Birkhoff-functions that are bounded on prescribed sets

We prove the existence of entire functions which are universal under translations and bounded on certain prescribed sets. It is also shown that the family of all these universal functions is a dense but not a G_δ-subset in the space of entire functions provided with a natural metric. Received: 24 November 2004; revised: 12 April 2005     

Holomorphic germs on certain locally convex spaces

Summary We study the bounded sets in the space of holomorphic germs defined on compact subsets of non-metrizable locally convex spaces. We relate this problem to the problem of existence of uniform Cauchy estimates for the bounded subsets. We show that the space of holomorphic germs defined on a compact subset of a reflexive dual Fréchet space is regular if the bounded subsets of the space of holomorphic germs defined at the origin have uniform Cauchy estimates.     

Fujita exponents for a space–time weighted parabolic problem in bounded domains

This paper deals with a semilinear weighted parabolic problem in general bounded domains, subject to zero Dirichlet boundary conditions, where the weighted functions depend not only on space variable but also on time variable. Fujita exponents for blow-up and global existence of solutions are determined by using semigroup methods and the comparison principle, which are composed by the dimensions of the space domains and the eight exponents in nonlinear coupled sources and the space–time weighted functions.     

On the strong CE-property of convex sets

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

Almost Chebyshev set with respect to bounded subsets

The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively). Project supported by the National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province, and the State Major Key Project for Basic Researchers of China.     

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.     

Common supports for pairs of sets

Sufficient conditions are given for the existence of a closed hyperplane which supports each member of a pair of closed and bounded subsets of a Banach space.     

On Global Attractors of Multivalued Semiflows Generated by the 3D Bénard System

In this paper we prove the existence of solutions for the 3D Bénard system in the class of functions which are strongly continuous with respect to the second component of the vector (that is, the one corresponding to the parabolic equation). We construct then a multivalued semiflow generated by such solutions and obtain the existence of a global φ −attractor for the weak-strong topology. Moreover, a family of multivalued semiflows is defined on suitable convex bounded subsets of the phase space, proving for them the existence of a global attractor (which is the same for every semiflow of the family) for the weak-strong topology.     

A model for operators with cyclic adjoint

It is proved that every bounded linear operator on a complex Banach space whose adjoint has aw ^*-cyclic vector, is similar to the differentiation operator on a Banach space of entire functions of finite exponential type. The relation of this model to the existence of non-trivial invariant subspaces is discussed.     

On Maps Of Bounded p-Variation With p>1

This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Hölder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Hölder continuous of exponent 0<<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection.     

Common supports of families of sets

For a familyA of closed bounded convex subsets of a Banach space, sufficient conditions are given for the existence of a closed hyperplane which supports each member ofA.     

On Order Bounded Subsets of Locally Solid Riesz Spaces

In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.     

Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems

A family of compact and positively invariant sets with uniformly bounded fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the process is constructed. The existence of such a family, called a pullback exponential attractor, is proved for a nonautonomous semilinear abstract parabolic Cauchy problem. Specific examples will be presented in the forthcoming Part II of this work.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

Compact Sets of Holomorphic Mappings

Compactness of locally bounded sets of holomorphic functions with infinite dimensional domains is connected, using Heinrich's density condition, to the Schwartz and semi-Montel properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.     

Some quantities related to monotonicity and bounded variation of functions

The aim of this paper is to introduce and study some quantities connected with monotonicity and bounded variation of real functions. These quantities will also be considered for bounded subsets of spaces of bounded or continuous functions on a compact interval, equipped with the standard supremum norm. Several properties of such quantities are established, and possible applications are briefly sketched. Moreover, some relation with the theory of measures of noncompactness is discussed as well.     

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction-diffusion equations with time-dependent external forces including the case of the FitzHugh-Nagumo system.     

A compactness result for differentiable functions with an application to boundary value problems

We characterize the relative compactness of subsets of the space ${\mathcal{BC}^m([0,+\infty [;E)}$ of bounded and m-differentiable functions defined on [0, +∞[ with values in a Banach space E. Moreover, we apply this characterization to prove the existence of solutions of a boundary value problem in Banach spaces.