scholarly journals Convergence Rate of Some Two-Step Iterative Schemes in Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
O. T. Wahab ◽  
R. O. Olawuyi ◽  
K. Rauf ◽  
I. F. Usamot
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Banach Spaces ◽  
Numerical Approach ◽  
Normed Spaces ◽  
Mann Iteration ◽  
Ishikawa Iteration ◽  
Iterative Schemes ◽  
Approximate Fixed Point

This article proves some theorems to approximate fixed point of Zamfirescu operators on normed spaces for some two-step iterative schemes, namely, Picard-Mann iteration, Ishikawa iteration, S-iteration, and Thianwan iteration, with their errors. We compare the aforementioned iterations using numerical approach; the results show that S-iteration converges faster than other iterations followed by Picard-Mann iteration, while Ishikawa iteration is the least in terms of convergence rate. These results also suggest the best among two-step iterative fixed point schemes in the literature.

scholarly journals Some fixed point theorems in n-normed spaces

2020 ◽  
Vol 25 (3) ◽  
pp. 1-15 ◽  
Author(s):  
Hanan Sabah Lazam ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Normed Space ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Weak Contraction ◽  
Contraction Mappings ◽  
Basic Properties ◽  
Definition Of

In this article, we recall the definition of a real n-normed space and some basic properties. fixed point theorems for types of Kannan, Chatterge, Zamfirescu, -Weak contraction and  - (,)-Weak contraction mappings in  Banach spaces.

scholarly journals An Iteration Scheme for Contraction Mappings with an Application to Synchronization of Discrete Logistic Maps

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Ke Ding ◽  
Jong Kyu Kim ◽  
Qiang Lu ◽  
Bin Du
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Contraction Mapping ◽  
Nonexpansive Mappings ◽  
Positive Influence ◽  
Iteration Scheme ◽  
Contraction Mappings ◽  
Iterative Schemes ◽  
Comparison Results ◽  
The Given

This paper deals with designing a new iteration scheme associated with a given scheme for contraction mappings. This new scheme has a similar structure to that of the given scheme, in which those two iterative schemes converge to the same fixed point of the given contraction mapping. The positive influence of feedback parameters on the convergence rate of this new scheme is investigated. Moreover, the derived convergence and comparison results can be extended to nonexpansive mappings. As an application, the derived results are utilized to study the synchronization of logistic maps. Two illustrated examples are used to reveal the effectiveness of our results.

scholarly journals Onn-normed spaces

2001 ◽  
Vol 27 (10) ◽  
pp. 631-639 ◽  
Author(s):  
Hendra Gunawan ◽  
M. Mashadi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Normed Space ◽  
Normed Spaces

Given ann-normed space withn≥2, we offer a simple way to derive an(n−1)-norm from then-norm and realize that anyn-normed space is an(n−1)-normed space. We also show that, in certain cases, the(n−1)-norm can be derived from then-norm in such a way that the convergence and completeness in then-norm is equivalent to those in the derived(n−1)-norm. Using this fact, we prove a fixed point theorem for somen-Banach spaces.

scholarly journals ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION

2017 ◽  
Vol 96 (1) ◽  
pp. 162-170 ◽  
Author(s):  
SHIN-YA MATSUSHITA
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Proximal Point Algorithm ◽  
Convergence Rates ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Mann Iteration

The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

scholarly journals A New Fixed Point Theorem and a New Generalized Hyers-Ulam-Rassias Stability in Incomplete Normed Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1117
Author(s):  
Maryam Ramezani ◽  
Ozgur Ege ◽  
Manuel De la Sen
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Fixed Point Method ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Normed Spaces ◽  
Rassias Stability

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results.

scholarly journals Fixed-Point Theorems for Mean Nonexpansive Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhanfei Zuo
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniqueness Theorems ◽  
Existence And Uniqueness Theorems ◽  
Approximate Fixed Point

We define a mean nonexpansive mappingTonXin the sense thatTx-Ty≤ax-y+bx-Ty,a,b≥0,a+b≤1. It is proved that mean nonexpansive mapping has approximate fixed-point sequence, and, under some suitable conditions, we get some existence and uniqueness theorems of fixed point.

