Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.     

The ω-limit set of a graph map

Let G be a graph and be continuous. Denote by P(f), , ω(f) and Ω(f) the set of periodic points, the closure of the set of periodic points, ω-limit set and non-wandering set of f, respectively. In this paper we show that: (1) v∈ω(f) if and only if v∈P(f) or there exists an open arc L=(v,w) contained in some edge of G such that every open arc U=(v,c)⊂L contains at least 2 points of some trajectory; (2) v∈ω(f) if and only if every open neighborhood of v contains at least r+1 points of some trajectory, where r is the valence of v; (3) ; (4) if , then x has an infinite orbit.     

Denjoy-Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators

Let K be a closed cone with nonempty interior in a Banach space X. Suppose that is order-preserving and homogeneous of degree one. Let be a continuous, homogeneous of degree one map such that q(x)>0 for all x∈K?{0}. Let T(x)=f(x)/q(f(x)). We give conditions on the cone K and the map f which imply that there is a convex subset of ∂K which contains the omega limit set ω(x;T) for every x∈intK. We show that these conditions are satisfied by reproduction-decimation operators. We also prove that ω(x;T)⊂∂K for a class of operator-valued means.     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x^′=f(x), x(0)=x₀, where f:X→X is Lipschitz, as being of three types I-III. We denote by SX the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x₀∈X. We say that S∈SX is of type I if there exists a Lipschitz function f and a solution x such that S=Ω(x) and . We say that S∈SX is of type II if it has non-empty interior. We say that S∈SX is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S=Ω(x) it holds . Our main results are the following: S is a type I set in SX if and only if S is a closed and separable subset of the topological boundary of an open and connected set U⊂X. Suppose that there exists an open separable and connected set U⊂X such that , then S is a type II set in SX. Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen Ck-smooth whenever the underlying Banach space is Ck-smooth.     

Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of 2n-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271-277] proved that for even integer k, if G-x has a k-factor for each x∈V(G), then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239-242] generalized Katerinis’ result to f-factors, and proved that if G-x has an f-factor for each x∈V(G), then G has an f-factor for an integer-valued function f defined on V(G) with even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether G-x has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on V(G) with even, |V(G)-X|?2. We prove that if and if G-x has an f-factor for each x∈V(G)-X, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition with . Furthermore the condition will be when |X|=1.     

Sensitivity of set-valued discrete systems

Consider the surjective, continuous map f:X→X and the continuous map of K(X) into itself induced by f, where X is a compact metric space and K(X) is the space of all non-empty compact subsets of X endowed with a Hausdorff metric. In this paper we give examples showing that sensitivity of f does not imply sensitivity of . Furthermore, we prove that if f is a surjective, continuous interval map, then is sensitive if and only if f is sensitive.     

Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x₀∈X, the set X_w={x∈X:lim_λ→∞w(λ,x,x₀)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case X_w coincides with the set of all x∈X such that w(λ,x,x₀)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Primitive elements in rings of continuous functions

Let be a surjective continuous map between compact Hausdorff spaces. The map π induces, by composition, an injective morphism C(Y)→C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with algebraic properties of the ring extension C(Y)⊆C(X) in relation to topological properties of the map . We prove that if the extension C(Y)⊆C(X) has a primitive element, i.e., C(X)=C(Y)[f], then it is a finite extension and, consequently, the map π is locally injective. Moreover, for each primitive element f we consider the ideal and prove that, for a connected space Y, If is a principal ideal if and only if is a trivial covering.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Brouwer's degree without properness

A degree deg(f,y) is defined for every continuous function , which possesses all the properties of Brouwer's degree provided that y is restricted to the complement of some closed set A(f) of “asymptotic” values. Sufficient conditions are given for A(f) to be nowhere dense. It is also shown that, in the opposite direction, A(f) having nonempty interior has a direct impact on the solutions of f(x)=y, which cannot be discovered by degree arguments.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .