Locally Accretive Mappings in Banach Spaces

Let X be a real Banach space for which the closed unit ballhas the fixed point property for nonexpansive self-mappings.Suppose that D is a bounded open subset of X, and T is a continuousmapping from the closure of D into X and locally accretive onD. Then T has a zero in D, provided that the following boundarycondition is fulfilled: there exists an element z in D so that||Tz|| < ||Tx|| for all x in the boundary of D.     

Omega Limit Sets in Positive Cones

The omega limit sets of a nonlinear operator T which is definedon a positive cone and satisfies certain ray-contractive typeconditions are discussed. Under the assumption that the restrictionof T to a compact subset is surjective, the following alternativesare proved: the omega limit set of a point in the cone eitherconsists of a fixed point or forms a 2-cycle. In addition, newproofs and extensions to relevant results are given. 2000 MathematicsSubject Classification 47H07, 47H09.     

Generalized Alexandroff-Urysohn squares and a characterization of the fixed point property

Given a Hausdorff continuum X, we introduce a topology on X×X that yields a Hausdorff continuum. We call the resulting space the Alexandroff-Urysohn square of X and prove that X has the fixed point property if and only if the Alexandroff-Urysohn square of X has the fixed point property.     

Weyl's Theorem, a-Weyl's Theorem, and Local Spectral Theory

Necessary and sufficient conditions are given for a Banach spaceoperator with the single-valued extension property to satisfyWeyl's theorem and a-Weyl's theorem. It is shown that if T orT^* has the single-valued extension property and T is transaloid,then Weyl's theorem holds for f(T)for every fH((T)). When T^*has the single-valued extension property, T is transaloid andT is a-isoloid, then a-Weyl's theorem holds for f(T) for everyfH((T)). It is also proved that if T or T^* has the single-valuedextension property, then the spectral mapping theorem holdsfor the Weyl spectrum and for the essential approximate pointspectrum.     

Almost isometric actions,property (<Emphasis Type="Italic">T</Emphasis>), and local rigidity

Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].One definition of property (T) is that a group Γ has property (T) if every isometric Γ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property (T). Some of these are used in the proofs of our local rigidity theorems. To Yakov G. Sinai on his 70th birthday     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

The Generic Local Spectrum of Any Operator is The Full Spectrum

Let T=(T₁, ..., T_n) be a commuting n-tuple of bounded linearoperators acting on some complex Banach space X. We show thatif T has the single-valued extension property, then the localspectrum _T(x) coincides with the spectrum (T), for all vectorsx X, except on a set of the first Baire category. 1991 MathematicsSubject Classification 47A11, 47A13.     

Block-size determination in fractured porous media

A flow-reversal method involving the injection and retrievalof fluid from a fractured, porous medium at a different temperatureto that of the system in its undisturbed state is describedand assessed as a potential technique for measuring the thermaltransients of blocks of fractured porous media. Formulae areobtained for the zeroth- and first-order time moments of thethermal history of the fluid when it is well mixed in the injection-retrievalhole, injected at a temperature T₁ (different to the uniformtemperature T₀ of the undisturbed system) at a fixed rate Q,for a time t₁ and then withdrawn through the same drill holeat the same fixed rate Q. These formulae have a potential fordetermining some geometric parameters of the block structure,including the block surface-to-volume ratio and the mean actiontime for conductive heating.     

On lifting properties for confluent mappings

Known results about lifting of paths for covering, light open and light confluent mappings are in some sense extended for all confluent mappings with the domain being a continuum having the arc property of Kelley. As an application we prove that each confluently tree-like continuum has the fixed point property.

     

Essential Equivariant Maps and Borsuk-Ulam Theorems

A full characterization is given of those compact Lie groupsG with the property that every G-map XX on a finite-dimensionalG-complex X of finite orbit type, X^G = Ø, is (non-equivariantly)essential. For arbitrary G, conditions are given on the G-spaceX which guarantee this property. Finally, conditions are givenfor the non-existence of a G-map XY inducing a homotopy equivalenceX^GY^G on the fixed point sets. These results have applicationsto critical point theory of almost G-invariant functionals.