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Let X, Y be real normed spaces and let $$\rho '_+$$ ρ + ′ , $$\rho '_-$$ ρ - ′ be norm derivatives. In this work, we solve a system of functional equations $$\begin{aligned} {\left\{ \begin{array}{ll}\rho '_+(f(x),f(y))=g(x)\rho '_+(x,y),\\ \rho '_-(f(x),f(y))=g(x)\rho '_-(x,y), \end{array}\right. } \end{aligned}$$ ρ + ′ ( f ( x ) , f ( y ) ) = g ( x ) ρ + ′ ( x , y ) , ρ - ′ ( f ( x ) , f ( y ) ) = g ( x ) ρ - ′ ( x , y ) , with unknown functions $$f:X\!\rightarrow \!Y$$ f : X → Y , $$g:X\rightarrow \mathbb {R}$$ g : X → R . Moreover, we give partial answer to open problem posed in 2010.

The purpose of this note is to prove several binomial-like formulas whose exponents are values of the function ω(n) counting distinct prime factors of n.

In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set.

In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the orbit in flows. With the aid of the Lefschetz fixed point theorem and the presence of a non-trivial limit set we prove the existence of asymptotically non-periodic orbits.

The investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

In this article we describe selected topological and dynamical properties of asymptotically periodic motions in continuous dynamical systems (flows). The main result is to show Lagrange stability (i.e. the closure of positive orbit is compact) of such motion with the aid of topological properties of limit sets. Two sufficient conditions for this kind of stability are provided: the value of asymptotic period (our approach to this notion differs from the usual one) and proper metric space property.