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.     

Realization of fixed point sets

Letf:XX be a selfmap of a compact connected polyhedron, andA a nonempty closed subset ofX. In this paper, we shall deal with the question whether or not there is a mapg:XX homotopic tof such that the fixed point set Fixg ofg equalsA. We introduce a necessary condition for the existence of such a mapg. It is shown that this condition is easy to check, and hence some sufficient conditions are obtained.Partially supported by the Natural Science Foundation of Liaoning University.     

An example in holomorphic fixed point theory

If is the open unit ball in the Cartesian product furnished with the -norm , where and , then a holomorphic self-mapping of has a fixed point if and only if for some

     

Asymptotic Lefschetz fixed point theory for ANES(compact) maps

A number of new Lefschetz fixed point theorems are established for ANES(compact) maps. Also compact absorbing contraction maps are discussed.      

Equivariant Nielsen fixed point theory for n-valued maps

We develop an equivariant Nielsen fixed point theory for n-valued G-maps by associating (as in Better (2010) [2]) an abstract simplicial complex to any equivariant n-valued map and defining, in terms of this complex, two n-valued continuous G-homotopy invariants that are lower bounds for the number of fixed points and of orbits in the n-valued continuous G-homotopy class of a given n-valued G-map. We also provide an equivariant Hopf construction for n-valued G-maps as well as a minimality result for the Nielsen numbers introduced in this setting.     

The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

In this paper we study, the Reidemeister zeta function. We prove rationality and functional equations of the Reidemeister zeta function of an endomorphism of finite group. We also obtain these results for eventually commutative endomorphisms. These results are applied to the theory of Reidemeister and Nielsen numbers of self-maps of topological spaces. Our method is to identify the Reidemeister number of a group endomorphism with the number of fixed points in the unitary dual. As a consequence, we show that the Reidemeister torsion of the mapping torus of the unitary dual is a special value of the Reidemeister zeta function. We also prove certain congruences for Reidemeister numbers which are equivalent to a Euler product formula for the Reidemeister zeta function. The congruences are the same as those found by Dold for Lefschetz numbers.     

Some problems in metric fixed point theory

Three papers, published coincidentally and independently by Felix Browder, Dietrich G?hde, and W. A. Kirk in 1965, triggered a branch of mathematical research now called metric fixed point theory. This is a survey of some of the highlights of that theory, with a special emphasis on some of the problems that remain open. Dedicated to Felix Browder on the occasion of his 80th birthday     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

不动点大类与 Nielsen数

通过引进 π１ （ Ｘ,ｘ０ ） 的同态 ｆπ 的不动子群 Ｆｉｘｆπ,在 Ｈ ＝ Ｆｉｘｆπ ·ｋｅｒｆ π为π１ （ Ｘ,ｘ０ ） 的正规子群时定义了不动点大类,得到不动点大类数是有限的．在曲面 Ｘ的正则复迭空间 Ｘ Ｈ 为有限叶时只要姜子 群相对于 Ｈ 极大,ｆ 的 Ｎｉｅｌｓｅｎ 数就可以用一维同调群计算     

On the fixed point sets of smooth involutions on the products of spheres

In this paper, we have, under some conditions on cohomology, that the fixed point set of a smooth involution on a product of spheres is of constant dimension.

     

Chebyshev centers and fixed point theorems

Brodskii and Milman proved that there is a point in C(K)$C(K)$, the set of all Chebyshev centers of K, which is fixed by every surjective isometry from K into K whenever K   is a nonempty weakly compact convex subset having normal structure in a Banach space. Motivated by this result, Lim et al. raised the following question namely “does there exist a point in C(K)$C(K)$ which is fixed by every isometry from K into K?”. In fact, Lim et al. proved that “if K is a nonempty weakly compact convex subset of a uniformly convex Banach space, then the Chebyshev center of K is fixed by every isometry T from K into K”. In this paper, we prove that if K   is a nonempty weakly compact convex set having normal structure in a strictly convex Banach space and F$\mathfrak{F}$ is a commuting family of isometry mappings on K   then there exists a point in C(K)$C(K)$ which is fixed by every mapping in F$\mathfrak{F}$.     

The Nielsen Method For Groups Acting on Trees

We use Nielsen methods to study generating sets of subgroupsof groups that act on simplicial trees and give several applications.In particular, we exhibit an explicit bound for the complexityof acylindrical splittings of a finitely generated group interms of its rank. This is applied to JSJ-splittings of word-hyperbolicgroups and 3-manifolds. As a last application we construct examplesof amalgamated products that show that there exists no non-trivialrank formula for amalgamated products. 2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.     

A forcing relation of braids from Nielsen fixed point theory

In this paper, we focus our attention on the connections between the braid group and Nielsen fixed point theory. A new forcing relation between braids is introduced, and we show that it can be fulfilled by using Nielsen fixed point theory.     

An example of application of the Nielsen theory to integro-differential equations

A new nontrivial example of an application of the Nielsen fixed-point theory is presented, this time, to integro-differential equations. The emphasis is on the parameter space so that no subdomain becomes invariant under the related solution (Hammerstein) operator. Thus, at least three (harmonic) periodic solutions are established to a planar integro-differential system.

     

A unifying model based on retraction for fixed point algorithms

We present a unifying model based on retraction for several restart fixed point algorithms. The model embraces the interpretation of the algorithms in terms of stationary point problem by van der Laan and Talman and fully explains the 2-ray method.     

A renorming in some Banach spaces with applications to fixed point theory

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(⋅)} which satisfy some special conditions. We define an equivalent norm ?⋅? on X such that if C is a convex bounded closed subset of (X,?⋅?) which is τ-relatively sequentially compact, then every nonexpansive mapping T:C→C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier-Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G=T, we recover P.K. Lin's renorming in the sequence space ?₁. Moreover, we give new norms in ?₁ with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L₁(μ) can be renormed to have the FPP.     

Permanence of two species and fixed point index

We present sufficient and necessary conditions for the permanence of discrete systems in the plane based on an index of fixed points on convex sets. In concrete models, a simple picture is sufficient to deduce whether our system is permanent or not.     

A quantitative version of Kirk's fixed point theorem for asymptotic contractions

In [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650], W.A. Kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings. Using techniques from proof mining, we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem.     

A Schauder-type fixed point theorem

In this brief note we study Schauder's second fixed point theorem in the space $(\mathit{BC},6\cdot 6)$ of bounded continuous functions $\varphi :[0,\infty )\to {\Re }^n$ with a view to reducing the requirement that there is a compact map to the requirement that the map is locally equicontinuous. Several examples are given, both motivating and applying the theory.     

Some geometric properties in modular spaces and application to fixed point theory

We prove that modular spaces L_ρ have the uniform Kadec-Klee property w.r.t. the convergence ρ-a.e. when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial condition w.r.t. the convergence ρ-a.e. for both the Luxemburg norm and the Amemiya norm. Some assumptions over the modular ρ need to be assumed. The above geometric properties will enable us to obtain some fixed point results in modular spaces for different kind of mappings.