COHOMOLOGY VANISHING IN HILBERT SPACES

§1. Introduction Oka [40] proved that any additive Cousin problem is solvable in a domain of holomorphyin Cn. Oka [41]-Cartan [2, 3]-Serre [46] generalized this as the theorem B for analyticcoherent sheaves over Stein spaces. J. P. Serre [47] proved…     

REGULAR RELATIONS AND MONOTONE NORMAL ORDERED SPACES

In this paper the classical theorem of Zareckii about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckii theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one.     

WEIERSTRASS REPRESENTATION FORSURFACES WITH PRESCRIBED NORMALGAUSS MAP AND GAUSS CURVATURE IN H~3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.     

EXISTENCE OF GENERALIZED WEAKLY SOLUTIONS OF DIFFERENTIAL EQUATIONS IN PRODUCT SPACES

In this paper, by using Lyapunov function and weak noncompactness conditions, we give an existence theorem of generalized weakly solutions of Cauchy problem of differential equations in product spaces.     

NONLOCAL INITIAL PROBLEM FOR NONLINEAR NONAUTONOMOUS DIFFERENTIAL EQUATIONS IN A BANACH SPACE

The nonlocal initial problem for nonlinear nonautonomous evolution equati-ons in a Banach space is considered. It is assumed that the nonlinearities havethe local Lipschitz properties. The existence and uniqueness of mild solutionsare proved. Applications to integro-differential equations are discussed.The main tool in the paper is the normalizing mapping (the generalizednorm).     

THE NONLOCAL INITIAL PROBLEMS OF A SEMILINEAR EVOLUTION EQUATION

The purpose of this paper is to investigate the existence of solutions to a nonlocal Cauchy problem for an evolution equation. The methods used here include the semigroup methods in proper spaces and Schauder's theorem. And the results are applied to a system of nonlinear partial differential equations with nonlinear boundary conditions.     

Connectedness of Cone-Efficient Solution Set for Cone-Quasiconvex Multiobjective Programming in Locally Convex Spaces

This paper deals with the connectedness of the cone-efficient solution set for vector optimization inlocally convex Hausdorff topological vector spaces.The connectedness of the cone-efficient solution set is provedfor multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set ofalternatives.The generalized saddle theorem plays a key role in the proof.     

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

The multiple solutions for one-dimensional cubic nonlinear problem u" u~3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ"_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k.     

绝对连续测度的Loeb代数

It is proved that if μ and v be finite measures on a measurable space (X,S) and v is absolutely continuous with respect to v, then it is holds that L(^*S,^*v) C L(^*S,^*v), while L(^*S,^*v) and L(^*S,^*v) are the Loeb algebras with respect to measure spaces (X,S,μ) and(X,S,μ).     

加权Dirichlet空间Dφ^2上的复合算子

设D是复平面中单位圆盘，φ：D→R是一个次调和函数，Dφ^2是D上的加权Dirichlet空间．对某类次调和函数φ文章研究了Dφ^2上的复合算子Cφ，分别得到了Cφ为Dφ^2上的有界、紧、Schatten p-类算子的特征．