Super-Relational Fixed Point Property and Products in Ordered Sets

Super-relational fixed point property for ordered sets is formulated. It is weaker than the strengthened fixed point property and stronger than the product property. Presented at the Summer School on General Algebra and Ordered Sets, Radějov, 3–9 September 2006.     

Examples of Powers of Ordered Sets with the Fixed Point Property

This note presents examples of non-dismantlable ordered sets P with the fixed point property such that P ^A with A an arbitrary antichain and P ^C with C being a four crown have the fixed point property.     

Fixed Point Property for Finite Ordered Sets that Contain No Crowns with 6 or More Elements

Order - We prove that, for a finite ordered set P that contains no crowns with 6 or more elements, it can be determined in polynomial time if P has the fixed point property. This result is obtained...     

Neocompact Sets and the Fixed Point Property

We use a newly introduced concept of neocompactness to study problems from metric fixed point theory. In particular, we give a sufficient condition for a superreflexive Banach space X to have the fixed point property and obtain shorter proofs of some well-known results in that theory.     

Fixed Point Theorems on Chain Complete Partially Ordered Sets

In this paper, we extend the results of Tarski, Abian and Brown on the completeness of lattices by fixed point theorems, to partially ordered sets or (Po sets) with chain completeness property. We also sharpen some of the existing results on completeness of lattices and fixed-point theorems. We also provide counter examples to some of the claims of R. Taskovic to disprove his theorem.Milan R. Taskovic claims to have proved this theorem but his proof is based on a false argument. In fact most of the contentions of Milan R. Taskovic in [4] are factually incorrect. Actually, reviewer of Taskovics paper has pointed out some of his inaccurate observations. See Mathematical Reviews No. 90b:06007, 06A10.     

序压缩映射的不动点定理

本文在序Banach空间中引入了几种按序压缩的压缩型映射,证明了相应的不动点定理.     

G-spaces and Fixed Point Sets

For a compact connected group G, in a compact free G-space M every closed subset is the fixed point set of an autohomeomorphism if the weight of M is not bigger than that of G. In a compact group with nonsingleton components, every closed subset is the fixed point set of an autohomeomorphism if and only if the weight of the whole group is not bigger than that of the component of the identity.     

序压缩映射的随机不动点定理

在Banach空间中利用一个随机Mann迭代序列组,讨论了随机映射的随机不动点的存在性问题,得出了几个随机不动点定理,改进了相关文献中的相应结果.     

The Zig-Zag Property and Exponential Cancellation of Ordered Sets

This paper outlines a chapter in the theory of ordered sets that concerns decompositions of ordered sets using various natural operations introduced by Garrett Birkhoff in 1940 – chiefly, the operations of sum (disjoint union), Cartesian product, and exponentiation. This area has seen two brief, four-year flowerings of research activity during the past sixty-four years. Ivan Rival, working with several collaborators around 1978, obtained beautiful results on exponentiation which attracted other researchers to this topic. Altogether, about a dozen people contributed a chain of significant new results between 1978–1982. The subject then lay dormant until a major contribution by Jonathan D. Farley appeared in 1996. This pre-figured the solution, four years later, of a central problem about exponentiation of finite ordered sets that Birkhoff had formulated in 1942. The theory of operations on ordered sets is a beautiful chapter of mathematics. We concentrate on those portions that are most relevant for the solution of the Birkhoff problem, offering (hopefully) accessible proofs of many results, including some new results. We would like to see a new renaissance of activity in this area, which continues to offer many attractive open problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Fixed Point Property for Posets of Small Width

The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly. The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved.     

Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

In this paper, we first show that for every mapping $f$ from a metric space $Ω$ to itself which is continuous off a countable subset of $Ω,$ there exists a nonempty closed separable subspace $S ⊂ Ω$ so that $f|_S$ is again a self mapping on $S.$ Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of $c_0(\Gamma)$ has a fixed point.     

不动点集和零点集公共元的强弱收敛性

在Hilbert空间中,为了找到渐近严格伪压缩映射的不动点集,极大单调算子与逆强单调映射和的零点集的公共元,文中引进两种迭代格式,在某些条件下得到迭代序列的强弱收敛定理.     

Fixed Point Curve for Weakly Inward Contractions and Approximate Fixed Point Property

In this paper, we discuss the concept of fixed point curve for linear interpolations of weakly inward contractions and establish necessary condition for a nonex- pansive mapping to have approximate fixed point property.     

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution. Research partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN     

Banach空间的L性与弱不动点性

该文将证明如果X是Banach空间,X     

序Banach空间中的随机不动点定理

该文获得了序Banach空间中随机序压缩映射存在不动点的充要条件.利用随机Mann迭代序列,给出了几个随机不动点收敛定理,改进了最近文献的相应结果.     

本原代换的非周期不动点的Hausdorff维数函数

设A={0,1,…,N-1},ξ是A上的一个本原代换,u是ξ的非周期不动点.记     

序Banach空间中集值混合单调映象的耦合不动点定理

在序Banach空间中,研究了一类集值混合单调映象,用不同方法证明了两个新的耦合不动点存在性定理,所做工作扩充了文[5]的研究成果.     

Order Extensions and the Fixed Point Property

The purpose of this paper is to investigate how the fixed point property and its negation behave when a covering relation is added to the order. We prove that every finite ordered set which is not totally ordered and which is dismantlable by retractables, respectively by irreducibles, has an upper cover (in its extension lattice) which is also dismantlable by retractables, respectively by irreducibles. We also provide examples of finite ordered sets having the fixed point property so that none of their upper covers has the fixed point property. Part of this work was done while the author was visiting Brandon University. The author thanks M. Roddy for his hospitality and financial support.     

半序锥度量空间中新公共不动点定理(英文)

该文在半序锥度量空间中研究了有关三个映射的公共不动点的存在唯一性,不要求映射的连续性和交换性,也不要求锥的正规性,其结果改进并推广了文献中的一些重要结论.