Normalized Duality Mapping

2021 ◽  
pp. 125-135
Author(s):  
Marek Galewski
Keyword(s):  
Duality Mapping ◽  
Normalized Duality Mapping

scholarly journals Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Songnian He ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Duality Mapping ◽  
Normalized Duality Mapping ◽  
Uniformly Smooth ◽  
Computational Work

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,  andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0,    p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

scholarly journals Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Duality Mapping ◽  
Nonlinear Mapping ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping ◽  
Weak Convergence Theorem

LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

scholarly journals Numerical range and orthogonality in normed spaces

Filomat ◽  
2009 ◽  
Vol 23 (1) ◽  
pp. 21-41 ◽  
Author(s):  
A. Bachir ◽  
A. Segres
Keyword(s):  
Normed Linear Space ◽  
Numerical Range ◽  
Positive Answer ◽  
Duality Mapping ◽  
Normed Algebra ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normalized Duality Mapping ◽  
James Orthogonality ◽  
Orthogonality In Normed Spaces

Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .

scholarly journals Convergence Theorems for a Maximal Monotone Operator and a -Strongly Nonexpansive Mapping in a Banach Space

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping

LetEbe a smooth Banach space with a norm . Let for any , where stands for the duality pair andJis the normalized duality mapping. With respect to this bifunction , a generalized nonexpansive mapping and a -strongly nonexpansive mapping are defined in . In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a -strongly nonexpansive mapping.

scholarly journals Orthogonality in smooth countably normed spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Tawfeek ◽  
Nashat Faried ◽  
H. A. El-Sharkawy
Keyword(s):  
Normed Space ◽  
Nonempty Subset ◽  
Metric Projection ◽  
Orthogonal Complement ◽  
Duality Mapping ◽  
Dual Cone ◽  
Birkhoff Orthogonality ◽  
Uniformly Convex ◽  
Normed Spaces ◽  
Normalized Duality Mapping

AbstractWe generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

COMPUTATION OF ZEROS OF MONOTONE TYPE MAPPINGS: ON CHIDUME’S OPEN PROBLEM

2020 ◽  
Vol 108 (2) ◽  
pp. 278-288 ◽  
Author(s):  
N. DJITTE ◽  
J. T. MENDY ◽  
T. M. M. SOW
Keyword(s):  
Open Problem ◽  
Monotone Mapping ◽  
Affirmative Answer ◽  
Duality Mapping ◽  
Positive Real ◽  
Normalized Duality Mapping ◽  
Strongly Monotone ◽  
Lipschitz Maps ◽  
Convex Minimization Problems ◽  
Monotone Type

For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.

On the Duality Mapping Sets in Orlicz Sequence Spaces

2001 ◽  
Vol 17 (4) ◽  
pp. 595-602
Author(s):  
Bao Xiang Wang
Keyword(s):  
Sequence Spaces ◽  
Duality Mapping ◽  
Orlicz Sequence Spaces

scholarly journals Nonlinear potentials in function spaces

2002 ◽  
Vol 165 ◽  
pp. 91-116 ◽  
Author(s):  
Murali Rao ◽  
Zoran Vondraćek
Keyword(s):  
Banach Space ◽  
Potential Theory ◽  
Finite Energy ◽  
Duality Mapping ◽  
Quadratic Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Nonlinear Potential Theory ◽  
Nonlinear Potential ◽  
Nonlinear Potentials

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

scholarly journals Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Yekini Shehu ◽  
Jerry N. Ezeora
Keyword(s):  
Real Banach Space ◽  
Iterative Scheme ◽  
Finite Family ◽  
Duality Mapping ◽  
Strong Convergence Theorem ◽  
Sunny Nonexpansive Retraction ◽  
Mild Conditions ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Accretive Mappings

LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.

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