Fixed points in products of ordered spaces

132, utilizes an example of a l-dimensional arcwise connected continuum X in 3 with the fixed point property.His question (5) asks if XXI has the fixed point property.The answer is yes, and the proof given uses standard techniques of point-set topology.A recent paper of Bing [l] contains a proof that the topological space X shown in the figure has the fixed point property (FPP), where X contains two sin 1/x curves c and / starting from c and/ respectively with closures cU [d, e] and fVJ [a, b], and an "expanding spiral" h starting at h, with closure h\j[a, c]\Jc^J[d, f\Uf.All the a A /\ ^--^19

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.

Based on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP.

The fixed point property of an M-retract and its applications

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.

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

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 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 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.

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

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.

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.