THE EXISTENCE OF SOLUTIONS OF NONLINEARFREDHOLM INTEGRAL EQUATIONS IN BANACH SPACES

This paper improves Tychonov ford point theorem and discusses the existence of solutions of nonlinear Fredholm integral equations on [0,+∞] in Banach spaces with Frechet space theory.

ON A LARGE SAMPLE PROBLEM IN NONLINEAR REGRESSION

Assume that in the nonlinear regression model, independent variable sequence {xi, i ≥ 1} is a known constant-vector sequence. This article proposes a condition on {xi}, which can be tested and verified easily. The condition is essential for proving the consistency and asymptotic normality of the estimator.

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.

The central limit theorem and chaos

Let X be a compact metric space studies some relationships between stochastic and f ： X→ X be a continuous map. This paper and topological properties of dynamical systems. It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and / is sensitively dependent on initial conditions if and only if / is neither minimal nor equicontinuous.

THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS

In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.

ALMOST PERIODICITY AND EQUICONTINUITY OF THE TOPOLOGICAL TRANSFORMATION GROUP

In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.

A NOTE ON MEASURE-THEORETIC EQUICONTINUITY AND RIGIDITY

Given a topological dynamical system(X,T)and a T-invariant measureμ,let B denote the Borel σ-algebra on X.This paper proves that(X,B,μ,T)is rigid if and only if(X,T)isμ-A-equicontinuous in the mean for some subsequence A of N,and a function f∈L^(2)(μ)is rigid if and only if f is μ-A-equicontinuous in the mean for some subsequence A of N.In particular,this gives a positive answer to Question 4.11 in[1].

THE PROXIMAL RELATION,REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS

In this paper,we study the proximal relation,regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions.A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation,and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

Equicontinuity and Sensitivity of Group Actions

Let(X,G)be a dynamical system(G-system for short),that is,X is a topological space and G is an infinite topological group continuously acting on X.In the paper,the authors introduce the concepts of Hausdorff sensitivity,Hausdorff equicontinuity and topological equicontinuity for G-systems and prove that a minimal G-system(X,G)is either topologically equicontinuous or Hausdorff sensitive under the assumption that X is a T_(3)-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs.In particular,under the condition that X is a Hausdorff uniform space,they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for G-systems admitting one transitive point.

Stronger Versions of Sensitivity for Minimal Group Actions

We study several stronger versions of sensitivity for minimal group actions,including nsensitivity,thick n-sensitivity and blockily thick n-sensitivity,and characterize them by the regionally proximal relation.

Regionally proximal relation of order d along arithmetic progressions and nilsystems

The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

Positive Solutions for the Initial Value Problems of Impulsive Evolution Equations

This paper deals with the existence of e-positive mild solutions（see Definition 1）for the initial value problem of impulsive evolution equation with noncompact semigroup u（t）＋ Au（t）= f（t,u（t））,t ∈ [0,＋∞）,t = tk,u-t=tk = Ik（u（tk））,k = 1,2,...,u（0）= x0 in an ordered Banach space E.By using operator semigroup theory and monotonic iterative technique,without any hypothesis on the impulsive functions,an existence result of e-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of f.Particularly,an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces,which is very convenient for application.An example is given to illustrate that our results are more valuable.

An Operator Arzelà-Ascoli Theorem

The author generalizes the Arzela-Ascoli theorem to the setting of matrix order unit spaces, extending the work of Antonescu-Christensen on unital C^＊-algebras. This gives an affirmative answer to a question of Antonescu and Christensen.

A NOTE ON THE EQUICONTINUITY IN SOME TOPOLOGICAL GROUPS

In this paper, we show that a theorem concerning the equicontinuity of theoperators in F-spaces can be generalized to a class of operators defined on the binaryfissionable topological groups.

Equicontinuity of Maps on a Dendrite with Finite Branch Points

Let （T, d） be a dendrite with finite branch points and f be a continuous map from T to T. Denote by w（x, f） and P（f） the w-limit set of x under f and the set of periodic points of f, respectively. Write Ω（x, f） = {yl there exist a sequence of points xk ∈ T and a sequence of positive integers n1 〈 n2 〈 … such that lim k→∞ Xk = X and lim k→∞ f nk （xk） = y}. In this paper, we show that the following statements are equivalent： （1） f is equicontinuous. （2） w（x, f） = Ω（x, f） for any x ∈ T. （3） ∩ ∞ n=1 f n（T） = P（f）, and w（x, f） is a periodic orbit for every x ∈ T and map h： x → w（x, f） （x ∈ T） is continuous. （4） Ω（x, f） is a periodic orbit for any x ∈ T.

Measure of Noncompactness and Semilinear Nonlocal Functional Differential Equations in Banach Spaces

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonloeal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups Of linear operators.

Density-equicontinuity and Density-sensitivity

In this paper we introduce the notions of(Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure;and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.

EQUICONTINUITY OF TRIANGULAR MAPS

In this paper we study the equicontinuity of a triangular map of the square,and obtain some equivalent conditions for it.