Subdifferentials of a minimum time function in Banach spaces

In general Banach space setting, we study the minimum time function determined by a closed convex set K and a closed set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function.     

A Geometric Characterization of Viable Sets for Controlled Degenerate Diffusions

For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a deterministic second-order normal cone to K and the data of the diffusion process. We also give an extension to stochastic differential games.     

The dual Radon–Nikodym property for finitely generated Banach <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-modules

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual \(X^\star \) has the RNP iff X does not contain a closed subspace isomorphic to \(\ell ^1\).     

Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces

In this paper, we generalize the concept of exceptional family of elements for a completely continuous field from Hilbert spaces to Banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a Banach space B to the dual space B^∗ by applying the generalized projection operator πK and by using the Leray-Schauder type alternative.     

Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

We consider the viability problem for nonlinear evolutions inclusions of the form u′(t)?∈?Au(t)?+?F(u(t)), where A is an m-dissipative (possible nonlinear and multi-valued) operator acting in a Banach space X, K?X is a nonempty, locally closed set and $F:K{\user1{ \rightsquigarrow }}X$ is with nonempty, convex, closed and bounded values. We define the concept of A-quasi-tangent set to K at a given point ξ?∈?K and we prove a necessary condition for C ⁰-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the necessary condition is also sufficient. We extend the results to the quasi-autonomous case, we deal with the existence of noncontinuable or even global C ⁰-solutions and, as applications, we deduce a comparison result and a sufficient condition for null controllability.     

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.     

Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type

We will show that an uniform treatment yields Wiener–Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/K. The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze–Stein phenomenon for noncompact semisimple Lie groups.     

A note in approximative compactness and midpoint locally k-uniform rotundity in Banach spaces

In this article, we prove the following results: (1) A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; (2) A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.     

Uncountable equilateral sets in Banach spaces of the form <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.     

Large sublattices in subsets of Banach lattices

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.     

On a characterization of Arakelian sets

Let K be a compact set in the complex plane, such that its complement in the Riemann sphere, (? ∪ {∞}) / K, is connected. Also, let U ? ? be an open set which contains K. Then there exists a simply connected open set V ? ? such that K ? V ? U. We show that if K is replaced by a closed set F ? ?, then the preceding result is equivalent to the fact that F is an Arakelian set in ?. This holds in more general case when ? is replaced by any simply connected open set Ω ? ?. In the case of an arbitrary open set Ω ? ?, the above extends to the one point compactification of Ω. If we do not require (? ∪ {∞}) /K to be connected, we can demand that each component of (? ∪ {∞}) / V intersects a prescribed set A containing one point in each component of (? ∪ {∞}) / K. Using the previous result, we prove that again if we replace K by a closed set F, the latter is equivalent to the fact that F is a set of uniform meromorphic approximation with poles lying entirely in A.     

Orbifold index cobordism invariance

We prove cobordism index invariance for pseudo-differential elliptic operators on closed orbifolds with K-theoretical methods.     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

On support points and support functionals of convex sets

Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set Σ₁(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set Σ ₁(K) is always uncountable, which answers a question posed by L. Zajíček.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.     

Stochastic control and compatible subsets of constraints

Given a stochastic differential control system and a closed set K in Rn, we study the that, with probability one, the associated solution of the control system remains for ever in the set K. This set is called the viability kernel of K. If N is equal to the whole set K, K is said to be viable. We prove that, in the general case, the viability kernel itself is viable and we characterize it through some partial differential equations. We prove that, under suitable assumptions, also the boundary of N is viable. As an application, we give a new characterization of the value function of some optimal control problem.     

On the Metric Projection Operator and Its Applications to Solving Variational Inequalities in Banach Spaces

In this paper, we investigate the characteristics of the metric projection operator P _K : B → K, where B is a Banach space with dual space B?, and K is a nonempty closed convex subset of B. Then we apply its properties to study the existence of solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.     

Approximation of fixed points and variational solutions for pseudo-contractive mappings in banach spaces

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ?-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.     

Small set in a large box

Let K ? ?^d be a compact convex set which is an intersection of half-spaces defined by at most two coordinates. Let Q be the smallest axes-parallel box containing K. We show that, as the dimension d grows, the ratio diam Q/ diam K can be arbitrarily large. We also give examples of compact sets in Banach spaces which are not contained in any compact contractive set.