scholarly journals Fixed Points and the Stability of an AQCQ-Functional Equation in Non-Archimedean Normed Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Choonkil Park
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Method ◽  
Point Method ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Quartic Functional Equation ◽  
The Stability

Using fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equationf(x+2y)+f(x−2y)=4f(x+y)+4f(x−y)−6f(x)+f(2y)+f(−2y)−4f(y)−4f(−y)in non-Archimedean Banach spaces.

scholarly journals Effective Uniform Bounds on the Krasnoselski-Mann Iteration

2000 ◽  
Vol 7 (9) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Convex Sets ◽  
Fixed Point Theory ◽  
Uniformly Convex ◽  
Asymptotic Regularity ◽  
Normed Spaces ◽  
Mann Iteration ◽  
Starting Point

This paper is a case study in proof mining applied to non-effective proofs<br />in nonlinear functional analysis. More specifically, we are concerned with the<br />fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex<br />spaces the rate of convergence is in general not computable in f (which is<br />related to the non-uniqueness of fixed points). However, the iterations yield<br />approximate fixed points of arbitrary quality for general normed spaces and<br />bounded C (asymptotic regularity).<br />In this paper we apply general proof theoretic results obtained in previous<br />papers to non-effective proofs of this regularity and extract uniform explicit<br />bounds on the rate of the asymptotic regularity. We start off with the classical<br />case of uniformly convex spaces treated already by Krasnoselski and show<br />how a logically motivated modification allows to obtain an improved bound. Already the analysis of the original proof (from 1955) yields an elementary<br />proof for a result which was obtained only in 1990 with the use of the deep<br />Browder-G¨ohde-Kirk fixed point theorem. The improved bound from the modified<br /> proof gives applied to various special spaces results which previously had<br />been obtained only by ad hoc calculations and which in some case are known<br />to be optimal.<br />The main section of the paper deals with the general case of arbitrary normed<br />spaces and yields new results including a quantitative analysis of a theorem<br />due to Borwein, Reich and Shafrir (1992) on the asymptotic behaviour of<br />the general Krasnoselski-Mann iteration in arbitrary normed spaces even for unbounded sets C. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform<br />bounds which do not depend on the starting point of the iteration and the<br />nonexpansive function and the normed space X and, in fact, only depend<br />on the error epsilon, an upper bound on the diameter of C and some very general information on the sequence of scalars k used in the iteration. Even non-effectively only the existence of bounds satisfying weaker uniformity conditions was known before except for the special situation, where lambda_k := lambda is constant. For the unbounded case, no quantitative information was known so far.

scholarly journals Strong convergence of the Ishikawa iteration for Lipschitz α-hemicontractive mappings

2015 ◽  
Vol 53 (1) ◽  
pp. 151-161
Author(s):  
Micah Okwuchukwu Osilike ◽  
Anthony Chibuike Onah
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iteration Scheme ◽  
Iteration Algorithm ◽  
Mann Iteration ◽  
Ishikawa Iteration ◽  
New Class ◽  
Hemicontractive Mappings ◽  
Demicontractive Mappings

Abstract A new class of α-hemicontractive maps T for which the strong convergence of the Ishikawa iteration algorithm to a fixed point of T is assured is introduced and studied. The study is a continuation of a recent study of a new class of α-demicontractive mappings T by L. Mărușter and Ș. Mărușter, Mathematical and Computer Modeling 54 (2011) 2486-2492 in which they proved strong convergence of the Mann iteration scheme to a fixed point of T. Our class of α-hemicontractive maps is more general than the class of α-demicontractive maps. No compactness assumption is imposed on the operator or it’s domain, and no additional requirement is imposed on the set of fixed points.

scholarly journals Fixed-point iteration processes for non-Lipschitzian mappings of asymptotically quasi-nonexpansive type

2003 ◽  
Vol 2003 (33) ◽  
pp. 2075-2081 ◽  
Author(s):  
Daya Ram Sahu ◽  
Jong Soo Jung
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Iteration ◽  
Ishikawa Iteration ◽  
Lipschitzian Mappings ◽  
Iteration Processes

We study convergences of Mann and Ishikawa iteration processes for mappings of asymptotically quasi-nonexpansive type in Banach spaces.

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