A Product Space with the Fixed Point Property

A topological space is said to have the fixed point property if every map (i.e. continuous function) from the space to itself leaves at least one point fixed. There is a rather well known conjecture (see e.g. [3]) asking if the fact that X and Y have the fixed point property implies that the product space X X Y has the property. This note contains an affirmative answer in the case that X and Y are compact ordered sets (with the order topology, of course). It is perhaps worth noting that there is an analogous result due to Ginsburg [1] for similarity transformations on ordered sets with the product ordered lexicographically. Since a compact ordered space has the fixed point property if and only if it is connected, we may rephrase our result to read as follows. If X and Y are compact connected ordered spaces and Z =X X Y, then Z has the fixed point property. We first establish several auxiliary properties of the space Z.

In 1979, Goebel and Kuczumow showed that very large class of non-weak* compact, closed, bounded and convex subsets of \(\ell ^1\) has the fixed point property (FPP) for nonexpansive mappings. Later, in 2008, Lin proved that \(\ell ^1\) can be renormed to have FPP for nonexpansive mappings. \(c_0\)-analogue of Lin’s result is still open. However, renorming \(c_0\), we prove that Goebel and Kuczumow analogy can be proved under affinity condition. That is, we prove that there exist a renorming of \(c_0\) and a very large class of non-weakly compact, closed, bounded and convex subsets of \(c_0\) with FPP for affine nonexpansive mappings whereas Dowling, Lennard and Turett proved in 2004 that weak compactness is equivalent to FPP for nonexpansive mappings when \(c_0\) is considered with its usual norm instead of any equivalent norm.

In this paper the preservation of the fixed point property under UV decompositions is studied. It is shown that if K is an n-dimensional complex with the fixed point property and G is U V n − 1 U{V^{n - 1}} decomposition of K, then K/G also will have the fixed point property. Furthermore, if X is a compact metric space with the fixed point property, and G is a U V n U{V^n} decomposition of X such that X/G may be embedded in a suitably small Euclidian space, R m {R^m} , then X/G retains the fixed point property.

Examples are given of polyhedra K and L which have the homotopy invariant fixed point property, in the sense that all polyhedra of the same homotopy type have the fixed point property (in fact K and L have no self-maps of zero Lefschetz number) but for which K X L fails to have the fixed point property.

An explosion point space without the fixed point property

We show that the l1 sum of the van Dulst space with itself possesses the fixed point property, although it fails most of the known conditions that imply (FPP). Mathematics Subject Classification: Primary, 46B20; 47H09.

We consider the fixed point property (FPP) in an ordered set of width two (every antichain contains at most two elements). The necessary condition of the FPP and a number of equivalent conditions to the FPP in such sets is established. The product theorem is proved, as well.

Chapter 31 - An update on the elusive fixed-point property

We prove that any group with the fixed point property actually leaves fixed points for measurable actions rather than only jointly continuous actions.

A Banach space X is said to have the fixed point property if for each nonexpansive mapping on a bounded closed convex subset E of X has a fixed point. We show that each infinite dimensional Abelian complex Banach algebra X satisfying: (i) property (A) defined in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010), (ii) for each such that for each , (iii) does not have the fixed point property. This result is a generalization of Theorem 4.3 in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010). MSC:46B20, 46J99.

Fourier–Stieltjes algebras and measure algebras on compact right topological groups

LetP, Q be ordered sets and leta∈P. IfP \ {a} is a retract ofP and setsP and {x∈P:x>p} (or its dual) have the fixed point property then, for each chain complete setP,P×Q has the fixed point property if and only if (P\{a})×Q has this property. This establishes the fixed point property for some products of ordered sets which are beyond the reach of all known product theorems.

We prove that every Banach space X with characteristic of uniform convexity less than 2 has the fixed point property whenever X satisfies a certain orthogonality condition.

Let ℤ6 act differentiably on ℝ n without fixed points. Such actions exist according to the constructions of E. E. Floyd and J. Kister. Extend this action to S n by adding a point at infinity. If we could choose an open invariant acyclic set S, then the complement C, which would also be acyclic, could not have the fixed point property. In particular, there exists no closed slice which is a generalized n-cell. (A generalized n-cell is an orientable, compact acyclic n-cm with boundary.) To see the last remark, it is only necessary to observe that the closure of the complement of the generalized cell would be a generalized n-cell. But a generalized cell does have the fixed point property.

In this paper we study the following problem (Problem 4.2 in, I.A. Rus, Sets with structure, mappings and fixed point property: fixed point structures, Fixed Point Theory 23, No. 2 (2022), 689-706): Let (𝒰, 𝒮, M ) be a fixed point structure on (𝒰, M ). We suppose that (𝒰, M ) is with cartesian product. In which conditions, we have that: X , Y ∈𝒮⇒ X × Y ∈𝒮? Some results in terms of exponential with fixed point property are also given. The basic results are illustrated by examples.