α-LIMIT SETS AND LYAPUNOV FUNCTION FOR MAPS WITH ONE TOPOLOGICAL ATTRACTOR

We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X.This kind of map is a generalization of maps such as topologically expansive Lorenz map,unimodal map without homtervals and so on.Under the finiteness and basin conditions,we provide a leveled A-R pair decomposition for such maps,and characterize α-limit set of each point.Based on weak Morse decomposition of X,we construct a bounded Lyapunov function V(x),which gives a clear description of orbit behavior of each point in X except a meager set.

ω-Limit Set of a Tree Map

Let T be a tree and f be a continuous map from T into itself. We show mainly in this paper that a point x of T is an w-limit point of f if and only if every open neighborhood of x in T contains at least nx + 1 points of some trajectory, where nx equals the number of connected components of T \ {x}. Then, for any open subset G w(f) in T, there exists a positive integer m = m(G) such that at most m points of any trajectory lie outside G.This result is a generalization of the related result for maps of the interval.

ON THE LIMITABILITY OF SOLUTIONS OF SOME DIFFERENCE EQUATIONS

We investigate limitability by the Cesaio (C, 1) and the Abel (A) methods of solutions of some third and fourth order difference equations. Also some results contained in [2],[3] are improved.

Stability for Cellular Neural Networks with Delay

The cellular neural networks with delay (DCNN’s) are investigated, and some new sufficient conditions on asymptotical stability of DCNN’s are derived by constructing the Liapunov functional and utilizing M ? matrixand theω?limit set. It is shown that the new conditions are not related to the delayed parameter.

NON-WANDERING SET OF A CONTINUOUS GRAPH MAP

The non-wandering set Ω(f) for a graph map f is investigated. It is showed that Ω(f) is contained in the closure of the set ER(f) of eventually recurrent points of f and ω-limit set ω(Ω(f)) of Ω(f) is contained in the closure of the set R(f) of recurrent points of f.

ON THE EXISTENCE OF COMMON FIXED POINTS FOR A PAIR OF LIPSCHITZIAN MAPPINGS IN BANACH SPACES

The existence of common fixed points for a pair of Lipschitzian mappings in Banach spaces is proved. By using this result, some common fixed point theorems are also established for these mappings in Hilbert spaces, in L p spaces, in Hardy spaces H p, and in Sobolev spaces H r,p , for 1<p<+∞ and r≥0.

Special α-limit points and unilateral γ-limit points for graph maps

Let G be a graph and f ： G → G be a continuous map. Denote by P（f）, R（f）, SA（f） and UF（f） the sets of periodic points, recurrent points, special α-limit points and unilateral γ-limit points of f, respectively. In this paper, we show that R（f） SA（f） = UP（f） ∪ P（f） R（f）.

On the Finite Basis Problem for Certain 2-limited Words

Let X＊ be a free monoid over an alphabet X and W be a finite language over X. Let S（W） be the Rees quotient X＊/I（W）, where I（W） is the ideal of X＊ consisting of all elements of X＊ that are not subwords of W. Then S（W） is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S（W） is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.

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.

Asymptotic Behavior of Solutions to a Class of Systems of Delay Differential Equations

The authors investigate the asymptotic behavior of solutions to a class of systems of delay differential equations. It is shown that every bounded solution of such a class of systems tends to a constant vector as t→∞. Our results improve and extend some corresponding ones already known.

Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.

Almost automorphically-forced flows on S^(1) or R in one-dimensional almost periodic semilinear heat equations

In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0<x<L(0.1)with the periodic boundary condition u(t,0)=u(t,L),u_(x)(t,0)=u_(x)(t,L),(0.2)where f is uniformly almost periodic in t.In particular,we study the topological structure of the limit sets of the skew-product semiflow.It is proved that any compact minimal invariant set(throughout this paper,we refer to it as a minimal set)can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S^(1)=R/LZ(see Definition 2.4 for“residually embedded”).Particularly,if f(t,u,p)=f(t,u,-p),then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on R(see Definition 2.4 for“embedded”).Moreover,it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation.In addition,we further consider the asymptotic dynamics of the skew-product semiflow generated by(0.1)with the Neumann boundary condition u_(x)(t,0)=u_(x)(t,L)=0 or the Dirichlet boundary condition u(t,0)=u(t,L)=0.For such a system,it has been known that theω-limit set of any bounded orbit contains at most two minimal sets.By applying the new results for(0.1)+(0.2),under certain direct assumptions on f,we prove in this paper that the flow on any minimal set of(0.1)with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on R.Finally,a counterexample is given to show that even for quasi-periodically-forced equations,the results we obtain here cannot be further improved in general.

Long Time Behavior of the Cahn-Hilliard Equation with Irregular Potentials and Dynamic Boundary Conditions

The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered.The existence of the global attractor is proved and the long time behavior of the trajectories,namely,the convergence to steady states,is studied.

THE ATTRACTING CENTRE OFA CONTINUOUS TREE MAP

Let X be a tree and f be a continuous map from X into itself. Denote by г*the union of the set of one-side γ-limit points and the set of periodic points of f. We show in this paper that the attracting centre of f is г*.

SOME PROPERTIES OF CHAIN RECURRENT POINTS OF FLOWS

In this paper, two geometric criteria for a point to be a non-chain recurrent point of flows on a compact Hausdorff topological space are given.

ATTRACTORS AND QUASI-ATTRACTORS

In this paper, some properties of attractor and quasi-attractor are given. Some of them generalize Conley's results in [1,2].