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 formulate a Covering Property Axiom $CP{A}_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨fₙ\in {ℝ}^{ℝ}{⟩}_{n<\omega }$ of Borel functions there are sequences:...

Pour trois fonctions non-négatives intégrables sur ${\mathbf{R}}^n$, $f,g$ et $h$, telles que $\left(h\right(x+y){)}^{-1/n}\le (f\left(x\right){)}^{-1/n}+\left(g\right(y){)}^{-1/n}$, Borelll a établi l’inégalité $\int h\left(z\right)dz\ge \min \int f\left(x\right)dx,\int g\left(y\right)dy)$. Nous déterminons les conditions précises où l’inégalité sera stricte. La clef de cette analyse est une nouvelle caractérisation des fonctions convexes mesurables.

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

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.