Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert’s metric or Thompson’s metric. We establish several Denjoy-Wolff type theorems which confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.     

A simple proof of Olatinwo’s theorems

In this paper we provide a simple proof of the existence coupled fixed point theorem in complete cone metric spaces due to Sabetghadam et al. (Fixed Point Theory Appl 2009:8, 2009) and due to Olatinwo (Annali Dell’Universita’Di Ferrara 57:173–180, 2011). In particular we prove that these results are spacial cases of Rezapour and Hamlbarani’s theorems (J Math Anal Appl 345(2):719–724, 2008).     

A weighted algebra analogues of Wiener’s and Lévy’s theorems

We obtain weighted algebra analogues of the classical theorems of Weiner and Lévy on absolutely convergent Fourier series.     

Wiener’s and Lévy’s theorems for some weighted power series

We obtain weighted version of the classical theorems of Wiener and Lévy on absolutely convergent power series.     

Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems

In the L _p spaces, 1 < _p < ∞, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff’s theorem in the presence of bounds on the convergence rate in von Neumann’s ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in L _p are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.     

Ball convergence theorems for Halley’s method in Banach space

We provide local convergence results for Halley??s method in order to approximate a locally unique zero of an operator in a Banach space setting using convex majorants. Kantorovich-type and Smale-type results are considered as applications and special cases.     

Kato’s inequality and Liouville theorems on locally finite graphs

In this paper, we study the Kato’s inequality on locally finite graphs. We also study the application of Kato’s inequality to Ginzburg-Landau equations on such graphs. Interesting properties of elliptic and parabolic equations on the graphs and a Liouville type theorem are also derived.     

A note on two Camina’s theorems on conjugacy class sizes

Let G be a finite group. We mainly investigate how certain arithmetical conditions on conjugacy class sizes of some elements of biprimary order of G influence the structure of G. Some known results are generalized.     

Darling-Erdős theorems for martingales

This paper considers Darlin-Erds theorems for sums of martingale differences. Our main theorem provides an optimal result for the case of bounded martingale difference sequences. A number of other results are presented, which deal with the unbounded case and which specialize to the case of independent summands. Previous related work on this problem has been based on deep strong approximation theorems. One of the novel features of our approach is that our methods rely on the more easily accessible Skorokhod-type embeddings.     

A quantitative version of Krein’s theorems for Fréchet spaces

For a Banach space E and its bidual space E ^′′, the following function ${k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}$ defined on bounded subsets H of E measures how far H is from being σ(E, E′)-relatively compact in E. This concept, introduced independently by Granero [10] and Cascales et al. [7], has been used to study a quantitative version of Krein’s theorem for Banach spaces E and spaces C _p (K) over compact K. In the present paper, a quantitative version of Krein’s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows ${k(H) := {\rm sup}\{d(h, E) : h \in \overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}\},}$ where d(h, E) is the natural distance of h to E in the bidual E ^′′. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds ${k(coH) < (2^{n+1} - 2) k(H) + \frac{1}{2^{n}}}$ for all ${n \in \mathbb{N}}$ . Consequently this yields also the following formula ${k(coH) \leq \sqrt{k(H)}(3 - 2\sqrt{k(H)})}$ . Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein’s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet space. We also define and discuss two other measures of weak non-compactness lk(H) and k′(H) for a Fréchet space and provide two quantitative versions of Krein’s theorem for both functions.     

A short proof of Fel’s theorems on 3-D Frobenius problem

The Frobenius number and the generating function for numerical additive semigroups with three generators is obtained in a way shorter with respect to the original proof by L. Fel.     

Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems

The known theorems by E.A. Barbashin and N.N. Krasovskii (1952) about the asymptotic and global stability of an equilibrium state for an autonomous system of differential equations are extended to nonautonomous differential inclusions with closed-valued (but not necessarily compact-valued) right-hand sides, where the equilibrium state is a weakly invariant (with respect to solutions of the inclusion) set. The statements are formulated in terms of the Hausdorff-Bebutov metric, the dynamical system of translations corresponding to the right-hand side of the differential inclusion, and the weakly invariant set corresponding to the inclusion.     

Sperner’s lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperner’s lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.     

Companion theorems to G. Szegö’s inequality for pairs of coefficients of bounded polynomials

We consider real univariate polynomials $P_n$ of degree $ \le n $ from class $$\begin{aligned} \mathbf {C}_n = \{P_n:|P_n \left( \cos \displaystyle \frac{(n -i)\pi }{n}\right) |\le 1 \; \text{ for }\; 0\le i\le n \} \end{aligned}$$ which encompasses the unit ball of polynomials with respect to the uniform norm on $[- 1, 1]$ . For pairs of consecutive coefficients of $ P_n(x) = \sum \nolimits _{k=0}^{n}a_kx^k$ there holds the inequality 1 $$\begin{aligned} |a_{k-1}|+|a_k|\le |t_{n,k}|, \quad \text{ if }\; k\equiv n\; \text{ mod }\; 2, \end{aligned}$$ where $T_n(x)=\sum \nolimits _{k=0}^{n} t_{n,k}x^k$ is the $n$ -th Chebyshev polynomial of the first kind. (1) implies Markov’s classical coefficient inequality of 1892 (Math. Ann. 77:213–258, 1916, p. 248) and goes back to Szegö, but was made public by P. Erdös (Bull. Am. Math. Soc. 53:1169–1176, 1947, p. 1176) in 1947. We ask here: will the (nonzero) coefficients of $T_n$ likewise majorize complementary pairs $|a_k| + |a_{k+1}|$ ? More generally: does there hold 2 $$\begin{aligned} |a_k| + |a_j| \le |t_{n,k}| \quad \text{ for } \text{ all }\; P_n \in \mathbf {C_n}, \end{aligned}$$ $\text{ where }\; k < j \;\text{ and }\; k\equiv n\mod 2\;\text{ but }\; j\not \equiv n\mod 2 ?$ We treat the marginal cases $n < 12$ separately, and for $n \ge 12$ we provide answers to this question with the aid of the explicitly determined optimal bound $K \sim \lceil \frac{n}{\sqrt{2}}\rceil $ which incorporates the height and the length of $ \frac{T'_n(x)}{n}$ . Theorem 2.1: (2) holds, provided $K \le k < j$ ; in particular, provided $\frac{n}{\sqrt{2}}<k<j$ . As a corollary we reveal new extremal properties of the leading coefficients of $\pm T_n$ . Theorem 2.4: (2) does not hold if $k < j < K$ . Theorem 2.5: If $k < K < j$ , then (2) holds for certain, but not for all, $k$ and $j$ . If we keep fixed $j = n-1> K$ , then (2) holds for all $k$ with $k_{*}\le k < j$ , where the bound $k_{*} < K$ is explicitly determined, and is optimal for $n\le 43$ . In Theorem 2.6 we return to G. Szegö’s original inequality (1) and constructively prove the non - uniqueness of its extremizer $\pm T_n$ .     

Remarks on orthocenters,Pappus’ theorem and Butterfly theorems

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.