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.

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 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.

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.

Two-step iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings

2015 ◽  
Vol 08 (03) ◽  
pp. 1550060
Author(s):  
Amit Singh ◽  
R. C. Dimri ◽  
Darshana J. Prajapati
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Iterative Approximation ◽  
Convergence Theorems ◽  
Strong And Weak Convergence

In this paper, we study an iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in a uniformly convex Banach space.

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 On Picard–Krasnoselskii Hybrid Iteration Process in Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-5 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Kifayat Ullah ◽  
Junaid Ahmad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Numerical Example ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Krasnoselskii Iteration ◽  
Iteration Processes

In this research, we prove strong and weak convergence results for a class of mappings which is much more general than that of Suzuki nonexpansive mappings on Banach space through the Picard–Krasnoselskii hybrid iteration process. Using a numerical example, we prove that the Picard–Krasnoselskii hybrid iteration process converges faster than both of the Picard and Krasnoselskii iteration processes. Our results are the extension and improvement of many well-known results of the 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).

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

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 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.

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