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DCDS

We study the problem of existence and multiplicity of subharmonic
solutions for a second order nonlinear ODE in presence of
lower and upper solutions. We show how such
additional information
can be used to obtain more precise multiplicity results.
Applications are given to pendulum type equations and to
Ambrosetti-Prodi results for parameter dependent equations.

PROC

It is proved the existence of infinitely many solutions to a superquadratic Dirac-type boundary value problem of the form $\tau z = \nabla_z F(t,z)$, $y(0) = y(\pi) = 0$ ($z=(x,y)\in \mathbb{R}^2 $). Solutions are distinguished by using the concept of rotation number. The proof is performed by a global bifurcation technique.

DCDS

We deal with positive solutions for the Neumann boundary value problem associated with
the scalar second order ODE
$$
u'' + q(t)g(u) = 0, \quad t \in [0, T],
$$
where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to
previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(u) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type
$$
x' = y, \qquad y' = h(x)y^2 + q(t),
$$
with $h(x)$ a continuous function defined on the whole real line.

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