scholarly journals Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Jingling Zhang ◽  
Yongfu Su ◽  
Qingqing Cheng
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Multivalued Mapping ◽  
Maximal Monotone ◽  
Multivalued Mappings ◽  
Convergence Theorems ◽  
Relatively Nonexpansive ◽  
Nonexpansive Multivalued Mapping

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

scholarly journals Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Ali Alghamdi ◽  
Naseer Shahzad ◽  
Habtu Zegeye
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Finite Family ◽  
Monotone Mappings ◽  
Reflexive Banach Spaces ◽  
Convergence Theorems

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

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 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 Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Multivalued Mapping ◽  
Iterative Process ◽  
Common Point ◽  
Finite Family ◽  
Relatively Nonexpansive ◽  
Nonexpansive Multivalued Mapping

We introduce an iterative process which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces.

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 Strong Convergence Theorems for Modifying Halpern Iterations for Quasi--Asymptotically Nonexpansive Multivalued Mapping in Banach Spaces with Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Li Yi
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Multivalued Mapping ◽  
Iterative Sequence ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Nonexpansive Multivalued Mapping

An iterative sequence for quasi--asymptotically nonexpansive multivalued mapping for modifying Halpern's iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by Chang et al. 2011.

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