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 A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-31 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Generalized Mixed Equilibrium Problem ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

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 Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Chin-Tzong Pang ◽  
Eskandar Naraghirad ◽  
Ching-Feng Wen
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Convergence Theorems ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

We study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. By exhibiting an example, we first show that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings which was investigated by Martín-Márques et al. (2013). We then prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature.

scholarly journals The convex function determined by a multifunction

1996 ◽  
Vol 54 (1) ◽  
pp. 87-97 ◽  
Author(s):  
M. Coodey ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Convex Function ◽  
Monotone Operator ◽  
Open Problem ◽  
Maximal Monotone Operator ◽  
Convex Functions ◽  
Maximal Monotone ◽  
Minimax Theorem ◽  
Local Boundedness

We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

scholarly journals Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Suthep Suantai ◽  
Wataru Takahashi
Keyword(s):  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Zero Point ◽  
Common Element ◽  
Fixed Point Set ◽  
Nonlinear Operators ◽  
Point Set

We establish strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

scholarly journals Stronger maximal monotonicity properties of linear operators

1999 ◽  
Vol 60 (1) ◽  
pp. 163-174 ◽  
Author(s):  
H.H. Bauschke ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Real Banach Space ◽  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Linear Mapping ◽  
Linear Operators ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function

The subdifferential mapping associated with a proper, convex lower semicontinuous function on a real Banach space is always a special kind of maximal monotone operator. Specifically, it is always “strongly maximal monotone” and of “type (ANA)”. In an attempt to find maximal monotone operators that do not satisfy these properties, we investigate (possibly discontinuous) maximal monotone linear operators from a subspace of a (possibly nonreflexive) real Banach space into its dual. Such a linear mapping is always “strongly maximal monotone”, but we are only able to prove that is of “type (ANA)” when it is continuous or surjective — the situation in general is unclear. In fact, every surjective linear maximal monotone operator is of “type (NA)”, a more restrictive condition than “type (ANA)”, while the zero operator, which is both continuous and linear and also a subdifferential, is never of “type (NA)” if the underlying space is not reflexive. We examine some examples based on the properties of derivatives.

Fixed point theorems and convergence theorems for monotone (α, β)-nonexpansive mappings in ordered Banach spaces

2017 ◽  
Vol 26 (2) ◽  
pp. 163-180
Author(s):  
KHANITIN MUANGCHOO-I ◽  
DAWUD THONGTHA ◽  
POOM KUMAM ◽  
YEOL JE CHO
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Nonexpansive Mappings ◽  
Ordered Banach Space ◽  
Weak And Strong Convergence ◽  
Convergence Theorems ◽  
Alpha Beta ◽  
Ordered Banach Spaces

In this paper, we introduce the notion of a monotone (\alpha,\beta)-nonexpansive mapping in an ordered Banach space E with the partial order \leq and prove some existence theorems of fixed points of a monotone (\alpha,\beta)-nonexpansive mapping in a uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of Ishikawa type iteration under the control condition \[\limsup_{n\to\infty}s_n(1-s_n) > 0\quad and \quad \liminf_{n\to\infty}s_n(1-s_n) > 0.\] Finally, we give an numerical example to illustrate the main result in this paper.

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