Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notions of equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables for solutions with partly controlled initial conditions. We obtain sufficient conditions for the equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables of solutions with partly controlled initial conditions. We introduce the notions of equiboundedness and uniform boundedness with respect to part of the variables for solutions of systems with partly controlled initial conditions. We obtain sufficient conditions for the equiboundedness and uniform boundedness with respect to part of the variables of solutions with partly controlled initial conditions.     

Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notion of partial uniform boundedness of solutions with partially controlled initial conditions in the general case, that is, the case in which the part of variables with respect to which the boundedness of solutions is studied is a subset of the part of variables with respect to which the initial conditions are controlled. We obtain a criterion for the partial uniform boundedness of solutions with partly controlled initial conditions. We introduce the notion of partial total boundedness of solutions with partly controlled initial conditions. We obtain a sufficient condition for the partial total boundedness of solutions with partly controlled initial conditions.     

变指标中心BMO空间

本文引进变指标中心有界平均振荡函数空间.作为应用,得到了Hardy算子及其共轭算子的交换子在变指标勒贝格空间上有界性的特征刻画.另外,还考虑了交换子在变指标Herz空间上的向量值不等式.     

Fuzzy bounded linear operators

In this paper, a notion of boundedness of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space is introduced and two types (strong and weak) of fuzzy bounded linear operators are defined. Relation between fuzzy continuity and fuzzy boundedness is studied. Definitions of fuzzy bounded linear functionals are given and the notions of fuzzy dual spaces are developed. The Hahn-Banach theorem, the Open mapping theorem, the Closed graph theorem and the Uniform boundedness principle theorem are established.     

单位球上的对偶Toeplitz算子

本文研究了单位球Bergman空间的直交补上的对偶Toeplitz算子的代数性质,首先我们给出了对偶Toeplitz算子的有界性和紧性的完全刻画,然后给出对偶Toeplitz算子的谱性质,最后证明了不存在以有界全纯或者反全纯函数为符号的拟正规对偶Toeplitz算子.     

On Metric Boundedness Structures

Many years ago, S.-T. Hu gave necessary and sufficient conditions for a family of subsets of a metrizable space X to be the family of bounded sets for some admissible metric for the space. In this article, we show that in any noncompact metrizable space there are uncountably many distinct metric boundedness structures. Also, given an initial metric d for X, we look carefully at the problem of characterizing those boundedness structures determined by metrics uniformly equivalent to d. Applications to hyperspaces are given. Throughout, we rely on a dual approach to the study of metric boundedness.     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for bi-parameter and bilinear Fourier integral operators

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L² case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.     

Boundedness of Singular Integral Operators on Local Hardy Spaces and Dual Spaces

Potential Analysis - The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the...     

Optimal simplex tableau characterization of unique and bounded solutions of linear programs

Uniqueness and boundedness of solutions of linear programs are characterized in terms of an optimal simplex tableau. LetM denote the submatrix in an optimal simplex tableau with columns corresponding to degenerate optimal dual basic variables. A primal optimal solution is unique iff there exists a nonvacuous nonnegative linear combination of the rows ofM, corresponding to degenerate optimal primal basic variables, which is positive. The set of primal optimal solutions is bounded iff there exists a nonnegative linear combination of the rows ofM which is positive. WhenM is empty, the primal optimal solution is unique.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024. This material is based upon work supported by the National Science Foundation under Grant No. MCS-79-01066.     

本刊英文版Vol.33(2017),No.2论文摘要（英文）

<正>L~p Estimates for Bi-parameter and Bilinear Fourier Integral Operators Qing HONG Lu ZHANG Abstract Fourier integral operators play an important role in Fourier analysis and partial differential equations.In this paper,we deal with the boundedness of the bilinear and biparameter Fourier integral operators,which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators.We consider such FIOs when they have compact support in spatial variables.If they contain a real-valued phaseφ(x,ξ,η)which is jointly homogeneous in the frequency variablesξ,η,and amplitudes of order zero supported away from the axes and the anti-diagonal,we can show that the boundedness holds in the local-L~2 case.Some stronger boundedness results are also obtained under more     

Norm duality for convex processes and applications

One can associate two norms with a Banach space convex process. These norms are dual to each other and the norm of a process agrees with the dual norm of its adjoint. This norm duality provides an extremely general and simple way of establishing surjectivity or boundedness properties of homogeneous (linear or convex) inequality systems.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Clark's theorem for semi-infinite convex programs

In 1961, Clark proved that if either the feasible region of a linear program or its dual is nonempty and bounded, then the other is unbounded. Recently, Duffin has extended this result to a convex program and its Lagrangian dual. Moreover, Duffin showed that under this boundedness assumption there is no duality gap. The purpose of this paper is to extend Duffin's results to semi-infinite programs.     

单位多圆柱上加权Bergman空间上的复合算子（英文）

We consider the boundedness of composition operators on the Bergman space,and shows that when it is induced by automorphism is always bounded.At first we got a change of variables formula,which is very important for the proof of the boundedness of composition operators,and then obtain an upper bound for the special operator norm on Bergman space.     

Primal-dual subgradient methods for convex problems

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.     

On Hardy Inequality in Variable Lebesgue Spaces with Mixed Norm

In this paper a two-weight boundedness of multidimensional Hardy operator and its dual operator acting from one weighted variable Lebesgue spaces with mixed norm into other weighted variable Lebesgue spaces with mixed norm spaces is proved. In particular, a new type two-weight criterion for multidimensional Hardy operator is obtained.     

Quasistable properties of the solutions of a multiply connected system of differential equations

We prove a theorem on the asymptotic stability of power type with respect to one part of the phase variables and on the uniform boundedness of the solutions of a multiply connected system of differential equations with respect to the other part of the variables.     

The geometry of solvability and duality in linear programming

Solvability and boundedness criteria for dual linear programming problems are given in terms of the problem data and the intersections of the nonnegative orthant with certain complementary orthogonal subspaces. This research was partly supported by the Office of Naval Research, contract Nonr-1228(10), project NR 047-021, and by the National Science Foundation, project G-14102.     

Some topological properties of the solution sets of parametrized minimax problems

Sufficient conditions are given to guarantee the closedness and uniform boundedness of the solution sets corresponding to a pair of dual parametrized minimax problems. The parameter of the minimax problems belongs to a metrizable space and affects not only the objective function, but also the feasible sets.This research was supported in part by Fondo Nacional de Ciencias, Proyecto No. 01273.     

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences

Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion. The author is supported in part by CNPq Grants 301508/2005-4, 490200/2005-2, 550317/2005-8, by PRONEX–Optimization, and by FAPERJ Grant E-26/151.942/2004.