scholarly journals Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Chang-He Xiang ◽  
Jiang-Hua Zhang ◽  
Zhe Chen
Keyword(s):  
Fixed Point ◽  
Normed Linear Space ◽  
Iterative Sequence ◽  
Mann Iteration ◽  
Mann Iterative Sequence ◽  
Lipschitzian Mappings ◽  
Nonempty Convex ◽  
Necessary And Sufficient ◽  
Increasing Function ◽  
Real Normed Linear Space

Suppose thatEis a real normed linear space,Cis a nonempty convex subset ofE,T:C→Cis a Lipschitzian mapping, andx*∈Cis a fixed point ofT. For givenx0∈C, suppose that the sequence{xn}⊂Cis the Mann iterative sequence defined byxn+1=(1-αn)xn+αnTxn,n≥0, where{αn}is a sequence in [0, 1],∑n=0∞αn2<∞,∑n=0∞αn=∞. We prove that the sequence{xn}strongly converges tox*if and only if there exists a strictly increasing functionΦ:[0,∞)→[0,∞)withΦ(0)=0such thatlimsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.

Fixed points of nearly weak uniformly L-Lipschitzian mappings in real Banach spaces

2018 ◽  
Vol 27 (1) ◽  
pp. 63-70
Author(s):  
Adesanmi Alao Mogbademu ◽  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Unique Fixed Point ◽  
Stability Result ◽  
Iteration Scheme ◽  
Lipschitzian Mapping ◽  
Nonempty Convex Subset ◽  
Lipschitzian Mappings ◽  
Nonempty Convex

Let K be a nonempty convex subset of a real Banach space X. Let T be a nearly weak uniformly L-Lipschitzian mapping. A modified Mann-type iteration scheme is proved to converge strongly to the unique fixed point of T. Our result is a significant improvement and generalization of several known results in this area of research. We give a specific example to support our result. Furthermore, an interesting equivalence of T-stability result between the convergence of modified Mann-type and modified Mann iterations is included.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

scholarly journals A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Songnian He ◽  
Wenlong Zhu
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Boundary Point ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Method ◽  
Minimum Norm ◽  
Mann Iteration ◽  
Boundary Point Method

LetHbe a real Hilbert space andC⊂H a closed convex subset. LetT:C→Cbe a nonexpansive mapping with the nonempty set of fixed pointsFix(T). Kim and Xu (2005) introduced a modified Mann iterationx0=x∈C,yn=αnxn+(1−αn)Txn,xn+1=βnu+(1−βn)yn, whereu∈Cis an arbitrary (but fixed) element, and{αn}and{βn}are two sequences in(0,1). In the case where0∈C, the minimum-norm fixed point ofTcan be obtained by takingu=0. But in the case where0∉C, this iteration process becomes invalid becausexnmay not belong toC. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of Tand prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projectionPC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

scholarly journals Strong Convergence Theorems for Asymptotically WeakG-Pseudo-Ψ-Contractive Non-Self-Mappings with the Generalized Projection in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Yuanheng Wang
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Approximation Method ◽  
Generalized Projection ◽  
Iterative Sequence ◽  
Successive Approximation Method ◽  
Convergence Theorems ◽  
Mann Iterative Sequence ◽  
Sequence Method ◽  
Generalized Projection Method

A new concept of the asymptotically weakG-pseudo-Ψ-contractive non-self-mappingT:G↦Bis introduced and some strong convergence theorems for the mapping are proved by using the generalized projection method combined with the modified successive approximation method or with the modified Mann iterative sequence method in a uniformly and smooth Banach space. The proof methods are also different from some past common methods.

scholarly journals On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 513-527
Author(s):  
JENJIRA PUIWONG ◽  
◽  
SATIT SAEJUNG ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Convergence Theorems ◽  
Convex Metric Spaces ◽  
Strict Fixed Point ◽  
Point Set

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

scholarly journals On Stability of Iterative Sequences with Error

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 765 ◽  
Author(s):  
Abed ◽  
Taresh
Keyword(s):  
Fixed Point ◽  
Unique Fixed Point ◽  
Contractive Mapping ◽  
Systems Of Equations ◽  
Operator Equations ◽  
Mann Iteration ◽  
Accretive Mappings ◽  
Non Linear Systems ◽  
Roots Of Equations ◽  
Linear Systems Of Equations

Iterative methods were employed to obtain solutions of linear and non-linear systems of equations, solutions of differential equations, and roots of equations. In this paper, it was proved that s-iteration with error and Picard–Mann iteration with error converge strongly to the unique fixed point of Lipschitzian strongly pseudo-contractive mapping. This convergence was almost F-stable and F-stable. Applications of these results have been given to the operator equations Fx=f and x+Fx=f, where F is a strongly accretive and accretive mappings of X into itself.

New Fixed Point Theorems via Contraction Mappings in Complete Intuitionistic Fuzzy Normed Linear Space

2018 ◽  
Vol 15 (01) ◽  
pp. 65-83
Author(s):  
Nabanita Konwar ◽  
Ayhan Esi ◽  
Pradip Debnath
Keyword(s):  
Fixed Point ◽  
Mathematical Models ◽  
Linear Space ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Existence Of A Solution ◽  
Linear Spaces ◽  
Contraction Mappings ◽  
Normed Linear Spaces ◽  
Intuitionistic Fuzzy

Contraction mappings provide us with one of the major sources of fixed point theorems. In many mathematical models, the existence of a solution may often be described by the existence of a fixed point for a suitable map. Therefore, study of such mappings and fixed point results becomes well motivated in the setting of intuitionistic fuzzy normed linear spaces (IFNLSs) as well. In this paper, we define some new contraction mappings and establish fixed point theorems in a complete IFNLS. Our results unify and generalize several classical results existing in the literature.

scholarly journals The Split Common Fixed Point Problem for Quasi-Total Asymptotically Nonexpansive Uniformly Lipschitzian Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Jing Na ◽  
Lin Wang ◽  
Zhaoli Ma
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Hilbert Spaces ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Lipschitzian Mappings ◽  
Common Fixed Point Problem ◽  
Asymptotically Nonexpansive ◽  
Uniformly Lipschitzian Mapping ◽  
Split Common Fixed Point

We introduce an algorithm for solving the split common fixed point problem for quasi-total asymptotically nonexpansive uniformly Lipschitzian mapping in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results.

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