Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.     

Crossed-products extensions,of Lp-bounds for amenable actions

We will extend earlier transference results due to Neuwirth and Ricard from the context of noncommutative $L_p$-spaces associated with amenable groups to that of noncommutative $L_p$-spaces associated with crossed-products of amenable actions. Namely, if $m:G\to \mathbb{C}$ is a completely bounded Fourier multiplier on $L_p$, then it extends to the crossed-product with similar bounds provided that the action θ is amenable and trace-preserving. Furthermore, our construction also allows to extend G-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Følner sets of the action θ satisfy certain accretivity property. As a corollary we obtain stability results for maximal $L_p$-bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the $L_p$-spaces of group algebras.     

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

一类自适应广义交替方向乘子法

广义交替方向乘子法是求解凸优化问题的有效算法.当实际问题中子问题难以求解时,可以采用在子问题中添加邻近项的方法处理,邻近矩阵正定时,算法收敛,然而这也会使迭代步长较小.最新研究表明,邻近矩阵可以有一定的不正定性.本文在基于不定邻近项的广义交替方向乘子法框架下,提出一种自适应的广义交替方向乘子法,动态地选择邻近矩阵,增大迭代步长.在一些较弱的假设下,证明了算法的全局收敛性.我们进行一些初等数值实验,验证了算法的有效性.     

Triebel-Lizorkin空间上二阶有限时滞退化微分方程的适定性

本文在周期Triebel-Lizorkin空间F_(p,q)~s(T;X)上研究二阶有限时滞退化微分方程(Mu')'(t)+αu'(t)=Au(t)Gu'_t+Fu_t+f(t)(t∈T:=[0,2π]),u(0)=u(2π),(Mu')(0)=(Mu')(2π)的适定性.利用Triebel-Lizorkin空间上算子值傅里叶乘子定理,给出上述方程是F_(p,q~-)~s适定的充要条件.     

Synchronization of coupled systems to periodic diagonal solutions with synchronized asymptotic phases

Using the change of coordinates, parameterization and characteristic multipliers, we prove the synchronization of a class of coupled nonlinear systems with nontrivial periodic solution. The periodic diagonal solution of the coupled system is asymptotically orbitally stable with asymptotic phase.Examples are given to illustrate the theorem.     

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1<p<∞. We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on Lp-spaces. We then apply these properties to the study of the pseudofunction algebra PFp(G), the pseudomeasure algebra PMp(G), and the Figà-Talamanca-Herz algebra Ap(G) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C^∗-algebra , the group von Neumann algebra VN(G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PFp(G), PMp(G), and Ap(G), respectively. The p-completely bounded multiplier algebra McbAp(G) plays an important role in this work.     

The multipliers‐free domain decomposition methods

This article concludes the development and summarizes a new approach to dual‐primal domain decomposition methods (DDM), generally referred to as “the multipliers‐free dual‐primal method.” Contrary to standard approaches, these new dual‐primal methods are formulated without recourse to Lagrange‐multipliers. In this manner, simple and unified matrix‐expressions, which include the most important dual‐primal methods that exist at present are obtained, which can be effectively applied to floating subdomains, as well. The derivation of such general matrix‐formulas is independent of the partial differential equations that originate them and of the number of dimensions of the problem. This yields robust and easy‐to‐construct computer codes. In particular, 2D codes can be easily transformed into 3D codes. The systematic use of the average and jump matrices, which are introduced in this approach as generalizations of the “average” and “jump” of a function, can be effectively applied not only at internal‐boundary‐nodes but also at edges and corners. Their use yields significant advantages because of their superior algebraic and computational properties. Furthermore, it is shown that some well‐known difficulties that occur when primal nodes are introduced are efficiently handled by the multipliers‐free dual‐primal method. The concept of the Steklov–Poincaré operator for matrices is revised by our theory and a new version of it, which has clear advantages over standard definitions, is given. Extensive numerical experiments that confirm the efficiency of the multipliers‐free dual‐primal methods are also reported here. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010