Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

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^′.     

Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if B_Eb is relatively compact with respect to the F topology and f:I×E_b→E_b is a measurable family of F-continuous maps then for each x₀∈E_b there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥_F norm) local solution to the initial valued problem u_t(t)=f(t,u(t)), u(t₀)=x₀. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.     

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,∞).     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^−tf(z,t)}_t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e^−tf(z,t)}_t?0 is a normal family on B. Finally we show that the set S⁰(B) of mappings which have parametric representation on B is compact.     

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).     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

Doubly nonlinear periodic problems with unbounded operators

The solvability of the evolution system v′(t)+B(t)u(t)∋f(t), v(t)∈A(t)u(t), 0<t<T, with the periodic condition v(0)=v(T) is investigated in the case where are bounded, possibly degenerate, subdifferentials and are unbounded subdifferentials.     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

Classical solutions to differential inclusions with totally disconnected right-hand side

Let F=F(t,x) be a bounded, Hausdorff continuous multifunction with compact, totally disconnected values. Given any y₀∈F(t₀,x₀), we show that the differential inclusion has a globally defined classical solution, with x(t₀)=x₀, .     

An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)ⁿx^(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x^(Δ∇)i(a)=0 and x^(Δ∇)i(c)=βx^(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

A bilinear Orlicz-Pettis Theorem

Suppose E,F,G are locally convex spaces, is a bilinear operator and λ is a scalar sequence space. A series j_∑xj in E is λ b multiplier convergent if for every t={tj}∈λ there exists xt∈E such that for every y∈F. Under continuity assumptions on the linear operators b(x,⋅), we establish several versions of the Orlicz-Pettis Theorem for multiplier convergent series. Applications to spaces of continuous linear operators are given.