Some results in asymptotic field point theory

In 1944, Levinson ([22]) introduced the concept of dissipativeness for a map T in a finite-dimensional space which leads to the existence of a fixed point of some iterate T ⁿ for n large, rather than a fixed point of T. Browder ([3]) gave an asymptotic field point theorem which proved that T itself had a field point. Although Browder’s result was a big step, it was not suitable for hyperbolic PDEs and neutral functional differential equations because, in those cases, the map T is not compact. For α-contraction maps the result was extended by Nussbaum ([25]) and Hale and Lopes ([13]) using different methods. In this paper, we review these ideas and some more recent applications. Dedicated to Felix Browder on the occasion of his 80th birthday     

Viscosity approximation methods for nonexpansive multimaps in Banach spaces

We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition ＂Tz = {z}＂ on the map T.     

Dynamics of acyclic interval maps

We investigate conditions under which a map f in a possibly non-compact interval is acyclic— the only periodic orbits are fixed points. Several earlier results are generalized to maps with multiple fixed points. The chief tools are convergence results due to Coppel and Sharkovski, and the Schwarzian derivative. Illustrative examples are given and open problems are suggested.

He who can digest a second or third fluxion . . . need not, methinks, be squeamish about any point in divinity. ―Bishop George Berkeley, “The Analyst,” 1734

     

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).     

Positively expansive differentiable maps

We study the space of positively expansive differentiable maps of a compact connected C ∞ Riemannian manifold without boundary. It is proved that （i） the C1-interior of the set of positively expansive differentiable maps coincides with the set of expanding maps, and （ii） Cl-generically, a differentiable map is positively expansive if and only if it is expanding.     

Multiplicities of fixed points of holomorphic maps in several complex variables

Let Δ^υ be the unit ball in ℂ^υ with center 0 (the origin of υ) and let F:Δ^υ→ℂ^υbe a holomorphic map withF(0) = 0. This paper is to study the fixed point multiplicities at the origin 0 of the iteratesF ⁱ =F∘⋯∘F (i times),i = 1,2,.... This problem is easy when υ = 1, but it is very complicated when υ > 1. We will study this problem generally.     

Boolean transformations with unique fixed points

We determine the Boolean transformations F: B ⁿ → B ⁿ which have unique fixed points.      

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S² V S² V ... →N. The main result is that the images of these renormalized maps converge in L^1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.     

Fixed point sets of deformations of polyhedra with local cut points

A locally finite simplicial complex is said to be 2-dimensionally connected if is connected. Such spaces exhibit ``classical' behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi's results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall's theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz's result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.

     

Subalgebras of the Stanley—Reisner Ring

In [2], Billera proved that the R -algebra of continuous piecewise polynomial functions (C ⁰ splines) on a d -dimensional simplicial complex Δ embedded in R ^d is a quotient of the Stanley—Reisner ring A _Δ of Δ. We derive a criterion to determine which elements of the Stanley—Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C ^r _k (Δ) is upper semicontinuous in the Zariski topology. Using the criterion, we give an algorithm for obtaining the defining equations of the set of vertex locations where the dimension jumps. Received June 2, 1997, and in revised form December 22, 1997, and March 24, 1998.     

Fixed point of contractive mappings in generalized metric spaces

We prove a fixed point theorem for contractive mappings of Boyd and Wong type in generalized metric spaces, a concept recently introduced in [BRANCIARI, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31–37].      

The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.     

Examples of expandingC 1 maps having no σ-finite invariant measure equivalent to Lebesgue

In this paper we construct aC ¹ expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on higher dimensional tori and the Riemann sphere. We also discuss recurrence of Lebesgue measure for the family of tent maps. Supported by the Deutsche Forschungsgemeinschaft (DFG). The research was carried out while HB was employed at the University of Erlangen-Nürnberg, Germany. Partially supported by NSF grant DMS # 9203489.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

Trace and Duality in Symmetric Monoidal Categories

Traces taking values in suitable ‘Hochschild complexes’ are defined in the context of symmetric monoidal categories and applied to various categories of chain complexes, simplicial abelian groups, and symmetric spectra. Topological applications to parametrized fixed point theory are given. (Received: October 2003) Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m.     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.     

A Short Simplicial h -Vector and the Upper Bound Theorem

   Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.