STATIONARY DISTRIBUTIONS OF INFINITE PARTICLE SYSTEMS WITH NONCOMPACT STATE SPACE.

This paper deals with the preblem of existence and uniqueness of the stationary distributions (abbr., s. d.'s) for the processes constructed in [4] .The main results are stated in § 1. For the reader's convenience we first restate the existence theorems (Theorem 1 and 2) of the processes given in [4]. Then two existence theorems (Theorem 3 and 4) and a uniqueness theorem (Theorem 5) for the s. d.'s of the processes are presented. The last result (Theorem 6), as an application of the previous ones, is about the Schlgl model which comes from nonequilibrium statisticali physics. The details of the proofs of Theorem 3—6 are given in § 2—4.     

ON EQUILIBRIA AND PERIODIC ORBITS OF COOPERATIVE SYSTEMS IN R~3

In the paper we present new criteria for existence of equilibria and periodic orbits for 3-dimensional systems which are cooperative. Particular attention is given to the relations of equilibria, cycle K and stable manifold of K. Meanwhile, the results of Hirsch [1, Theorem 9] and Jiang [2] are improved.     

PERIODIC TRAVELLING-WAVE SOLUTION OF BRUSSELATOR

The Hopf bifurcation Theorem requires that one at first knows all eigenvalues of the linearized system;and a diffieulty in applying the Hopf Theorem is that in the general case we are unable to solve an algebraic equation of degree n (n≥3) with parameter coefficients.This paper firstly proves Theorem 1 which enables us to know the existence of the Hopf bifuecation by direct use of the coefficients of the characteristic equ ation of the linearized system for the case where the characteristic equation of the linearized system has degree 2,3 or 4.Then by using Theorem 1,we gave out the condition for existing a periodic traveling-wave solution of Brusselator in diffusion by Hopf bifurcat on.Finally,we discuss the problem of bifurcation direction and the problem of stability of the bifurcation.     

RADIUS OF STARLIKENESS FOR THE CLASS S(α,n)

In this paper we shall investigate the bound of starlikeness r for the classS(α, n) in [1]. We obtain the following sharp result:where α_0 is the unique solution of the equationα[1+(1-2α)r~(2n)]/1-r~(2n)={1-(1-2α)r~n/1+r~n}~2with respect to α in the open interval (0, 1). This result is also the extension of somewell-known ones in [2], [3, Theorem 1] and [4, Theorem 2]     

ON THE CENTER OF GENERALIZED LIENARD SYSTEM

This paper discusses the local and global center of the generalized Lienard system, gives two theorems which extend and improve the relative results of [1]-[3]. Moreover the lemmas in this paper are also useful to the study for the oscillation, stability and the existence of limit cycles of the generalized Lienard system.     

SOLVABILITY OF SINGULAR CANTILEVER BEAM EQUATION

Two local existence theorems are established for a class of fourth-order two-point boundary value problems with all order derivatives and singularity.Main ingredients are Green function and integral equation.In mechanics,such class of problems is called cantilever beam equation which describes the deflection of an elastic beam fixed at left and freed at right.     

On Alternating Sequences Relative to a c-Matching of a Hypergraph

In[1], C.Berge has modified Theorem 1 of [2]to a theorem which is a charac-terization of maximum c-Matchings in a hypergraph (see[1], P.416, Theorem 2),and at the same time can be considered as a generalization of Berge's theorem on     

CALDER6N-ZYGMUND OPERAORS ON THE HARDY SPACES OF WEIGHTED HERZ TYPE

Applying the decomposition theorems in [1] and [2] , we obtain the boundedness theorem of Calderbn-Zygmund operator of type 6 on the Hardy spaces of weighted Herz type and establish interpolation theorem of linear operators on the weighted Herz spaces. -     

SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF ARRAYS OF NOD RANDOM VARIABLES

In this paper the authors study the complete,weak and almost sure convergence for weighted sums of NOD random variables and obtain some new limit theorems for weighted sums of NOD random variables,which extend the corresponding theorems of Stout [1],Thrum [2] and Hu et al.[3].     

