SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS UNDER WEAK TOPOLOGY

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill’s equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.

Continuity in Weak Topology: First Order Linear Systems of ODE

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.

Weak Anti-Localization and Quantum Oscillations in Topological Crystalline Insulator PbTe

Topological crystalline insulators （TCIs） have attracted worldwide interest since their theoretical predication and have created exciting opportunities for studying topological quantum physics and for exploring spintronic appli- cations. In this work, we successfully synthesize PbTe nanowires via the chemical vapor deposition method and demonstrate the existence of topological surface states by their 2D weak anti-localization effect and Shubnikov-de Haas oscillations. More importantly, the surface state contributes ~61% of the total conduction, suggesting dom- inant surface transport in PbTe nanowires at low temperatures. Our work provides an experimental groundwork for researching TCIs and is a step forward for the applications of PbTe nanowires in spintronic devices.

Space Topologies and Their Dual Space Topologies for Conventional Functional Space Topologies

In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(Rn) of functions of class C∞ on Rn which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C∞ with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.

Locally Asymptotic-norming Property and Kadec Property

In this paper we study the three new asymptotic-norming properties which are called locally asymptotic-norming property k,k=Ⅰ, Ⅱ,Ⅲ, and discuss the relationship between the locally asymptotic-norming property and the Kadec Property.

A Riesz Product Type Measure on the Cantor Group

A Riesz type product as Pn=nЛj=1（1＋awj＋bwj＋1）is studied, where a, b are two real numbers with ｜a｜ ＋ ｜b｜ 〈 1, and {wj} are indepen- dent random variables taking values in （-1, 1} with equal probability. Let dw be the normalized Haar measure on the Cantor group Ω = （-1, 1}^N. The sequence of P,~dw 1 probability measures {Pndw/E（Pn） } is showed to converge weakly to a unique continuous measure on/2, and the obtained measure is singular with respect to dw.

Weakly Compact Sets and Riesz Representation Theorem in Musielak Sequence Spaces

In this work,we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function.Finally,as an immediate consequence of the criteria considered in this paper,the criteria of the weakly compact sets of Orlicz sequence spaces are deduced.

Some Dynamical Properties in Set-valued Discrete Systems

1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).

The Asymptotic Average Shadowing Property and Its Applications

Let (X, d) be a bounded metric space and f : X → X be a uniformly continuous surjection. For a given dynamical system (X, f) which may not be compact, we investigate the relation between the asymptotic average shadowing property(AASP), transitivity and mixing. If f has the AASP, then the following statements hold: (1) f n is chain transitive for every positive integer n; (2) If X is compact and f is an expansive homeomorphism, then f is topologically weakly mixing; (3) If f is equicontinuous, then f is topologically weakly mixing; (4) If X is compact and f is equicontinuous, then f ×f is a minimal homeomorphism. We also show that the one-sided shift map has the AASP and the identity map 1 X does not have the AASP. Furthermore, as its applications, some examples are given.

Weakly Compact Sets and Orlicz Spaces

In this paper, We give the simple criteria of weakly compact sets in L1 and l1, which perfects Auto's result [1]， Also as a corollary, we get Shur's theorem， In view of weak compactness, we give another proof of the reflexivity of Orlicz spaces，

Projections on Weak*-closed Subspace of Dual Banach Algebras

Our first purpose in this paper is to provide necessary conditions for a weak＊-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak＊-closed left translation invariant subspace in Ma,（S）＊ of a compact cancellative foundation semigroup S is the range of a weak＊-weak＊ continuous projection on M~,（S）＊ commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A＊ and /3（A＊） which commutes with translations and convolution.