy Problem ◽  
Bounded Operator ◽  
Maximal Monotone ◽  
Global Solvability ◽  
Continuous Functions ◽  
Linear Operators ◽  
Evolutionary Equation ◽  
Bounded Linear ◽  
Weakly Continuous ◽  
The Cauchy Problem

Let $X$ be a Hilbert space, $U$ be a Banach space, $G\colon X\to X$ be a linear operator such that the operator $B_\lambda=\lambda I-G$ is maximal monotone with some (arbitrary given) $\lambda\in\mathbb{R}$. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows \[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\] where $u=u(t)\colon[0;T]\to U$ is a control, $x(t)$ is unknown function with values in $X$, we prove the totally (with respect to a set of admissible controls) global solvability subject to global solvability of the Cauchy problem associated with some ordinary differential equation in the space $\mathbb{R}$. Solution $x$ is treated in weak sense and is sought in the space $\mathbb{C}_w\bigl([0;T];X\bigr)$ of weakly continuous functions. In fact, we generalize a similar result having been proved by the author formerly for the case of bounded operator $G$. The essence of this generalization consists in that postulated properties of the operator $B_\lambda$ give us the possibility to construct Yosida approximations for it by bounded linear operators and thus to extend required estimates from “bounded” to “unbounded” case. As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.

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