Necessary and sufficient condition on the weights will be derived under which a $k$-th order Hardy inequality holds on classes of functions satisfying more than $k$ “boundary” conditions.

Let $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n\left(p\right)$ such that the inequality $$\sum _{i=1}^n{\left|a_i\right|}^p\ge c_n\left(p\right)$$ holds for all real numbers $a_1,...,a_n$ which are pairwise distinct and satisfy ${min}_{i\ne j}|a_i-a_j|=1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n\left(p\right)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.

We prove that if f: → is Darboux and has a point of prime period different from $2^i$, i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ${\nabla }_k$ is the gradient of order $k$, $ℳ$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m\left({ℝ}^n\right)\to W_p^l\left({ℝ}^n\right))$ is described.

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$, = φ: ℝ → ℝ|φ is continuous, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$. We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of integration, we consider the fractional q–derivative of Caputo type. Especially, its applications to q-exponential functions allow us to introduce q–analogues of the Mittag–Leffler function. Vice versa, those functions can be used for defining generalized operators in fractional q–calculus.