scholarly journals Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 147
Author(s):  
Murtala Haruna Harbau ◽  
Godwin Chidi Ugwunnadi ◽  
Lateef Olakunle Jolaoso ◽  
Ahmad Abdulwahab
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Banach Space ◽  
Asymptotically Nonexpansive Mapping ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive ◽  
Mann Algorithm

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature.

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 Convergence Analysis of an Accelerated Iteration for Monotone Generalizedα-Nonexpansive Mappings with a Partial Order

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Yi-An Chen ◽  
Dao-Jun Wen
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Partial Order ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Ordered Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence

In this paper, we introduce a new accelerated iteration for finding a fixed point of monotone generalizedα-nonexpansive mapping in an ordered Banach space. We establish some weak and strong convergence theorems of fixed point for monotone generalizedα-nonexpansive mapping in a uniformly convex Banach space with a partial order. Further, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration process.

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 Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

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 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.

scholarly journals Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-19
Author(s):  
Murat Ozdemir ◽  
Sezgin Akbulut ◽  
Hukmi Kiziltunc
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Iterative Scheme ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive

We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).

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 New Iterative Algorithm for Two Infinite Families of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Fang Zhang ◽  
Huan Zhang ◽  
Yulong Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Weak Convergence Theorem

We introduce a new iterative scheme for finding a common fixed point of two countable families of multivalued quasi-nonexpansive mappings and prove a weak convergence theorem under the suitable control conditions in a uniformly convex Banach space. We also give a new proof method to the iteration in the paper of Abbas et al. (2011).

scholarly journals A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-ϕ-Asymptotically Nonexpansive Mapping in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Preedaporn Kanjanasamranwong ◽  
Poom Kumam ◽  
Siwaporn Saewan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Asymptotically Nonexpansive Mapping ◽  
Point Problem ◽  
Common Solution ◽  
Asymptotically Nonexpansive ◽  
System Of Equilibrium Problems

The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalizedf-projection operator for finding a common solution of fixed-point problem of a totally quasi-ϕ-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.

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