AN ITERATIVE SCHEME FOR FIXED POINT PROBLEMS

2021 ◽  
Vol 10 (5) ◽  
pp. 2295-2316
Author(s):  
F. Akutsah ◽  
O. K. Narain ◽  
K. Afassinou ◽  
A. A. Mebawondu
Keyword(s):  
Fixed Point ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Satisfying Condition ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Examples ◽  
Convergence Results ◽  
The Stability

In this paper, we introduce a new three steps iteration process, prove that our newly proposed iterative scheme can be used to approximate the fixed point of a contractive-like mapping and establish some convergence results for our newly proposed iterative scheme generated by a mapping satisfying condition (E) in the framework of uniformly convex Banach space. In addition, with the aid of numerical examples, we established that our newly proposed iterative scheme is faster than the iterative process introduced by Ullah et al., [26], Karakaya et al., [16], Abass et. al. [1] and some existing iterative scheme in literature. More so, the stability of our newly proposed iterative process is presented and we also gave some numerical examples to display the efficiency of our proposed algorithm.

scholarly journals Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 273
Author(s):  
Mujahid Abbas ◽  
Muhammad Waseem Asghar ◽  
Manuel De la Sen
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Contraction Mapping ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Examples ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Fractional Differential

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

scholarly journals Fixed Point Approximation for a Class of Generalized Nonexpansive Mappings in Hadamard Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Akbar Ali Khan ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Satisfying Condition ◽  
Iteration Process ◽  
Hadamard Space ◽  
Point Approximation ◽  
Convergence Results

In this paper, we establish strong and Δ convergence results for mappings satisfying condition B γ , μ through a newly introduced iterative process called JA iteration process. A nonlinear Hadamard space is used the ground space for establishing our main results. A novel example is provided for the support of our main results and claims. The presented results are the good extension of the corresponding results present in the literature.

scholarly journals A New Modified Fixed-Point Iteration Process

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3109
Author(s):  
Chanchal Garodia ◽  
Afrah A. N. Abdou ◽  
Izhar Uddin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Weak Convergence ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Fixed Point Iteration Process

In this paper, we present a new modified iteration process in the setting of uniformly convex Banach space. The newly obtained iteration process can be used to approximate a common fixed point of three nonexpansive mappings. We have obtained strong and weak convergence results for three nonexpansive mappings. Additionally, we have provided an example to support the theoretical proof. In the process, several relevant results are improved and generalized.

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 Convergence of Common Random Fixed Point of Finite Family of Asymptotically Quasi-Nonexpansive-Type Mappings by an Implicit Random Iterative Scheme

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
A. S. Saluja ◽  
Pankaj kumar Jhade
Keyword(s):  
Banach Space ◽  
Iterative Scheme ◽  
Sufficient Conditions ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Random Fixed Point ◽  
Random Iteration ◽  
Necessary And Sufficient

We introduce a new implicit random iteration process generated by a finite family of asymptotically quasi-nonexpansive-type mappings and study necessary and sufficient conditions for the convergence of this process in a uniformly convex Banach space. The results presented in this paper extend and improve the recent ones announced by Plubtieng et al. (2007), Beg and Thakur (2009), and Saluja and Nashine (2012).

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 Approximation of the Fixed Point for Unified Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 26
Author(s):  
Hassan Almusawa ◽  
Hasanen A. Hammad ◽  
Nisha Sharma
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Iterative Schemes ◽  
Convergence Results

In this manuscript, a new three-step iterative scheme to approximate fixed points in the setting of Busemann spaces is introduced. The proposed algorithms unify and extend most of the existing iterative schemes. Thereafter, by making consequent use of this method, strong and Δ-convergence results of mappings that satisfy the condition (Eμ) in the framework of uniformly convex Busemann space are obtained. Our results generalize several existing results in the same direction.

scholarly journals Common Fixed Points for Weak and Strong Convergence Results

2016 ◽  
Vol 4 (2) ◽  
pp. 340
Author(s):  
S. C. Shrivastava
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Convergence Results

<div><p> <em>In this paper, we study the approximation of common fixed points for more general classes of mappings through weak and strong convergence results of an iterative scheme in a uniformly convex Banach space. Our results extend and improve some known recent results.</em></p></div>

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 Fixed Point Approximation of Nonexpansive Mappings on a Nonlinear Domain

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Safeer Hussain Khan
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hyperbolic Space ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
Uniformly Convex Hyperbolic Space

We use a three-step iterative process to prove some strong andΔ-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.

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