Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

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.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

The boundedness of the composition operator C_φf(z)=f(φ(z)) from the Hardy space , where X is the upper half-plane or the unit disk D={z∈C:|z|<1} in the complex plane C, to the nth weighted-type space, where φ is an analytic self-map of X, is characterized.     

A link between the Perron-Frobenius theorem and Perron’s theorem for difference equations

Let A_n,n∈N, be a sequence of k×k matrices which converge to a matrix A as n→∞. It is shown that if x_n,n∈N, is a sequence of nonnegative nonzero vectors such that

     

Tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means

In this paper, we introduce a new type of slow oscillation and slow decrease conditions. We prove that these or their variants are Tauberian conditions from to smn→s. We also prove that they are Tauberian conditions from to smn→s, where are the weighted means of the double sequence . Our results not only generalize well-known results, but also solve the conjecture of Móricz posed in [F. Móricz, Tauberian theorems for double sequences that are statistically summable (C,1,1), J. Math. Anal. Appl. 286 (2003) 340-350].     

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R₊×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H^∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H²(D)→H²(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable.     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying n_k+1/n_k?λ>1 for all k∈N, belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that lim_r→1(1-r²)^α‖Rf_r‖_p=0 if and only if . These results confirm two conjectures from the following recent paper [S. Stevi?, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, , respectively. Suppose {xn} is generated iteratively by

     

Convergence in mean of some random Fourier series

For a symmetric stable process X(t,ω) with index α∈(1,2], f∈Lp[0,2π], p?α, and , we establish that the random Fourier-Stieltjes (RFS) series converges in the mean to the stochastic integral , where fβ is the fractional integral of order β of the function f for . Further it is proved that the RFS series is Abel summable to . Also we define fractional derivative of the sum of order β for an, An(ω) as above and . We have shown that the formal fractional derivative of the series of order β exists in the sense of mean.     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and