We present a new theorem on the differential inequality $u^{\left(m\right)}\le w\left(u\right)$. Next, we apply this result to obtain existence theorems for the equation $x^{\left(m\right)}=f(t,x)$.

In this paper, we employ some new techniques to study the existence of positive periodic solution of $n$-species neutral delay system $$N_i^{\text{'}}\left(t\right)=N_i\left(t\right)\left[a_i\left(t\right)-\sum _{j=1}^n{\beta }_{ij}\left(t\right)N_j\left(t\right)-\sum _{j=1}^n{b}_{ij}\left(t\right)N_j(t-{\tau }_{ij}\left(t\right))-\sum _{j=1}^n{c}_{ij}\left(t\right)N_j^{\text{'}}(t-{\tau }_{ij}\left(t\right))\right].$$ As a corollary, we answer an open problem proposed by Y. Kuang.

In questa conferenza descrivo alcuni recenti sviluppi relativi al problema dell'unicità per l'equazione differenziale ordinaria e per l'equazione di continuità per campi vettoriali debolmente differenziabili. Descrivo infine un'applicazione di questi risultati a un sistema di leggi di conservazione.

Let ${\ell }_j:=-d²/dx²+k²q_j\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_j\left(x\right)\le esssup{q}_j\left(x\right)<\infty$. Suppose that (*) ${\int }_0^{1}p\left(x\right)u₁(x,k)u₂(x,k)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem ${\ell }_j{u}_j=0$, 0 ≤ x ≤ 1, $u_j^{\text{'}}(0,k)=0$, $u_j(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

In this paper we introduce the definition of coupled point with respect to a (scalar) quadratic functional on a noncompact interval. In terms of coupled points we prove necessary (and sufficient) conditions for the nonnegativity of these functionals.

In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.

In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.