Fixed Point Property Research Articles

Proximate fixed point property and operations

Klee introduced the proximate fixed point property for compacta which is stronger than fixed point property. We consider relations between proximate fixed point property of spaces being result of application of different operations to continua. As an application we show this property for products, cones, suspensions and joins of span 0 continua.

Temporal Distinctiveness in Task Switching: Assessing the Mixture-Distribution Assumption

In task switching, increasing the response - cue interval has been shown to reduce the switch cost. This has been attributed to a time-based decay process influencing the activation of memory representations of tasks (task-sets). Recently, an alternative account based on interference rather than decay has been successfully applied to this data (Horoufchin et al., 2011a). In this account, variation of the RCI is thought to influence the temporal distinctiveness (TD) of episodic traces in memory, thus affecting their retrieval probability. This can affect performance as retrieval probability influences response time: If retrieval succeeds, responding is fast due to positive priming; if retrieval fails, responding is slow, due to having to perform the task via a slow algorithmic process. This account - and a recent formal model (Grange & Cross, 2015) - makes the strong prediction that all RTs are a mixture of one of two processes: a fast process when retrieval succeeds, and a slow process when retrieval fails. The present paper assesses the evidence for this mixture-distribution assumption in TD data. In a first section, statistical evidence for mixture-distributions is found using the fixed-point property test. In a second section, a mathematical process model with mixture-distributions at its core is fitted to the response time distribution data. Both approaches provide good evidence in support of the mixture-distribution assumption, and thus support temporal distinctiveness accounts of the data.

The fixed point property in every weak homotopy type

We prove that for any connected compact CW-complex $K$ there exists a space $X$ weak homotopy equivalent to $K$ which has the fixed point property, that is, every continuous map $X\to X$ has a fixed point. The result is known to be false if we require $X$ to be a polyhedron. The space $X$ we construct is a non-Hausdorff space with finitely many points.

On strong property (T) and fixed point properties for Lie groups

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.

Symplectic Tate homology

For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical map $\kappa \colon \underrightarrow{H}\underleftarrow{T}(W) \to \underleftarrow{H}\underrightarrow{T}(W)$. Our geometric approach to Tate homology uses the moduli space of finite energy gradient flow lines of the Rabinowitz action functional for a circle in the complex plane as a classifying space for $S^1$-equivariant Tate homology. For rational coefficients the symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ has the fixed point property and is therefore isomorphic to $H(W;\mathbb{Q}) \otimes \mathrm{Q}[u,u^{-1}]$, where $\mathbb{Q}[u,u^{-1}]$ is the ring of Laurent polynomials over the rationals. Using a deep theorem of Goodwillie, we construct examples of Liouville domains where the canonical map $\kappa$ is not surjective and examples where it is not injective.

Geometric properties of the second-order Cesàro spaces

We prove that, for any p∈(1,∞), the second-order Cesaro sequence space Ces2(p) has the (β)-property and the k-NUC property for k≥2. In addition, we show that Ces2(p) has the Kadec–Klee, rotundity, and uniform convexity properties. For any positive integer k, we also investigate the uniform Opial and (L) properties of the sequence space. We also establish that Ces2(p) is reflexive and has the fixed-point property. Finally, we calculate the packing constant (C) of the space.

Some results of approximate fixed points in fuzzy Banach spaces

In this paper, first we introduce some aspects of approximate fixed points, such as approximate fixed point property for subsets of a fuzzy Banach space and approximate eigenvalues of a self mapping on a fuzzy normed space. Then as their consequences, we prove more results for Fixed Point Theory on Fuzzy Banach and Banach Spaces.

Is There Evidence for a Mixture of Processes in Speed-Accuracy Trade-Off Behavior?

The speed-accuracy trade-off (SAT) effect refers to the behavioral trade-off between fast yet error-prone respones and accurate but slow responses. Multiple theories on the cognitive mechanisms behind SAT exist. One theory assumes that SAT is a consequence of strategically adjusting the amount of evidence required for overt behaviors, such as perceptual choices. Another theory hypothesizes that SAT is the consequence of the mixture of multiple categorically different cognitive processes. In this paper, these theories are disambiguated by assessing whether the fixed-point property of mixture distributions holds, in both simulations and data. I conclude that, at least for perceptual decision making, there is no evidence for a mixture of different cognitive processes to trade off accuracy of responding for speed.

