We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...

Many authors have recently studied compact and weakly compact homomorphisms between function algebras. Among them, Lindström and Llavona [2] treat weakly compact continuous homomorphisms between algebras of type C(T) when T is a completely regular Hausdorff space.Llavona asked wether the results in [2] are valid in the case of algebras of differentiable functions on Banach spaces. The purpose of this note is to give an affirmative answer to this question, by proving that weakly compact homomorphisms...

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 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.

A new construction method for aggregation operators based on a composition of aggregation operators is proposed. Several general properties of this construction method are recalled. Further, several special cases are discussed. It is also shown, that this construction generalizes a recently introduced twofold integral, which is exactly a composition of the Choquet and Sugeno integral by means of a min operator.

In this paper we shall establish a result concerning the covering dimension of a set of the type $\{x\in X:\Phi (x)\cap \Psi (x)\ne \varnothing \}$, where $\Phi$, $\Psi$ are two multifunctions from $X$ into $Y$ and $X$, $Y$ are real Banach spaces. Moreover, some applications to the differential inclusions will be given.

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\mathrm{BV}}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X={ℝ}^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\mathrm{BV}}$ and $C^2$ parametrization problems are equivalent for $X=ℝ$ but are not equivalent for $X={ℝ}^2$.

In this paper we consider the Darboux type properties for the paratingent. We review some of the standard facts on the multivalued functions and the paratingent. We prove that the paratingent has always the Darboux property but the property D* holds only when the paratingent is a multivalued function.