scholarly journals Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
C. Diop ◽  
M. Sene ◽  
N. Djitté
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Uniformly Convex ◽  
Convergence Theorems

Let K be a nonempty closed and convex subset of a uniformly convex real Banach space E and let T1,…,Tm:K→2K be m multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence xn is proved to be an approximating fixed point sequence of each Ti; that is, limdxn;Txn=0. Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).

scholarly journals Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Yekini Shehu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Iterative Process ◽  
Real Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Multivalued Maps ◽  
Convergence Theorems ◽  
Uniformly Convex Banach Spaces

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.

scholarly journals Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Point Problem ◽  
Uniformly Convex ◽  
Differentiable Norm

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

2013 ◽  
Vol 756-759 ◽  
pp. 3628-3633
Author(s):  
Yuan Heng Wang ◽  
Wei Wei Sun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

scholarly journals Fixed point iteration for asymptotically quasi-nonexpansive mappings in Banach spaces

2005 ◽  
Vol 2005 (11) ◽  
pp. 1685-1692 ◽  
Author(s):  
Somyot Plubtieng ◽  
Rabian Wangkeeree
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Subset ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Completely Continuous

Suppose thatCis a nonempty closed convex subset of a real uniformly convex Banach spaceX. LetT:C→Cbe an asymptotically quasi-nonexpansive mapping. In this paper, we introduce the three-step iterative scheme for such map with error members. Moreover, we prove that ifTis uniformlyL-Lipschitzian and completely continuous, then the iterative scheme converges strongly to some fixed point ofT.

scholarly journals Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yuanheng Wang ◽  
Weifeng Xuan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Finite Family ◽  
Convergence Theorems ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Generalized Projection Method ◽  
Hybrid Iterative Method

We establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space. Our results improve and extend the corresponding results by many others.

scholarly journals Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces

2005 ◽  
Vol 2005 (10) ◽  
pp. 1643-1653 ◽  
Author(s):  
Hafiz Fukhar-Ud-Din ◽  
Abdul Rahim Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unbounded Domain ◽  
Iterative Process ◽  
Error Term ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Finite Family ◽  
Iterative Sequence ◽  
Uniformly Convex

We prove that an implicit iterative process with errors converges weakly and strongly to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings on unbounded sets in a uniformly convex Banach space. Our results generalize and improve upon, among others, the corresponding recent results of Sun (2003) in the following two different directions: (i) domain of the mappings is unbounded, (ii) the iterative sequence contains an error term.

An ergodic theorem for asymptotically nonexpansive mappings

1994 ◽  
Vol 124 (1) ◽  
pp. 23-31
Author(s):  
Manfred Krüppel ◽  
Jaroslaw Górnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Ergodic Theorem ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Image Position

The purpose of this paper is to prove the following (nonlinear) mean ergodic theorem: Let E be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E and let T: C → C be an asymptotically nonexpansive mapping. Ifexists uniformly in r = 0, 1, 2,…, then the sequence {Tnx} is strongly almost-convergent to a fixed point y of T, that is,uniformly in i = 0, 1, 2, ….

scholarly journals Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Aunyarat Bunyawat ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Method ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Banach Space ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Uniformly Convex Banach Spaces

We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping{Ti}in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of eachTi. Some strong convergence theorems of the proposed method are also obtained for the following cases: allTiare continuous and one ofTiis hemicompact, and the domainKis compact.

scholarly journals Strong Convergence of Approximating Fixed Point Sequences for Nonexpansive Mappings

2006 ◽  
Vol 74 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Contraction Principle ◽  
Uniformly Convex ◽  
Banach’S Contraction Principle

Consider a nonexpansive self-mapping T of a bounded closed convex subset of a Banach space. Banach's contraction principle guarantees the existence of approximating fixed point sequences for T. However such sequences may not be strongly convergent, in general, even in a Hilbert space. It is shown in this paper that in a real smooth and uniformly convex Banach space, appropriately constructed approximating fixed point sequences can be strongly convergent.

scholarly journals Iterative Algorithm for Mappings Satisfying Bγ,μ Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Kifayat Ullah ◽  
Junaid Ahmad
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Weak Convergence ◽  
Iterative Algorithm ◽  
Iterative Algorithms ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Example ◽  
Strong And Weak Convergence ◽  
Convergence Results

In this article, we introduce a novel iterative algorithm to approximate fixed point of mappings with Bγ,μ condition. We establish some strong and weak convergence results in a uniformly convex Banach space. Using a numerical example, we compare the speed of the proposed algorithm with some leading iterative algorithms.

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