Stability of Jensen functional equation on semigroups

In this paper we introduce a Jensen type functional equation on semigroups and study the Hyers-Ulam stability of this equation. It is proved that every semigroup can be embedded into a semigroup in which the Jensen equation is stable.     

Musielak-Orlicz-Sobolev空间关于最大值范数的端点

自20世纪以来,Sobolev空间作为有着重要价值的数学模型而受到广泛关注.它在偏微分方程中有着非常重要的作用.而Musielak-Orlicz-Sobolev空间是将Sobolev空间中的L~p(Ω)空间推广到Musielak-Orlicz空间L_M(Ω)之后形成的空间.因而Musielak-Orlicz-Sobolev空间具有Musielak-Orlicz空间和Sobolev空间中的一些性质.讨论了赋最大值范数的Musielak-Orlicz-Sobolev空间端点的性质.这里的最大值范数指:最大值Luxemburg范数和最大值Amemiya-Orlicz范数.主要得到了Musielak-Orlicz-Sobolev空间关于最大值范数端点的充分条件,并指出这类空间都不是严格凸的.     

On the minimal time function for distributed control systems in Banach spaces

In this paper, it is shown that the minimal time function is locally Lipschitz continuous for the control systemx=Ax+u in a Banach spadeE, under either of two conditions:A is linear and generates aC ₀-semigroup of bounded linear operators; orA is nonlinear, possibly multivalued, and dissipative. The main tool used for the nonlinear case is a result of Barbu concerning the null controllability of the system.     

The braid groups of the projective plane and the Fadell–Neuwirth short exact sequence

We study the pure braid groups of the real projective plane , and in particular the possible splitting of the Fadell–Neuwirth short exact sequence , where n ≥ 2 and m ≥ 1, and p _* is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration of configuration spaces. Van Buskirk proved (1966, Trans. Am. Math. Soc., 122:81–97) that p and p _* admit a section if n = 2 and m = 1. Our main result in this paper is to prove that there is no section if n ≥ 3. As a corollary, it follows that n = 2 and m = 1 are the only values for which a section exists. As part of the proof, we derive a presentation of : this appears to be the first time that such a presentation has been given in the literature.      

Notes on symmetric g-functions

A number of generalized metric spaces have been defined or characterized in terms of g-functions. Symmetric g-functions are discussed by C. Good, D. Jennings and A. M. Mohamad. In this paper, some questions about symmetric g-functions are answered, particularly it is shown that every sym-wg-space is expandable. The project is supported by the NNSF (10571151) and NSF (2006J0397) of Fujian Province of China.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

Combined effects for magnetic Laplace equations with lack of compactness

We are concerned with qualitative analysis of weak solutions for a class of magnetic Laplace equations with lack of compactness. By variational arguments, we establish related nonexistence, existence and multiplicity results.     

An Elementary Approach to Exponential Spaces

In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.