FIXED POINT THEOREMS FOR DIGITAL IMAGES

In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

Normal structure and invariance of Chebyshev center under isometries

A normed space is said to have the weak fixed point property (WFPP) if every nonexpansive self map on a weakly compact convex set has a fixed point. Kirk proved that if a normed space has normal structure, then it has WFPP. It is known that if a normed space is uniformly convex in every direction (UCED), then it has normal structure. Also known is that every normed space that is uniformly convex in all but countably many directions has normal structure. We show that a normed space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a countable union of n-dimensional subspaces of X for some positive integer n. We also show that in such a space, the Chebyshev center C(K) of a weakly compact convex set K is a common invariant set for the collection of all isometries that map K into K and also that there is a common fixed point in C(K) for this collection of maps. This was previously known to be true only in the case of a normed space that is UCED. Another observation made in this paper is that a Banach space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a linear subspace with a countable Hamel basis.

Nonexpansive mappings on complex C*-algebras and their fixed points

A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E $ on a nonempty bounded closed convex subset $ E $ of $ mathfrak{X} $ has a fixed point. In this paper, we first show that if $ X $ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (iii) $C_0 (X)$ does not have the fixed point property. We also show that if $A$ is a commutative complex $ mathsf{C}^star$--algebra with nonempty carrier space, then the following statements are equivalent: (i) Carrier space of $ A $ is infinite, (ii) $ A $ is infinite dimensional, (iii) $ A $ does not have the fixed point property. Moreover, we show that if $ A $ is an infinite complex $ mathsf{C}^star$--algebra (not necessarily commutative), then $ A $ does not have the fixed point property.

On generalized García-Falset coefficient in Musielak-Orlicz sequence spaces

We introduce a new geometric coefficient which is related Garcia-Falset coefficient and weak star fixed point property. The Garcia-Falset coefficient that was introduced by Benavides in (Houst. J. Math. 22:835-849, 1996) is calculated in this paper for Musielak-Orlicz sequence spaces equipped with the Luxemburg norm. Specifically, in reflexive Banach spaces, the new geometric coefficient and the Garcia-Falset coefficient are the same.

Common endpoints of generalized weak contractive mappings via separation theorem with applications

In this paper, first, we give the separation theorem which is an extension of the separation theorem due to Jachymski and Joźwik (J. Math. Anal. Appl. 300:147-159, 2004). Then, by using this and the related results, we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property. This result is an extension and correct version of the main result given by Khojasteh and Rakocevic (Appl. Math. Lett. 25:289-293, 2012).

Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times

We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For $\varepsilon >0$ small, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as $\varepsilon \searrow0$. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that essentially depends only on the power exponent associated to the fixed point. Such random variables, or laws, have therefore a universal character and they arise of course in a variety of contexts. We then obtain quantitative sharp estimates, notably tail properties, on these universal laws.

The fixed point property and unbounded sets in spaces of negative curvature

Motivated by the well-known cases of the real Hilbert ball and complete ℝ-trees, both being particular cases of CAT(-1) spaces, we give an affirmative answer to the question of whether the geodesic boundedness property is a necessary and sufficient condition for a closed convex subset K of a complete CAT(κ) space, with κ < 0, to have the fixed point property for nonexpansive mappings.

Fixed point properties for semigroups of nonlinear mappings on unbounded sets

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T: C→C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c∈C with a bounded orbit and assuming that certain subspace of Cb(S) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5]. In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi.

The fixed point property and the Opial condition on tree-like Banach spaces

We introduce some new tree-like Banach spaces, belonging to the class of separable Banach spaces not containing $\ell_1$ with non-separable dual, each one of which satisfies the following: $(1)$~the space has the fixed point property and $(2)$~the space does not satisfy the Opial condition. In addition, one of these spaces contains subspaces isomorphic to $c_0$, whose Banach-Mazur distance from $c_0$ becomes arbitrarily large.

The τ-fixed point property for left reversible semigroups

In this article we use the generalized Gossez-Lami Dozo property and the Opial condition to study the fixed point property for left reversible semigroups in separable Banach spaces. As a consequence, some previous results will be deduced and new examples of Banach spaces satisfying the fixed point property for left reversible semigroups are shown. We will also extend some previous theorems when we consider the semigroup formed by a unique nonexpansive mapping and its iterates.

Finite-dimensional invariant subspace property and amenability for a class of Banach algebras

Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author. We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra.

A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability

AbstractIn this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.