scholarly journals Estimation of Newly Established Iterative Scheme for Generalized Nonexpansive Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Aftab Hussain ◽  
Nawab Hussain ◽  
Danish Ali
Keyword(s):  
Banach Space ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Space Environment ◽  
Iterative Approach ◽  
Uniformly Convex ◽  
Prior Art ◽  
Nonexpansive Maps ◽  
New Iterative Method

We introduce a new iterative method in this article, called the D iterative approach for fixed point approximation. Analytically, and also numerically, we demonstrate that our established D I.P is faster than the well-known I.P of the prior art. Finally, in a uniformly convex Banach space environment, we present weak as well as strong convergence theorems for Suzuki’s generalized nonexpansive maps. Our findings are an extension, refinement, and induction of several existing iterative literatures.

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 results of a faster iterative scheme including multi-valued mappings in Banach spaces

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1359-1368
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Khan ◽  
Naseer Muhammad
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Current Literature ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Efficiency ◽  
Convergence Results

In this paper, we study M-iterative scheme in the new context of multi-valued generalized ?-nonexpansive mappings. A uniformly convex Banach space is used as underlying setting for our approach. We also provide a new example of generalized ?-nonexpasive mappings. We connect M iterative scheme and other well known schemes with this example, to show the numerical efficiency of our results. Our results improve and extend many existing results in the current literature.

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 On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
M. De la Sen ◽  
Mujahid Abbas
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
The Self ◽  
Uniformly Convex ◽  
Cyclic Structure ◽  
Best Proximity Points ◽  
Uniformly Convex Banach Spaces

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

scholarly journals Weak convergence theorem for Passty type asymptotically nonexpansive mappings

1999 ◽  
Vol 22 (1) ◽  
pp. 217-220
Author(s):  
B. K. Sharma ◽  
B. S. Thakur ◽  
Y. J. Cho
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Weak Convergence Theorem ◽  
Fréchet Differentiable

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

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 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.

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