Weak chain-completeness and fixed point property for pseudo-ordered sets

In this paper the notion of weak chain-completeness is introduced for pseudo-ordered sets as an extension of the notion of chain-completeness of posets (see [3]) and it is shown that every isotone map of a weakly chain-complete pseudo-ordered set into itself has a least fixed point.     

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 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.     

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.

     

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.     

Multifunctions and the fixed point property for products of ordered sets

Sufficient conditions for the fixed point property for products of two partially ordered sets are proved. These conditions are formulated in terms of multifunctions (functions with non-empty sets as values).     

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.     

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.

     

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.     

Fixed point theorems in ordered abstract spaces

We extend some fixed point theorems in -spaces, obtaining extensions of the Banach fixed point theorem to partially ordered sets.

     

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.

     

The fixed point property in ordered sets of width two

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.     

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.     

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).     

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.     

Some fixed point results without monotone property in partially ordered metric-like spaces

The purpose of this paper is to obtain the fixed point results for F-type contractions which satisfies a weaker condition than the monotonicity of self-mapping of a partially ordered metric-like space. A fixed point result for F-expansive mapping is also proved. Therefore, several well known results are generalized. Some examples are included which illustrate the results.     

Banach空间中一类序压缩映射的不动点定理

在Banach空间中,利用迭代方法,研究了满足一定条件的序压缩算子的一些性质,获得了一类序压缩映射的不动点定理,证明了相应的结果,推广和改进了原有的结论,使其应用范围更加广泛.     

Some fixed point theorems on ordered cone metric spaces

In the present work, two fixed point theorems for self maps on ordered cone metric spaces are proved motivated by [7, L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468–1476] and [15, A. C. M. Ran and M. C. B. Reuring, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132, (2004), 1435–1443]