Fixed Point Property Research Articles

Weak compactness is not equivalent to the fixed point property in c

We show that there exists a non-weakly compact, closed, bounded, convex subset W of the Banach space of convergent sequences (c,‖⋅‖∞), such that every nonexpansive mapping T:W⟶W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space X isomorphic to c0, for which W has the fixed point property for nonexpansive mappings. We also prove that the sets W may be perturbed to a large family of non-weakly compact, closed, bounded, convex subsets Wq of (c,‖⋅‖∞) with the fixed point property for nonexpansive mappings; and we discuss similarities and differences with work of Goebel and Kuczumow concerning analogous subsets of ℓ1.

Uniform convexity, normal structure and the fixed point property of metric spaces

We say that a complete metric space X has the fixed point property if every group of isometric automorphisms of X with a bounded orbit has a fixed point in X. We prove that if X is uniformly convex then the family of admissible subsets of X possesses uniformly normal structure and if so then it has the fixed point property. We also show that from other weaker assumptions than uniform convexity, the fixed point property follows. Our formulation of uniform convexity and its generalization can be applied not only to geodesic metric spaces.

Degrees of maps between locally symmetric spaces

Let X be a locally symmetric space Γ\\G/K where G is a connected non-compact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G, and Γ⊂G is a torsion-free irreducible lattice in G. Let Y=Λ\\H/L be another such space having the same dimension as X. Suppose that real rank of G is at least 2. We show that any f:X→Y is either null-homotopic or it is homotopic to a covering projection of degree an integer that depends only on Γ and Λ. As a corollary we obtain that the set [X,Y] of homotopy classes of maps from X to Y is finite.We obtain results on the (non-)existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X.

The fixed point property for some generalized nonexpansive mappings and renormings

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, Cλ-mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2,5,9,10,15]). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for Cλ-mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for Cλ-mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for Cλ-mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance.

ON ACTION OF LAU ALGEBRAS ON VON NEUMANN ALGEBRAS

Let <TEX>$\mathbb{G}$</TEX> be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that <TEX>$\hat{\mathbb{G}}$</TEX>, the dual of <TEX>$\mathbb{G}$</TEX>, is co-amenable if and only if there is a state <TEX>$m{\in}L^{\infty}(\hat{\mathbb{G}})^*$</TEX> which is invariant under a left module action of <TEX>$L^1(\mathbb{G})$</TEX> on <TEX>$L^{\infty}(\hat{\mathbb{G}})^*$</TEX>. This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.

On super fixed point property and super weak compactness of convex subsets in Banach spaces

For a nonempty convex set C of a Banach space X, a self-mapping on C is said to a linear (respectively, affine) isometry if it is the restriction of a linear (respectively, affine) isometry defined on the whole space X. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a Banach space has super fixed point property for affine (or, equivalently, linear) isometries if and only if it is super weakly compact; and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a Banach space under equivalent renorming sense. With the application of Fabian–Montesinos–Zizler's renorming theorem, we finally show that every strongly super weakly compact generated Banach space can be renormed so that every weakly compact convex set has super fixed point property.

Generalized von Neumann-Jordan constant and its relationship to the fixed point property

We introduce a new geometric constant \(C_{NJ}^{(p)}(X)\) for a Banach space X, called a generalized von Neumann-Jordan constant. Next, it is shown that \(1\leq C_{NJ}^{(p)}(X)\leq2\) for any Banach space X and that the right hand side inequality is sharp if and only if X is uniformly non-square. Moreover, a relationship between the James constant \(J(X)\) and \(C_{NJ}^{(p)}(X)\) is presented. Finally, the generalized von Neumann-Jordan constant of the Lebesgue space \(L_{r}([0,1])\) is calculated and a relationship between \(C_{NJ}^{(p)}(X)\) and the fixed point property is found.

Pseudo-rigidity and products with the fixed-point property

We introduce the concept of pseudo-rigidity and prove that if $$X$$ is a pseudo-rigid continuum, and $$X$$ and $$Y$$ are continua with the fixed-point property, then $$X\times Y$$ has the fixed-point property.

