In this paper, we develop monotone iterative technique to obtain the extremal solutions of a second order periodic boundary value problem (PBVP) with impulsive effects. We present a maximum principle for ``impulsive functions'' and then we use it to develop the monotone iterative method. Finally, we consider the monotone iterates as orbits of a (discrete) dynamical system.

The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent...

Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\overline{x}=\varphi \left(x\right)$, $\overline{y}=L\left(x\right)y$, $\overline{z}=M\left(x\right)z,...$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\text{'}}=f(x,y,z,z^{\text{'}})$) and certain differential equations with deviation (if $z=y\left(\xi \right(x\left)\right)$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the...

We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $$y^{\left(n\right)}+r\left(t\right)f(y,y^{\text{'}},\cdots ,y^{(n-1)})=0.$$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

We study the large-time behaviour of the nonlinear oscillator$$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$where $m,k\>0$ and $f$ is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$ with $\alpha$ real, $A\>0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

We study the large-time behaviour of the nonlinear oscillator $$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$ where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case  $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

Let (P,Q) be a C1 vector field defined in a open subset U ⊂ R2. We call a null divergence factor a C1 solution V (x, y) of the equation P ∂V/∂x + Q ∂V/ ∂y = ( ∂P/∂x + ∂Q/∂y ) V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method...

The period function of a planar parameter-depending Hamiltonian system is examined. It is proved that, depending on the value of the parameter, it is either monotone or has exactly one critical point.