scholarly journals Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

2002 ◽  
Vol 31 (4) ◽  
pp. 251-257 ◽  
Author(s):  
Wei-Shih Du ◽  
Young-Ye Huang ◽  
Chi-Lin Yen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Finite Union ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Nonconvex Sets ◽  
Firmly Nonexpansive ◽  
Asymptotically Regular

It is shown that every asymptotically regular orλ-firmly nonexpansive mappingT:C→Chas a fixed point wheneverCis a finite union of nonempty weakly compact convex subsets of a Banach spaceXwhich is uniformly convex in every direction. Furthermore, if{T i}i∈Iis any compatible family of strongly nonexpansive self-mappings on such aCand the graphs ofT i,i∈I, have a nonempty intersection, thenT i,i∈I, have a common fixed point inC.

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 Convergence of Ishikawa iterates of generalized nonexpansive mappings

1997 ◽  
Vol 20 (3) ◽  
pp. 517-520 ◽  
Author(s):  
M. K. Ghosh ◽  
L. Debnath
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Strictly Convex ◽  
Strictly Convex Banach Spaces

This paper is concerned with the convergence of Ishikawa iterates of generalized nonexpansive mappings in both uniformly convex and strictly convex Banach spaces. Several fixed point theorems are discussed.

scholarly journals Iterated nonexpansive mappings in Hilbert spaces

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Bozena Piatek
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Normal Structure ◽  
Point Theory ◽  
Affirmative Answer ◽  
Weakly Compact ◽  
Special Case

AbstractIn [T. Dominguez Benavides and E. Llorens-Fuster, Iterated nonexpansive mappings, J. Fixed Point Theory Appl. 20 (2018), no. 3, Paper No. 104, 18 pp.], the authors raised the question about the existence of a fixed point free continuous INEA mapping T defined on a closed convex and bounded subset (or on a weakly compact convex subset) of a Banach space with normal structure. Our main goal is to give the affirmative answer to this problem in the very special case of a Hilbert space.

scholarly journals Fixed Point Theorems for Noncyclic Monotone Relatively ρ-Nonexpansive Mappings in Modular Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Nour-eddine El Harmouchi ◽  
Karim Chaira ◽  
El Miloudi Marhrani
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Modular Spaces

In this paper, we obtain some fixed point results for noncyclic monotone ρ-nonexpansive mappings in uniformly convex modular spaces and uniformly convex in every direction modular spaces. As an application, we prove the existence of the solution of an integral equation.

scholarly journals Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

2003 ◽  
Vol 2003 (2) ◽  
pp. 83-91 ◽  
Author(s):  
Wieslawa Kaczor
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Pairwise Disjoint ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Nonconvex Sets ◽  
Opial’S Property ◽  
Asymptotically Regular

It is shown that ifXis a Banach space andCis a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets{Ci:1≤i≤n }ofX, and eachCihas the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping ofChas a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.

scholarly journals Some fixed point theorems on non-convex sets

2017 ◽  
Vol 18 (2) ◽  
pp. 377
Author(s):  
Mohanasundaram Radhakrishnan ◽  
S. Rajesh ◽  
Sushama Agrawal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Compact Set ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Weakly Compact Set

<span style="color: #000000;">In this paper, we prove that if </span><span style="color: #008000;">$K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonempty</span><span style="color: #000000;"> weakly compact set in a </span><span style="text-decoration: underline; color: #000000;">Banach</span><span style="color: #000000;"> space </span><span style="color: #008000;">$X$</span><span style="color: #000000;">, </span><span style="color: #008000;">$T:K\to K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonexpansive</span><span style="color: #000000;"> map satisfying </span><span style="color: #008000;">$\frac{x+Tx}{2}\in K$</span><span style="color: #000000;"> for all </span><span style="color: #008000;">$x\in K$</span><span style="color: #000000;"> and if </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> is </span><span style="color: #008000;">$3-$</span><span style="color: #000000;">uniformly convex or </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> has the </span><span style="text-decoration: underline; color: #000000;">Opial</span><span style="color: #000000;"> property, then </span><span style="color: #008000;">$T$</span><span style="color: #000000;"> has a fixed point in </span><span style="color: #008000;">$K.$ <br /></span>

A Set-Valued Generalization of Fan's Best Approximation Theorem

1992 ◽  
Vol 44 (4) ◽  
pp. 784-796 ◽  
Author(s):  
Xie Ping Ding ◽  
Kok-Keong Tan
Keyword(s):  
Fixed Point ◽  
Convex Subset ◽  
Best Approximation ◽  
Fixed Point Theorems ◽  
Approximation Theorem ◽  
Compact Convex ◽  
Convex Approximation ◽  
Weakly Compact ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

AbstractLet (E, T) be a Hausdorff topological vector space whose topological dual separates points of E, X be a non-empty weakly compact convex subset of E and W be the relative weak topology on X. If F: (X, W) → 2(E,T) is continuous (respectively, upper semi-continuous if £ is locally convex), approximation and fixed point theorems are obtained which generalize the corresponding results of Fan, Park, Reich and Sehgal-Singh-Smithson (respectively, Ha, Reich, Park, Browder and Fan) in several aspects.

scholarly journals Iterative Methods for Pseudocontractive Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jong Soo Jung
Keyword(s):  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Initial Value ◽  
Weakly Compact ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm

LetEa reflexive Banach space having a uniformly Gâteaux differentiable norm. LetCbe a nonempty closed convex subset ofE,T:C→Ca continuous pseudocontractive mapping withF(T)≠∅, andA:C→Ca continuous bounded strongly pseudocontractive mapping with a pseudocontractive constantk∈(0,1). Let{αn}and{βn}be sequences in(0,1)satisfying suitable conditions and for arbitrary initial valuex0∈C, let the sequence{xn}be generated byxn=αnAxn+βnxn-1+(1-αn-βn)Txn,  n≥1.If either every weakly compact convex subset ofEhas the fixed point property for nonexpansive mappings orEis strictly convex, then{xn}converges strongly to a fixed point ofT, which solves a certain variational inequality related toA.

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