scholarly journals Properties of the solutions of those equations for which the Krasnoselskii iteration converges

2012 ◽  
Vol 28 (2) ◽  
pp. 329-336
Author(s):  
IOAN A. RUS ◽  
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Data Dependence ◽  
Picard Operator ◽  
Nonempty Convex Subset ◽  
Krasnoselskii Iteration ◽  
Nonempty Convex ◽  
Weakly Picard Operator

Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.

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 Some remarks on the location of fixed points

1984 ◽  
Vol 37 (3) ◽  
pp. 358-365
Author(s):  
Michael Edelstein ◽  
Daryl Tingley
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Chebyshev Center ◽  
Weakly Compact ◽  
Asymptotic Center

AbstractSeveral procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

scholarly journals An application of KKM-map principle

1992 ◽  
Vol 15 (4) ◽  
pp. 659-661 ◽  
Author(s):  
A. Carbone
Keyword(s):  
Convex Subset ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Compact Convex ◽  
Nonempty Convex Subset ◽  
Nonempty Convex ◽  
Coincidence Theorems ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset ◽  
Kkm Map

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. LetCbe a nonempty convex subset of a normed linear spaceX,f:C→Xa continuous function,g:C→Ccontinuous, onto and almost quasi-convex. Assume thatChas a nonempty compact convex subsetDsuch that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖   for   all   x∈D}is compact.Then there is a pointy0∈Csuch that‖g(y0)−f(y0)‖=d(f(y0),C).

scholarly journals Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-Like Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vatan Karakaya ◽  
Faik Gürsoy ◽  
Kadri Doğan ◽  
Müzeyyen Ertürk
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Data Dependence ◽  
Iterative Schemes ◽  
Iterative Processes ◽  
Space Setting

We intend to establish some results on the data dependence of fixed points of certain contractive-like operators for the multistep and CR iterative processes in a Banach space setting. One of our results generalizes the corresponding results of Soltuz and Grosan (2008) and Chugh and Kumar (2011).

scholarly journals A new minimax theorem and a perturbed James's theorem

2002 ◽  
Vol 66 (1) ◽  
pp. 43-56 ◽  
Author(s):  
M. Ruiz Galán ◽  
S. Simons
Keyword(s):  
Bilinear Form ◽  
Convex Subset ◽  
Direct Proof ◽  
Minimax Theorem ◽  
Locally Convex Space ◽  
Sufficient Condition ◽  
Nonempty Convex Subset ◽  
Nonempty Convex ◽  
Locally Convex ◽  
Special Case

The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y, where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of James's (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstine's theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

scholarly journals Construction of sunny nonexpansive retractions in Banach spaces

2002 ◽  
Vol 66 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Tomas Dominguez Benavides ◽  
Genaro López Acedo ◽  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convex Subset ◽  
Regularity Condition ◽  
Common Fixed Points ◽  
Sunny Nonexpansive Retraction ◽  
Uniformly Smooth Banach Space ◽  
Uniformly Smooth ◽  
Commutative Family

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.

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.

scholarly journals COMMON FIXED POINTS FOR SEMIGROUPS OF POINTWISE LIPSCHITZIAN MAPPINGS IN BANACH SPACES

2011 ◽  
Vol 84 (3) ◽  
pp. 353-361 ◽  
Author(s):  
W. M. KOZLOWSKI
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convex Subset ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Lipschitzian Mappings ◽  
Nonlinear Mappings

AbstractLet C be a bounded, closed, convex subset of a uniformly convex Banach space X. We investigate the existence of common fixed points for pointwise Lipschitzian semigroups of nonlinear mappings Tt:C→C, where each Tt is pointwise Lipschitzian. The latter means that there exists a family of functions αt:C→[0,∞) such that $\|T_t(x)-T_t(y)\| \leq \alpha _{t}(x)\|x-y\|$ for x,y∈C. We also demonstrate how the asymptotic aspect of the pointwise Lipschitzian semigroups can be expressed in terms of the respective Fréchet derivatives.

scholarly journals Stability And Data Dependence Results For The Mann Iteration Schemes on n-Banach Space

2020 ◽  
pp. 1456-1460
Author(s):  
Mustafa Mohamed Hamed ◽  
Zeana Zaki Jamil
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Data Dependence ◽  
Mann Iteration ◽  
The Stability

Let  be an n-Banach space, M be a nonempty closed convex subset of , and S:M→M be a mapping that belongs to the class  mapping. The purpose of this paper is to study the stability and data dependence results of a Mann iteration scheme on n-Banach space

scholarly journals Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space

2007 ◽  
Vol 38 (1) ◽  
pp. 85-92 ◽  
Author(s):  
G. S. Saluja
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Convex Subset ◽  
Convergence Theorem ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Strong Convergence Theorem

In this paper, we study strong convergence of common fixed points of two asymptotically quasi-nonexpansive mappings and prove that if $K$ is a nonempty closed convex subset of a real Banach space $E$ and let $ S, T\colon K\to K $ be two asymptotically quasi-nonexpansive mappings with sequences $ \{u_n\}$, $\{v_n\}\subset [0,\infty) $ such that $ \sum_{n=1}^{\infty}u_n

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