ON LIAPUNOV FUNCTIONAL IN THEORY OF STABILITY OF SYSTEMS WITH TIME-LAG

It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.     

POSITIVE FIXED POINT THEOREMS FOR β-SEMIDIFFERENTIABLE κ -SET-CONTRACTIVE MAPPINGS WITH APPLICATION TO AN INFECTIOUS DISEASE MODEL

In this paper we define the new concept of β -semidifferentiability of a mapping and obtain the positive fixed point theorems for β -semidifferentiable k -set-contractive mappings. Then the theorems are ap-plied to the nonlinear integral equation, x(t) = (s,x(s))ds , which can be interpreted as a modelfor the spread of certain infectious diseases, and the existence theorems of the positive solutions and positive eigenvalues for the equation are obtained.     

EXISTENCE OF WEAK SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

0 IntroductionRecently, the study of ordinary differential equations in a Banach spacehas been developed extensively by using the strong topology, see [2],13],[4]. Atpresent, however, there is no reference or paper which systematically studiesweak solutions of the Cauchy problem of ordinary differential equation in theweak topology. In references [l],15],[7] and [8], several results of weak solutionsare proved. In this paper, we shall give the other results of weak solutions,some of which are…     

THE FUBINI THEOREMS OF STOCHASTIC MEASURES

This paper discusses the existence of product measures of two specific kinds of stoehastic measures,and expands the integrable classes from classes of real measurable functions to ones which consist of some Banach space valued mappings.With the help of this newly defined stochastie integral, which is called S.B.integral,we have proved three Fubini Theorems:Theorem 1.7,Theorem 1.8 and Theorem 2.7.     

SEMILINEAR RANDOM EQUATIONS WITH A-PROPER OPERATORS IN BANACH SPACE AND THEIR APPLICATION

In this paper the existence of random solutions of operator equations with A-proper Operators is studied. The random analog of deterministic results obtained by Petryshyn in [9] and [10] for such operators is the main theorems. It is then followed by existence reuslts for the Second order differential equation with Newmann boundary value problem.     

DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION

This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian^[7] and by Kulczycki M, Tabor J.^[3] are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.     

On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslov’s Theorem.We also establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices,and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well.The particular solutions of the adjoint equation are then obtained.As an appliction of these particular solutions,we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.     

APPROXIMATING COMMON FIXED POINTS OF NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

We use an iteration scheme to approximate common fixed points of nearly asymptotically nonexpansive mappings.We generalize corresponding theorems of [1] to the case of two nearly asymptotically nonexpansive mappings and those of [9] not only to a larger class of mappings but also with better rate of convergence.     

CHARACTER, DENSITY AND CELLULARITY OF L-FUZZY TOPOLOGICAL SPACES

In this paper we generalize the notions of weight,character, density and cellularity of topological spaces to L-fuzzy topogical spaces. We prove theorems corresponding to some of usual theorems. In particular we generalize Hewitt-Marczewski-Pondiczery theorem on the density of the product of topological spaces to the case of L-fuzzy topological spaces (Theorem 4. 4). We also discuss the relations between topological spaces and its induced fuzzy topological spaces for these cardinal functions.     

INSTABILITY THEOREMS OF YANG MILLS FIELDS

Yang-Mills connections (potentials) are critical points of Yang-Mills functional. Its corresponding curvatures are called Yang-Mills fields. It is natrual to ask that under certain conditions they are stable or unstable which are dependent on the nonnegativity or negativity of their second variations,respectively.J.P.Bourguignonon, H. B. Laws and J. simons[1][2]firstly considered such a problem and obtained some results in this direction. The purpose of this note is to generalize such a theorem.The main result of this article is to prove that any Yang-Mills fields over compact submanifolds M~m in the Euclidean space R~(m+p) is unstable, provided the second fundamental forms of M~m in R~(m+P) satisfy certain conditions (Theore m 1). Conseqnently, on such manifolds there is no selfdual or auti-selfdual solution of Yang-Mills equation.From Theorem 1 we obtained several interesting consequence. Among them is BourguignonLawson-Simons'theorem.     

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C＊-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.