Let $F$ be a disjoint iteration semigroup of $C^n$ diffeomorphisms mapping a real open interval $I\ne ⌀$ onto $I$. It is proved that if $F$ has a dense orbit possesing a subset of the second category with the Baire property, then $F=\{f_t{f}_t\left(x\right)=f^{-1}(f\left(x\right)+t)\text{for}\text{every}x\in I,t\in R\}$ for some $C^n$ diffeomorphism $f$ of $I$ onto the set of all reals $R$. The paper generalizes some results of J.A.Baker and G.Blanton [3].

Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates $fⁿ:X\times {\Omega }^{\mathbb{N}}\to X$ defined by f¹(x,ω ) = f(x,ω₁), $f^{n+1}(x,\omega )=f(fⁿ(x,\omega ),{\omega }_{n+1})$.

Let x be an indeterminate over ℂ. We investigate solutions $$\begin{array}{c}\hfill \alpha (s,x)=\sum _{n\ge 0}{\alpha }_n\left(s\right)x^n,\end{array}$$αn : ℂ → ℂ, n ≥ 0, of the first cocycle equation $$\begin{array}{c}\hfill \alpha (s+t,x)=\alpha (s,x)\alpha \left(t,F(s,x)\right),s,t\in ,\left(\mathrm{Co}1\right)\end{array}$$in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation $$\begin{array}{c}\hfill F(s+t,x)=F(s,F(t,x\left)\right),s,t\in ,\left(\mathrm{T}\right)\end{array}$$of the form F(s,x) ≡ x + ck(s)xk mod xk+1, where k ≥ 2 and ck ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions αn(s) of $$\begin{array}{c}\hfill \alpha (s,x)=1+\sum _{n\ge 1}{\alpha }_n\left(s\right)x^n\end{array}$$are polynomials in ck(s).It is possible to replace...

We consider the problem of the vanishing of non-negative continuous solutions ψ of the functional inequalities (1)   ψ(f(x)) ≤ β(x,ψ(x)) and (2)   α(x,ψ(x)) ≤ ψ(f(x)) ≤ β(x,ψ(x)), where x varies in a fixed real interval I. As a consequence we obtain some results on the uniqueness of continuous solutions φ :I → Y of the equation (3)  φ(f(x)) = g(x,φ(x)), where Y denotes an arbitrary metric space.

Let f be a transcendental meromorphic function and $g\left(z\right)=f(z+c₁)+\cdots +f(z+c_k)-kf\left(z\right)$ and $g_k\left(z\right)=f(z+c₁)\cdots f(z+c_k)-f^k\left(z\right)$. A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_k\left(z\right)$, g(z)/f(z), and $g_k\left(z\right)/f^k\left(z\right)$.

The paper contains sufficient conditions under which all solutions of linear functional equations of the higher order are oscillatory.