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

Curves in Banach spaces which allow a $C^{1, BV}$ parametrization or a parametrization with finite convexity

Jakub Duda, Luděk Zajíček (2013)

Czechoslovak Mathematical Journal

We give a complete characterization of those $f : [0, 1] \to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1, BV}$ 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 $C^{1, BV}$ and $C^{2}$ parametrization problems are equivalent for $X = ℝ$ but are not equivalent for $X = ℝ^{2}$ .

Cyclic mean-value inequalities for the gamma function

Horst Alzer (2013)

Colloquium Mathematicae

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