Let $\Lambda \subset {ℝ}^n\times {ℝ}^m$ and $k$ be a positive integer. Let $f:{ℝ}^n\to {ℝ}^m$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda$, the derivatives $D_{\xi }^{j}f\left(x\right):=\frac{d^j}{d{t}^j}f(x+t\xi ){|}_{t=0}$, $j=1,2,\cdots k$, 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 $\Lambda$ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =({\eta }_1,{\eta }_2,\cdots ,{\eta }_m)$ and homogeneous of degree $k$ in $\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)$. 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...

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) $\psi \left(x\right)=(x,y_1\left(x\right)-y_2\left(x\right))$, $x\in [0,\alpha ]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha ]$. In other words: singularities propagate along arcs with finite turn. ...

The present paper studies the following constrained vector optimization problem: ${min}_{C}f\left(x\right)$, $g\left(x\right)\in -K$, $h\left(x\right)=0$, 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...

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 $T:L\left(f\right)\equiv \overline{span}f\left(X\right)\to X$ 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...

We deal with the integral equation $u\left(t\right)=f\left({\int }_{I}g(t,z)u\left(z\right)dz\right)$, with $t\in I=[0,1]$, $f:{𝐑}^n\to {𝐑}^n$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in L^{\infty }(I,{𝐑}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.