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.

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 Approximating common fixed points of families of quasi-nonexpansive mappings

1995 ◽  
Vol 18 (2) ◽  
pp. 287-292 ◽  
Author(s):  
M. K. Ghosh ◽  
Lokenath Debnath
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Special Cases

This paper deals with a family of quasi-nonexpansive mappings in a uniformly convex Banach space, and the convergence of iterates generated by this family. A fixed point theorem for two quasi-nonexpansive mappings is then proved. This theorem is then extended for a finite family of quasinonexpansive mappings. It is shown that Ishikawa's [1] result follows as special cases of results proved in this paper.

scholarly journals Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

1991 ◽  
Vol 43 (1) ◽  
pp. 153-159 ◽  
Author(s):  
J. Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive ◽  
Fourth Kind

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

scholarly journals Approximating fixed points by ishikawa iterates

1989 ◽  
Vol 40 (1) ◽  
pp. 113-117 ◽  
Author(s):  
M. Maiti ◽  
M.K. Ghosh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

scholarly journals Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

2001 ◽  
Vol 27 (11) ◽  
pp. 653-662 ◽  
Author(s):  
Jui-Chi Huang
Keyword(s):  
Banach Space ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Bounded Sequences

LetEbe a uniformly convex Banach space,Ca nonempty closed convex subset ofE. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by{Tj:C→C}j=1ras follows:Un(j)=an(j)I+bn(j)TjnUn(j−1)+cn(j)un(j),j=1,2,…,r,x1∈C,xn+1=an(r)xn+bn(r)TrnUn(r−1)xn+cn(r)un(r),n≥1, whereUn(0):=I,Ithe identity map; and{un(j)}are bounded sequences inC; and{an(j)},{bn(j)}, and{cn(j)}are suitable sequences in[0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.

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 Monotone asymptotic pointwise contractions

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3291-3294 ◽  
Author(s):  
Dehaish Bin ◽  
Mohamed Khamsi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Partial Order ◽  
Metric Space ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Convex Metric Space ◽  
Uniformly Convex Metric Space

In this work, we extend the fixed point result of Kirk and Xu for asymptotic pointwise nonexpansive mappings in a uniformly convex Banach space to monotone mappings defined in a hyperbolic uniformly convex metric space endowed with a partial order.

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 Common Fixed Point of a Finite-step Iteration Algorithm Under Total Asymptotically Quasi-nonexpansive Maps

2019 ◽  
Vol 16 (3) ◽  
pp. 0654
Author(s):  
Abed Et al.
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Banach Space ◽  
Finite Family ◽  
Iteration Algorithm ◽  
Uniformly Convex ◽  
Nonexpansive Maps ◽  
Finite Step

      Throughout this paper, a generic iteration algorithm for a finite family of total asymptotically quasi-nonexpansive maps in uniformly convex Banach space is suggested. As well as weak / strong convergence theorems of this algorithm to a common fixed point are established. Finally, illustrative numerical example by using Matlab is presented.

scholarly journals A New Iterative Scheme of Modified Mann Iteration in Banach Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jinzuo Chen ◽  
Dingping Wu ◽  
Caifen Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Accretive Operators ◽  
Mann Iteration ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

We introduce the modified iterations of Mann's type for nonexpansive mappings and asymptotically nonexpansive mappings to have the strong convergence in a uniformly convex Banach space. We study approximation of common fixed point of asymptotically nonexpansive mappings in Banach space by using a new iterative scheme. Applications to the accretive operators are also included.

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