Dans ce travail on s’intéresse aux opérateurs de composition $T_f\left(g\right):=f\circ g$ sur certains espaces de Besov et de Lizorkin-Triebel à valeurs dans ${ℝ}^m$. Dans le but de caractériser les fonctions qui opèrent, on établit que la condition de Lipschitz, locale ou globale suivant que l’espace $B_{p,q}^s({ℝ}^n,{ℝ}^m)$ ou $F_{p,q}^s({ℝ}^n,{ℝ}^m)$ se plonge ou non dans $L_{\infty }({ℝ}^n,{ℝ}^m)$, est nécessaire pour $s\>0$, et que l’appartenance locale au même espace l’est aussi pour $m\le n$. Nous étudions enfin la régularité de l’opérateur $T_f$.

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f:\Omega \times E\to 2^F$, denote by $N_f\left(x\right)$ the set of all measurable selections of the multifunction $f(·,x\left(·\right)):\Omega \to 2^F$, s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_f$ mapping some open domain G ⊂ X into $2^Y$, where X and Y are Köthe-Bochner...

We study the superposition operator f on on the space ac 0 of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac 0 with a = f 0, b ∈ D(ac 0), g a superposition operator from ℓ∞ into I(ac 0), D(ac 0) = {z: zx ∈ ac 0 for all x ∈ ac 0}, and I(ac 0) the maximal ideal in ac 0. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f...

We characterize the set of all functions f of R to itself such that the associated superposition operator Tf: g → f º g maps the class BVp1(R) into itself. Here BVp1(R), 1 ≤ p < ∞, denotes the set of primitives of functions of bounded p-variation, endowed with a suitable norm. It turns out that such an operator is always bounded and sublinear. Also, consequences for the boundedness of superposition operators defined on Besov spaces Bp,qs are discussed.

Let $\lambda$ and $\mu$ be solid sequence spaces. For a sequence of modulus functions $\Phi =\left({\varphi }_k\right)$ let $\lambda \left(\Phi \right)=\{x=\left(x_k\right)({\varphi }_k\left(\right|x_k\left|\right))\in \lambda \}$. Given another sequence of modulus functions $\Psi =\left({\psi }_k\right)$, we characterize the continuity of the superposition operators $P_f$ from $\lambda \left(\Phi \right)$ into $\mu \left(\Psi \right)$ for some Banach sequence spaces $\lambda$ and $\mu$ under the assumptions that the moduli ${\varphi }_k$$(k\in ...$

We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination principle. This is applied for the characterization of 1-summing Hammerstein operators on C(S,X)-spaces. For p-summing Hammerstein operators we derive the existence of control measures and p-summing extensions to B(Σ,X)-spaces.