Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem

1981 ◽  
Vol 24 (3) ◽  
pp. 351-357 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Convex Set ◽  
Necessary Conditions ◽  
Unique Fixed Point

AbstractKakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

scholarly journals Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Luo Yi Shi ◽  
Ru Dong Chen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Subset ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Viscosity Approximation Methods

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT(0) spaceX. Suppose that the set Fix(T)of fixed points ofTis nonempty. For a contractionfonCandt∈(0,1), letxt∈Cbe the unique fixed point of the contractionx↦tf(x)⊕(1-t)Tx. We will show that ifXis a CAT(0) space satisfying some property, then{xt}converge strongly to a fixed point ofTwhich solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn+1=αnf(xn)⊕(1-αn)Txnforn≥1, where{αn}⊂(0,1). It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofTwhich solves some variational inequality.

scholarly journals Inner composition of analytic mappings on the unit disk

1991 ◽  
Vol 14 (2) ◽  
pp. 221-226 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Unique Fixed Point ◽  
Iteration Theory ◽  
Basic Theorem ◽  
Closed Unit Disk ◽  
Compositional Structure ◽  
Compositional Structures

A basic theorem of iteration theory (Henrici [6]) states thatfanalytic on the interior of the closed unit diskDand continuous onDwithInt(D)f(D)carries any pointz ϵ Dto the unique fixed pointα ϵ Doff. That is to say,fn(z)→αasn→∞. In [3] and [5] the author generalized this result in the following way: LetFn(z):=f1∘…∘fn(z). Thenfn→funiformly onDimpliesFn(z)λ, a constant, for allz ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structuresf1∘…∘fn(z)where thefj's are analytic onInt(D)and continuous onDwithInt(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

scholarly journals Common fixed point theorems for contractive mappings satisfying Φ-maps in S-metric spaces

2016 ◽  
Vol 8 (2) ◽  
pp. 298-311 ◽  
Author(s):  
Shaban Sedghi ◽  
Mohammad Mahdi Rezaee ◽  
Tatjana Došenović ◽  
Stojan Radenović
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Contractive Mappings ◽  
Weakly Compatible ◽  
Common Fixed Point Theorems

Abstract In this paper we prove the existence of the unique fixed point for the pair of weakly compatible self-mappings satisfying some Ф-type contractive conditions in the framework of S-metric spaces. Our results generalize, extend, unify, complement and enrich recently fixed point results in existing literature.

The Equivariant Grothendieck Groups of the Russell-Koras Threefolds

2001 ◽  
Vol 53 (1) ◽  
pp. 3-32 ◽  
Author(s):  
J. P. Bell
Keyword(s):  
Fixed Point ◽  
Tangent Space ◽  
Unique Fixed Point ◽  
Linear Action ◽  
Grothendieck Groups ◽  
Smooth Affine

AbstractThe Russell-Koras contractible threefolds are the smooth affine threefolds having a hyperbolic *-action with quotient isomorphic to the corresponding quotient of the linear action on the tangent space at the unique fixed point. Koras and Russell gave a concrete description of all such threefolds and determined many interesting properties they possess. We use this description and these properties to compute the equivariant Grothendieck groups of these threefolds. In addition, we give certain equivariant invariants of these rings.

A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

1994 ◽  
Vol 37 (4) ◽  
pp. 552-555 ◽  
Author(s):  
Juris Steprans ◽  
Stephen Watson ◽  
Winfried Just
Keyword(s):  
Fixed Point ◽  
Hausdorff Space ◽  
Unique Fixed Point ◽  
Banach Fixed Point Theorem ◽  
Hausdorff Spaces ◽  
Compact Metric Space ◽  
Banach Contraction ◽  
Compact Metrizable Space ◽  
Admissible Metric ◽  
The Banach Contraction Principle

AbstractWe propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

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