Strengthened fixed point property and products in ordered sets

Strengthened fixed point property for ordered sets is formulated. It is weaker than the strong fixed point property due to Duffus and Sauer and stronger than the product property meaning that A × Y has the fixed point property whenever A has the former and Y has the latter. In particular, doubly chain complete ordered sets with no infinite antichain have the strengthened fixed point property whenever they have the fixed point property, which yields a transparent proof of the well-known theorem saying that doubly chain complete ordered sets with no infinite antichain have the product property whenever they have the fixed point property. The new proof does not require the axiom of choice. Presented at the Summer School on General Algebra and Ordered Sets, Malá Morávka, 4–10 September 2005.     

Compact groups and fixed point sets

Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.

     

Absolute fixed point sets for continuum-valued maps

The notion of an absolute fixed point set in the setting of continuum-valued maps will be defined and characterized.

     

Absolute fixed point sets for multi-valued maps

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.

     

Retracts, fixed point property and existence of periodic points

Let X be a metric space, f ∈ C⁰( X), andV ⊂X. The set-trajectory ( V, f( V), …,f ⁿ (V)) is investigated and some conditions for f to have periodic points with given periods are obtained.     

b-度量空间中压缩型映象的公共不动点定理

在完备的b-空间中,建立一个压缩型映象,研究公共不动点的存在性和唯一性,得到了一个新的公共不动点定理.举例证明了在b-空间中的新结果,改进了当前文献中的结果.     

Renorming of and the fixed point property

For any , let P_k denote the natural projections on ℓ₁. Let |||||| be an equivalent norm of ℓ₁ that satisfies all of the following four conditions:

(1) There are α>4 and a positive (decreasing) sequence (α_n) in (0,1) such that for any normalized block basis {f_n} of (ℓ₁,||||||) and xℓ₁ with P_k−1(x)=x and |||x|||<α_k,

(2) There are two strictly decreasing sequences {β_k} and {γ_k} with

such that for any normalized block basis {f_n} of (ℓ₁,||||||) and x with (I−P_k)(x)=x,

(3) For any , I−P_k=1.

(4) The unit ball of (ℓ₁,||||||) is σ(ℓ₁,c₀)-closed.

In this article, we prove that the space (ℓ₁,||||||) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.

Virtually stable maps and their fixed point sets

We introduce the concept of virtually stable selfmaps of Hausdorff spaces, which generalizes virtually nonexpansive selfmaps of metric spaces introduced in the previous work by the first author, and explore various properties of their convergence sets and fixed point sets. We also prove that the fixed point set of a virtually stable selfmap satisfying a certain kind of homogeneity is always star-convex.     

A combinatorial proof of a fixed point property

A class of finite simplicial complexes, called pseudo cones, is developed that has a number of useful combinatorial properties. A partially ordered set is a pseudo cone if its order complex is a pseudo cone. Pseudo cones can be constructed from other pseudo cones in a number of ways. Pseudo cone ordered sets include finite dismantlable ordered sets and finite truncated noncomplemented lattices. The main result of the paper is a combinatorial proof of the fixed simplex property for finite pseudo cones in which a combinatorial structure is constructed that relates fixed simplices to one another. This gives combinatorial proofs of some well known non-constructive results in the fixed point theory of finite partially ordered sets.     

On the super fixed point property in product spaces

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for nonexpansive mappings, then F⊕X has the super fixed point property with respect to a large class of norms including all lp norms, 1?p<∞. This provides a solution to the “super-version” of the problem of Khamsi (1989).     

Jungck＇s Common Fixed Point Theorem and E.A Property

We prove that （E.A） property buys the required containment of range of one mapping into the range of other in common fixed point considerations up to a pair of mappings. While proving our results, we utilize the idea of implicit functions due to Popa, keeping in view their unifying power.     

Coupled fixed point results for mappings without mixed monotone property in partially ordered G-metric spaces

In this paper, we prove coupled fixed point results for mappings without mixed monotone property in partially ordered G-metric spaces. Also we showed that if $(X\text{,}G)$ is regular, these fixed point results holds.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Nielsen fixed point theory for partially ordered sets

In this paper, we introduce a Nielsen type number for any selfmap f of a partially ordered set of spaces. This Nielsen theory relates to various existing Nielsen type fixed point theories for different settings such as maps of pairs of spaces, maps of triads, fibre preserving maps, equivariant maps and iterates of maps, by exploring their underlying poset structures.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.     

Weak separation properties for self-similar sets

We develop a theory for self-similar sets in that fulfil the weak separation property of Lau and Ngai, which is weaker than the open set condition of Hutchinson.

     

Fixed point theorems for multifunctions having KKM property on almost convex sets

In this paper, for two nonempty subsets X and Y of a linear space E, we define the class KKM(X,Y) and investigate the fixed point problem for T∈KKM(X,X) with X an almost convex subset of a locally convex space. Our fixed point theorem contains Lassonde fixed point theorem for Kakutani factorizable multifunctions as special case.     

Weak compactness and fixed point property for affine mappings

It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M>1 in the case of c₀, the class of affine mappings which are uniformly Lipschitzian with some constant in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.     

A combinatorial Lefschetz fixed point formula

We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdⁿK →K is expanding directions preserving.