Coincidence theorem for admissible set-valued mappings and its applications in FC-spaces

A new coincidence theorem for admissible set-valued mappings is proved in FC-spaces with a more general convexity structure. As applications, an abstract variational inequality, a KKM type theorem and a fixed point theorem are obtained. Our results generalize and improve the corresponding results in the literature.     

模糊逻辑系统中的演绎定理和HS规则

Cancelled the first axiom L1) or the third axiom L3) of the classical formal logic system we established two kinds of quasi-formal deductive system, LG-Rand LG, respectively. In LG-R we proved that neither the deduction theorem nor the hypothetical syllogism (HS) rule held but a deduction theorem and an HS rule are obtained in a weak sense. We also proved that both the deduction theorem and the hypothetical syllogism(HS) rule hold in LG.     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

THE EXTENSION THEOREM OF-LINEAR RANDOM FUNCTIONALS AND ITS APPLICATIONS

The extension theorem for linear random functionals in linear apaces is proved. And the representation theorem for bounded linear random functionals and the continuous linear random functionals are studied     

ON THE EXISTENCE AND UNIQUENESS OF CLASSICAL GLOBAL SOLUTION TO A TYPE OF B0USSINESQ EQUATIONS

The existence and uniqueness of classical global solutions to a type of Boussinesq equations withinitial and boundary values are studied in this paper. The existence of such solutions is proved bymeans of compactness theorem and Schauders fixed point theorem, and its uniqueness by the so calledenergy method.     

一般化凸空间上的几乎不动点, 不动点和拟变分不等式

The definitions of S-KKM property and Γ-invariable property for multi-valued map- ping are established, and by which, a new almost fixed point theorem and several fixed point theorems on Haudorff locally G-convex uniform space are obtained, and a quasi-variational inequality theorem for acyclic map on Hausdorff Φ-space is proved. Our results improve and generalize the corresponding results in recent literatures.     

关于多目标分式规划

Abstract. In this paper some optimality criteria are proved and some Mond-Weir type duality theorem for multiobjective fractional programming problems defined in a Banach space is obtained.     

THE EXTENSION OPERATOR IN BANACH SPACES FOR LOCALLY BIHOLOMORPHIC MAPPINGS

In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.     

On the Trace Embedding and Compact Properties of Finite Element Spaces

Ⅰ. Introduction In paper, the embedding theorem and the compact theorem of Rellich and Kondrachev for the Sobolev spaces are generalized to finite element spaces with certain properties. By means of them, the convergence of finite element methods for a class of nonlinear problems are proved in papers.In this paper, the trace embedding and compact theorems for Sobolev spaces will be generalized     

On π-Subpairs of π-Blocks

Alperin and Broug have given the p-subpairs in a finite group, and proved that there is a Sylow theorem for p-subpairs. For a π-separable group with π-Hall subgroup nilpotent, we prove that there is a π-Sylow theorem for π-subpairs. Note that our π-subpairs are different from what Robinson and Staszewski gave.     

On the Kantorovich–Rubinstein theorem

The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W₁ on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved.     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

Generalized Solutions of Linear Parabolic Stochastic Partial Differential Equations

Existence and uniqueness theorems for parabolic stochastic partial differential equations with space—time white noise are proved. The method is a combination of the characterization theorem for Hida distributions with the Feynman—Kac and Girsanov formulae. Accepted 3 December 1996     

Spectral measures,II: Characterization of scalar operators

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be spectral. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.     

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3$3\times 3$-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations — The sub-half-linear case

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas-Fermi equation presented in this article are proved. It is assumed that coefficient q(t) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder-Tychonoff fixed point theorem are the main tools for the proofs.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

Lp Solutions for Multidimensional BDSDEs with Locally Weak Monotonicity Coefficients

In this paper, the authors establish the existence and uniqueness theorem of L^p (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of L^p solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.