On the uniform boundedness of solutions of contingent equations

It is proved that, under standard conditions (in particular the convexity of the velocity set), the boundedness of the individual solutions of a contingent equation implies the uniform boundedness of the solutions, and consequently the boundedness of the reachable sets. This result is used to obtain an existence theorem in optimal control theory. By means of an example, it is shown that the convexity condition cannot be omitted.     

多复变数星形映射在某方向上精确的偏差定理

研究了Cn中单位多圆柱上星形映射在某方向上精确的偏差定理．给出了复Banach空间单位球的某方向上精确的偏差的上界,同时给出了下界的猜测．     

Some Intersection Theorems and Minimax Inequalities

This paper proves some intersection theorems. The proofs are new in the sense that they do not require the finite intersection property. As an application, some of the Fan minimax inequalities are proved.     

Limit theorems for polylinear forms

The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal.     

Noncommutative maximal ergodic theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

     

矩条件下的任意随机变量序列的一类强极限定理

本文利用随机变量的截尾方法和条件三级数定理,研究了任意随机变量序列在矩条件下的一类强极限定理,改进了与此相应的一些结果的条件.     

Fixed Point Theorems for Non-convex Valued Multifunctions

The purpose of this paper is to establish fixed point theorems for non-convex valued multifunctions which generalize known results in the literature. We also derive coincidence theorems in the non-compact setting.AMS Subject Classification (2000): 47H, 54H     

Remarks on lower semicontinuity and lower closure

In the context of a recent geometric condition of Cesari, used in the reduction of seminormality requirements in lower closure theorems, this paper shows that the existence of a strongly convergent selection from the sequence of orientor fields, under Kuratowski property (K), is adequate to guarantee lower closure theorems. This generalization is justified through examples. Several related remarks are made.This work was done in the framework of Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan. The author wishes to thank Professor L. Cesari for helpful discussions during the writing of this paper.     

A linear hyperbolic system and an optimal control problem

We prove a theorem of existence, uniqueness, and continuous dependence for a linear hyperbolic system with Darboux-type conditions under assumptions on the coefficients, which are in a sense the most general possible. Moreover, an application of this result to an optimal control problem is given.     

A note on the use of direct sufficient conditions in optimal control problems

It is always possible to transform a nonautonomous optimal control problem into an autonomous one. However, the direct sufficient conditions may yield no information when applied to this autonomous problem, even though they do allow one to conclude sufficiency when applied to the original nonautonomous problem.This research was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-76-2923.     

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ² into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem. The author thanks Professor F. Giannessi for valuable comments, especially on Gale and Klee-type separation theorems. This research was partially supported by Kyushu University 21st Century COE Program (Development of Dynamic Mathematics with High Functionality) and by the Grant-in-Aid for General Scientific Research from the Japan Society for the Promotion of Science 14340037.     

Pontryagin functions for multiple integral control problems

An optimal control problem is considered whose performance index is represented by anm-fold multiple integral, the state equations being given by a system of first-order partial differential equations. The concept of a field of optimal control variables with respect to an independent integral is introduced, the significance of these fields being due to the fact that a control pair is optimal whenever it is imbedded in such a field. Since there arem distinctm-fold independent integrals, it is possible to constructm distinct fields of this kind. For each of these, a Pontryagin function is defined, and it is shown that, if an optimal pair is embedded in one of these fields, it satisfies a corresponding Pontryagin maximum principle.This research was supported in part by NSF Grant No. GP-32830     

Tauberian Theorems for Sequences Satisfying Certain Convolution Relations

The purpose of this paper is to prove qualitative and optimal quantitative Tauberian theorems for the sequence λ defined by the relation g′ = λ∗︁g ,where ∗︁ denotes the Cauchy convolution, g is a given sequence with g(0) = 0 satisfying suitable growth conditions, and g′ is defined by g′(n) = (n + 1)g(n + 1) for n ∈ ℕ = {0, 1, 2, …}. The problem originates in the analytic theory of the distribution of prime elements in additive arithmetic semigroups developed by Knopfmacher , and in recent work of Indlekofer , Manstavičius and Warlimont , of Zhang , and of Warlimont . The new results are based on the approximation of g by recurrent sequences and do not assume λ ≥ 0; the method of proof consists in the transition to generating power series and utilizes a weighted version of a Wiener type inversion theorem due to Lucht and Reifenrath .     

一类m-相依随机变量序列加权和的极值分布

我们将证明一类m-相依随机变量序列加权和的极值分布定理．该定理既无需i．i．d．这一假设，也不必计算协变量部分和的极限值，更没有繁杂的有关条件分布方面的假设．更重要的是该定理的结论有许多统计方面的直接应用．     

Controllability of nonlinear systems

Some sufficient conditions are presented for the controllability of general nonlinear systems. First, the controllability problem is transformed into the problem of existence of fixed points for some operator; using Schauder's theorem, it is derived that a sufficient condition for controllability is the existence of a subsetS inC _n+m (T) which is invariant for a derived operator. Secondly, with the aid of the notion of comparison principle, the existence of the subsetS is guaranteed by the existence of solutions for some nonlinear integral inequality or equality equations. For example, one solution for such nonlinear integral equations is obtained under the assumption of the uniform boundedness for a nonlinear term of the differential equation.     

The fundamental continuity theory of optimization on a compact space. I

Optimizers on a compact feasible region are abstractly specified and, as set-valued mappings, are studied for sufficient conditions yielding them (as well as certain associated maps and certain restrictions of all these) continuous, using function space methods. In particular, the study concerns the continuity of the set of optimal solutions as a function of the three arguments: (i) objective function used, (ii) an incentive (or penalty/reward) function imposed, and (iii) an abstract parameter. An interpretation of the mathematical apparatus is suggested and a brief game-theoretic illustration is given.     

Variational Principles, Minimization Theorems, and Fixed-Point Theorems on Generalized Metric Spaces

In this paper, we prove a new minimization theorem by using the generalized Ekeland variational principle. We apply our minimization theorem to obtain some fixed-point theorems. Our results extend, improve, and unify many known results due to Kui, Ekeland, Takahashi, Caristi, iri, and others.     

Sufficiency conditions for multiple integral control problems

The construction of multiple integrals which are independent, in the sense that they depend solely on the values of their integrands on the boundary of the domain of integration is described. These integrals are applied to the derivation of a sufficiency condition for multiple integral optimal control problems.This research was supported in part by NSF Grant No. GP-32830.