The linear homogeneous differential equation with variable delays $ẏ\left(t\right)={\sum }_{j=1}^n{\alpha }_j\left(t\right)[y\left(t\right)-y(t-{\tau }_j\left(t\right))]$ is considered, where ${\alpha }_j\in C(I,ℝ͞͞⁺)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum }_{j=1}^n{\alpha }_j\left(t\right)>0$ on I, ${\tau }_j\in C(I,ℝ⁺),$ the functions $t-{\tau }_j\left(t\right)$, j=1,...,n, are increasing and the delays ${\tau }_j$ are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.

Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^p$-distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies$$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$where $\Omega$ is an open set of ${ℝ}^2$ and $E$ is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies $$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$ where Ω is an open set of ${ℝ}^2$ and E is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on φ.

We study differential equations $F(y,...,y^{\left(n\right)})=0$ where $F$ is a formal series in $y,y^{\prime },...,y^{\left(n\right)}$ with coefficients in some field of generalized power series${𝕂}_r$ with finite rank $r\in {\mathbb{N}}^{*}$. Our purpose is to express the support $\mathrm{Supp}{y}_0$, i.e. the set of exponents, of the elements $y_0\in {𝕂}_r$ that are solutions, in terms of the supports of the coefficients of the equation, namely $\mathrm{Supp}F$.

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded $H^{\infty }$-calculus and is based on elementary analysis.