On local fixed or periodic point properties

A space X has the local fixed point property LFPP, (local periodic point property LPPP) if it has an open basis $\mathcal{B}$ such that, for each $B\in \mathcal{B}$, the closure $\overline{B}$ has the fixed (periodic) point property. Weaker versions wLFPP, wLPPP are also considered and examples of metric continua that distinguish all these properties are constructed. We show that for planar or one-dimensional locally connected metric continua the properties are equivalent.

Characterization of compact geodesic spaces

Driven by Klee's well-known work about the topological properties of convex sets in locally convex linear spaces, we give a complete characterization of compact geodesic spaces with curvature bounded below in terms of the fixed point property for continuous functions. Furthermore, we provide an example which highlights the role of the sectional curvature in our result.

A weakly chainable uniquely arcwise connected continuum without the fixed point property

A weakly chainable uniquely arcwise connected continuum without the fixed point property

Non-additivity of the fixed point property for tree-like continua

We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's

Positivity, Betweenness, and Strictness of Operator Means

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality, and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions, and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, operator monotone functions, Borel measures, and certain operator equations.

A property for locally convex*-algebras related to property (T) and character amenability

For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions on a second countable locally compact group $G$ with compact supports, equipped with the convolution $^*$-algebra structure and a certain inductive topology. We show that $(C_c(G), \varepsilon_G)$ has property $(FH)$ if and only if $G$ has property $(T)$. On the other hand, many Banach algebras equipped with canonical characters have property $(FH)$ (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property $(FH)$ and character amenablility, we obtain characterizations of property $(T)$, amenability and compactness of $G$ in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These characterizations can be regarded as analogues of one another. Moreover, we show that $G$ is compact if and only if the normed algebra $\big\{f\in C_c(G): \int_G f(t)dt =0\big\}$ (under $\|\cdot\|_{L^1(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.

Some Topological and Geometric Properties of Some New Spaces ofλ-Convergent and Bounded Series

The main purpose of this study is to introduce the spacescsλ,cs0λ, andbsλwhich areBK-spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spacescs,cs0, andbs, respectively, and derive some inclusion relations. Additionally, Schauder bases of the spacescsλandcs0λhave been constructed and theα-,β-, andγ-duals of these spaces have been computed. Besides, we characterize some matrix classes from the spacescsλ,cs0λ, andbsλto the spaceslp,c, andc0, where1≤p≤∞. Finally, we examine some geometric properties of these spaces as Gurarǐ’s modulus of convexity, propertym∞, property(M), property WORTH, nonstrict Opial property, and weak fixed point property.

Mean Field Stackelberg Games: Aggregation of Delayed Instructions

In this paper, we consider an $N$-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite $N$-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an $\epsilon$-Nash equilibrium for the corresponding finite $N$-player game. Second, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of...

Fixed-point property of random quotients by plain words

We show a fixed-point property of certain random groups for a wide class of CAT(0) spaces. Th e model of random groups under consideration is given as the set of presentations (S, R) , where S is a generating set and the set of relation s R is a subset of the set of all plain words of the same length with suitably xed density. Our main theorem says that, with high probability, groups obtained by such presentations have the fixed-point property for all CAT (0) spaces having bounded singularities.

New families of nonreflexive Banach spaces with the fixed point property

In this paper we define the concept of sequentially separating norm and we relate it to the fulfillment of the fixed point property for nonexpansive mappings. We develop a technique to construct families of nonreflexive Banach spaces with the fixed point property. Different examples will be exhibited. We also study the geometry of the Banach spaces that can be renormed with a sequentially separating norm.

Cohomology of deformations

In this paper we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.

Applications of the Lefschetz Number to Digital Images

The goal of this paper is to develop some applications of the Lefschetz fixed point theorem to digital images. We also deal with relative and reduced Lefschetz fixed point theorem for digital complexes. We give some examples related to the topic. We calculate the degree of the antipodal map for sphere-like digital images using fixed point properties.