Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).

We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator $D$ and we develop a technique based on homological algebra to prove index theorems for $D$.As particular cases, we obtain index theorems for $D$ acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal...

We investigate the existence of solutions to first order initial value problems for differential inclusions subject to impulsive effects. We shall rely on a fixed point theorem for condensing maps to prove our results.

The main goal of the paper is to formulate some new properties of the Ishlinskii hysteresis operator $F$, which characterizes e.g. the relation between the deformation and the stress in a non-perfectly elastic (elastico-plastic) material. We introduce two energy functionals and derive the energy inequalities. As an example we investigate the equation $u^{\text{'}\text{'}}+F\left(u\right)=0$ describing the motion of a mass point at the extremity of an elastico-plastic spring.

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill x^{\left(n\right)}\left(t\right)\in -A_1(t,x\left(t\right),...,x^{(n-1)}\left(t\right))x^{(n-1)}\left(t\right)-...-A_n(t,x\left(t\right),...,& x^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\text{a.a.}t\in [a,\infty ),\end{array}$$ where $a\in (0,\infty )$, and $A_i:[a,\infty )\times {ℝ}^n\to ℝ$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of $n$th order with complex coefficients $M\left[y\right]-\lambda wy=wf(t,y^{\left[0\right]},...,y^{[n-1]})$, $t\in [a,b)$ provided that all $r$th quasi-derivatives of solutions of $M\left[y\right]-\lambda wy=0$ and all solutions of its normal adjoint $M^{+}\left[z\right]-\overline{\lambda }wz=0$ are in $L_w^2(a,b)$ and under suitable conditions on the function $f$.

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we...