DCDS
Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
Alberto Boscaggin Fabio Zanolin
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 89-110 doi: 10.3934/dcds.2013.33.89
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.
keywords: parameter dependent equations. lower and upper solutions subharmonic solutions Periodic solutions Poincaré-Birkhoff twist theorem
PROC
Infinitely many solutions to superquadratic planar Dirac-type systems
Alberto Boscaggin Anna Capietto
Conference Publications 2009, 2009(Special): 72-81 doi: 10.3934/proc.2009.2009.72
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.
keywords: Dirac-type systems Boundary value problem Rotation number
DCDS
Positive solutions to indefinite Neumann problems when the weight has positive average
Alberto Boscaggin Maurizio Garrione
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5231-5244 doi: 10.3934/dcds.2016028
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.
keywords: average condition Neumann problem Indefinite weight shooting method.

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