We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.

MSC 2010: 26A33, 34D05, 37C25In the paper, long-time behavior of solutions of autonomous two-component incommensurate fractional dynamical systems with derivatives in the Caputo sense is investigated. It is shown that both the characteristic times of the systems and the orders of fractional derivatives play an important role for the instability conditions and system dynamics. For these systems, stationary solutions can be unstable for wider range of parameters compared to ones in the systems with...

Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas être rendus Γ par changement des deux topologies polonaises.

We evaluate the descriptive set theoretic complexity of the space of continuous surjections from ${ℝ}^m$ to ${ℝ}^n$.

We describe the average behaviour of the Brjuno function Φ in the neighbourhood of any given point of the unit interval. In particular, we show that the Lebesgue set of Φ is the set of Brjuno numbers and we find the asymptotic behaviour of the modulus of continuity of the integral of Φ.

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=[a,b]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy $h_D\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.

Let F t: t ≥ 0 be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $$G\left(x\right)=\underset{s\to 0}{lim}\left(F^0\left(x\right)-F^s\left(x\right)\right)/\phantom{\left(F^0\left(x\right)-F^s\left(x\right)\right)\left(-s\right)}\left(-s\right)$$ for x ∈ K.

Let $\Omega \subset {ℝ}^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1{L}^n{log}^{\alpha }L\left(\Omega \right)$ into an Orlicz space corresponding to a Young function which behaves like $exp\left(t^{n/(n-1-\alpha )}\right)$ for large $t$. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very...