FREUD-BUTZER-HAHN TYPE QUANTITATIVE THEOREMS FOR PROBABILISTIC REPRESENTATIONS OF(C_0)OPERATOR SEMIGROUPS

In this paper,we shall exploit the Freud method in the Classical operator approximation theory to im- prove known quantitative estimalions with an emphasis on Calculah' on the generic constants.

Infinite Dimensional Compound Operator Equation in Hilbert Space

In this article, we consider compound matrices and compound operator equations in a Hilbert space. First, we recall some concepts and main results introduced by Muldowney and by Roger Temam. After that we establish the rule of compound matrices in a Hilbert space, and obtain the expression of solution to a compound operator equation by using the method of operator semigroup. Our brief results generalize the corresponding results in a finite space.

Some properties of Riesz potential associated with Schrdinger operator

Let L = --△ ＋ Ⅴ be the SchrSdinger operator on Rd, d ≥ 3, where A is the Laplacian on Rd and V ≠ 0 is a nonnegative function satisfying the reverse HSlder inequality. In this article, the author investigates some properties of the Riesz potential IaL associated with L on the Campanato-type spaces ∧Lβ and the Hardy-type spaces HLP.

SPECTRA OF COMPOSITION GROUPS ON THE WEIGHTED DIRICHLET SPACE OF THE UPPER HALF-PLANE

We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane U is strongly continuous on the weighted Dirichlet space of U,Dα(U).Further,we investigate when they are isometries on Dα(U).In each case,we determine the semigroup properties while in the case that the induced composition group is an isometry,we apply similarity theory to determine the spectral properties of the group.

EXISTENCE AND REGULARITY OF SOLUTIONS TO MODEL FOR LIQUID MIXTURE OF ^3HE-^4HE

Existence and regularity of solutions to model for liquid mixture of 3He-4He is considered in this paper. First, it is proved that this system possesses a unique global weak solution in H^1 （Ω, C ×R） by using Galerkin method. Secondly, by using an iteration procedure, regularity estimates for the linear semigroups, it is proved that the model for liquid mixture of 3He-4He has a unique solution in H^k（Ω, C × R） for all k ≥ 1.

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE H^k

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.

COMPLEX WIENER-ITO CHAOS DECOMPOSITION REVISITED

In this article, some proper ties of complex Wiener-Ito multiple integrals and complex Ornstein-Uhlenbeck opera tors and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula, and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-Ito multiple integrals are given.

PERTURBED NONLINEAR EVOLUTION INCLUSIONS IN BANACH SPACES

In this paper we discuss tLhe existence results of the integral solutions to nonlinear evolution inclusion: u' (t) ∈ Au(t) +F(t,u(t)), where A is m-dissipative and F is a set valued map in separable Banach spaces, and extend the relative results in references.

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in W pl,γ（; E0, E） is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

Existence and uniqueness results for mild solutions of random impulsive abstract neutral partial differential equation over real axis

In this paper, we discuss the existence and uniqueness of mild solutions of random impulsive abstract neutral partial differential equations in a real separable Hilbert space. The results are obtained by using Leray-Schauder Alternative and Banach Contraction Principle.Finally an example is given to illustrate our problem.

Relative locations of operated semigroups subwords in free and Motzkin words

Bracketed words are basic structures both in mathematics （such as Rota-Baxter algebras） and mathematical physics （such as rooted trees） where the locations of the substructures are important. In this paper, we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested, and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.

Distances between Elements of a Semigroup and Estimates for Derivatives

This paper is concerned first with the behaviour of differences T（t） - T（s） near the origin, where （T（t）） is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane （in which case it is assumed analytic）. For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T（t） -T（2t） have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.

The Existence of Global Attractors for 2D Navier-Stokes Equations in H^k Spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.

Stability for a competing system of hierarchical age-structured populations

In this paper,we are concerned with the stability for a model in the form of system of integro-partial differential equations,which governs the evolution of two competing age-structured populations.The age-specified environment is incorporated in the vital rates,which displays the hierarchy of ages.By a non-zero fixed-point result,we show the existence of positive equilibria.Some conditions for the stability of steady states are derived by means of semigroup method.Furthermore,numerical experiments are also presented.

Existence and Regularity of Solution of the Liquid^(4)He Model Coupling with an Applied Magnetic Field

In this paper,we derive a time-dependent Ginzburg-Landau model for liquid^(4)He coupling with an applied magnetic field basing on the Le Chatlier principle.We also obtain the existence and uniqueness of global weak solution for this model.In addition,by utilizing the regularity estimates for linear semigroup,we prove that the model possesses a global classical solution.

THE ABSTRACT CAUCHY PROBLEM AND A GENERALIZATION OF THE LUMER-PHILLIPS THEOREM

For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.

ON AGE-STRUCTURED SIS EPIDEMIC MODEL FOR TIME DEPENDENT POPULATION

A model which describes the dynamics of an SIS epidemic in an age-structured and time dependent population is considered. The global behavior of the model with a special form for the infective force is obtained. The existence and uniqueness of the solution and nontrivial endemic equilibrium state are proved.

APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL BILINEAR SYSTEMS

The aim of this paper is to study approximate controllability for bilinear systems in the general case. The existence and uniqueness of solutions to bilinear evolution equations are obtained. The paper shows that the approximate controllability of the bilinear control problem is equivalent to the approximate controllability of a discrete problem, using the method developed by Loreti and Siconolfi to approximate the bilinear control problem in an infinite-dimensional Banach space by means of a sequence of discrete problems. Finally, the necessary conditions for the bilinear system to be approximately controllable are stste and proved.