By applying the Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three unbounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature.

In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks...). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro.Finally, we reformulate some uncertainty principles in terms of properties of the free heat and shrödinger equations.

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi's theorem, Beurling's theorem, and Donoho-Stark's uncertainty principle are obtained for the Weinstein transform.

The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the $A_n$-acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary...

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_0\in X$, where $X\in \{M_{2,q}^s\left(ℝ\right),H^{\sigma }\left(𝕋\right),H^{s_1}\left(ℝ\right)+H^{s_2}\left(𝕋\right)\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s\left(ℝ\right)\hookrightarrow L^3\left(ℝ\right)$, or if $\sigma \ge \frac{1}{6}$, or if $s_1\ge \frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...

On construit par voie géométrique une classe de symboles classiques en dehors d’une sous-variété. La classe d’opérateurs pseudodifférentiels associée contient les paramétrix d’opérateurs tels que ${\sum }_{i=1}^{n-1}{\left(\frac{\partial }{\partial x_i}\right)}^4+{\left(\frac{\partial }{\partial x_n}\right)}^3$ ou ${\left(\frac{\partial }{\partial x_n}\right)}^3+{\sum }_{i=1}^{n-1}{\left(\frac{\partial }{\partial x_i}\right)}^2.$