Fixed point structures on a set-mapping pair and cartesian product

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

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

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.

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.

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.

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

It is well known that the propositional modal logic $\mathbf{GL}$ of provability satisfies the de Jongh-Sambin fixed-point property. On the other hand, Montagna showed that the predicate modal system $\mathbf{QGL}$, which is the natural variant of $\mathbf{GL}$, loses the fixed-point property. In this paper, we discuss some versions of the fixed-point property for predicate modal logics. First, we prove that several extensions of $\mathbf{QGL}$ including $\mathbf{NQGL}$ do not have the fixed-point property. Secondly, we prove the fixed-point theorem for the logic $\mathbf{QK} + \Box^{n+1} \bot$. As a consequence, we obtain that the class $\mathsf{FH}$ of Kripke frames which are transitive and finite height satisfies the fixed-point property locally. We also show the failure of the Craig interpolation property for $\mathbf{NQGL}$. Finally, we give a sufficient condition for formulas to have a fixed-point in $\mathbf{QGL}$.

Weak* fixed point property of reduced Fourier–Stieltjes algebra and generalization of Baggett's theorem

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.

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

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

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