A characterization of $C^{1,1}$ functions via lower directional derivatives

Dušan Bednařík; Karel Pastor

Displaying similar documents to “A characterization of $C^{1,1}$ functions via lower directional derivatives”

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

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Let $Λ \subset ℝ^{n} \times ℝ^{m}$ and $k$ be a positive integer. Let $f : ℝ^{n} \to ℝ^{m}$ be a locally bounded map such that for each $(ξ, η) \in Λ$ , the derivatives $D_{ξ}^{j} f (x) : = \frac{d^{j}}{d t^{j}} f (x + t ξ) |_{t = 0}$ , $j = 1, 2, \dots k$ , exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^{k}$ it is necessary and sufficient that $Λ$ be not contained in the zero-set of a nonzero homogenous polynomial $Φ (ξ, η)$ which is linear in $η = (η_{1}, η_{2}, \dots, η_{m})$ and homogeneous of degree $k$ in $ξ = (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ . This generalizes a result of J. Boman for the case $k = 1$ . The statement and the proof of a theorem of Boman for the case $k = \infty$ is...

A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

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P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $ℝ^{n}$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n = 2$ , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $ψ (x) = (x, y_{1} (x) - y_{2} (x))$ , $x \in [0, α]$ , where $y_{1}$ , $y_{2}$ are convex and Lipschitz on $[0, α]$ . In other words: singularities propagate along arcs with finite turn. ...

Derivatives of Hadamard type in scalar constrained optimization

Karel Pastor (2017)

Kybernetika

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Vsevolod I. Ivanov stated (Nonlinear Analysis 125 (2015), 270-289) the general second-order optimality condition for the constrained vector problem in terms of Hadamard derivatives. We will consider its special case for a scalar problem and show some corollaries for example for $ℓ$ -stable at feasible point functions. Then we show the advantages of obtained results with respect to the previously obtained results.

Locally Lipschitz vector optimization with inequality and equality constraints

Ivan Ginchev, Angelo Guerraggio, Matteo Rocca (2010)

Applications of Mathematics

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The present paper studies the following constrained vector optimization problem: ${min}_{C} f (x)$ , $g (x) \in - K$ , $h (x) = 0$ , where $f : ℝ^{n} \to ℝ^{m}$ , $g : ℝ^{n} \to ℝ^{p}$ are locally Lipschitz functions, $h : ℝ^{n} \to ℝ^{q}$ is $C^{1}$ function, and $C \subset ℝ^{m}$ and $K \subset ℝ^{p}$ are closed convex cones. Two types of solutions are important for the consideration, namely $w$ -minimizers (weakly efficient points) and $i$ -minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point $x^{0}$ to be a $w$ -minimizer and first-order sufficient conditions...

Universal stability of Banach spaces for ε -isometries

Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou (2014)

Studia Mathematica

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Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T : L (f) \equiv \bar{s p a n} f (X) \to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally...