Unbounded Solutions to Systems of Differential Equations at Resonance

Journal of Dynamics and Differential Equations - We deal with a weakly coupled system of ODEs of the type $$\begin{aligned} x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots ,x_d) = p_j(t), \qquad...     

A note on the existence of multiple solutions for a class of systems of second order ODEs

We prove the existence of multiple solutions for some systems of second order ODEs with Dirichlet boundary conditions. Such systems are obtained by coupling scalar ODEs with different growth conditions. The proof relies on a global continuation technique.     

Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition

We consider the planar Hamiltonian system $$Ju^{\prime} = \nabla F(u) + \nabla_u R(t,u), \quad t \in [0,T], \,u \in \mathbb{R}^2,$$ with F(u) positive and positively 2-homogeneous and ${\nabla_{u}R(t, u)}$ sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.     

Multiple Solutions to Neumann Problems with Indefinite Weight and Bounded Nonlinearities

We study the Neumann boundary value problem for the second order ODE

$$\begin{aligned} u^{\prime \prime } + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{aligned}$$

(1)

where \(g \in {\mathcal {C}}^1({\mathbb {R}})\) is a bounded function of constant sign, \(a^+,a^-: [0,T] \rightarrow {\mathbb {R}}^+\) are the positive/negative part of a sign-changing weight \(a(t)\) and \(\mu > 0\) is a real parameter. Depending on the sign of \(g^{\prime }(u)\) at infinity, we find existence/multiplicity of solutions for \(\mu \) in a “small” interval near the value

$$\begin{aligned} \mu _c = \frac{\int _0^T a^+(t) \, dt}{\int _0^T a^-(t) \, dt}\,. \end{aligned}$$

The proof exploits a change of variables, transforming the sign-indefinite Eq. (1) into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for \(\mu \rightarrow 0^+\) and \(\mu \rightarrow +\infty \) are given, as well.

     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].     

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE$u^{″}+\lambda a(t)g(u)=0,\lambda >0,$ where $g(x)$ is a positive function on ${\mathbb{R}}^{+}$, superlinear at zero and sublinear at infinity, and $a(t)$ is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities $g(x)$ when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.     

Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré-Birkhoff theorem

In the general setting of a planar first order system

(0.1)     

Subharmonic solutions of the forced pendulum equation: a symplectic approach

Using the Poincaré–Birkhoff fixed point theorem, we prove that for every β > 0 and for a large (both in the sense of prevalence and of category) set of continuous and T-periodic functions \({f: \mathbb{R} \to \mathbb{R}}\) with \({\int_0^T f(t)\,dt = 0}\) , the forced pendulum equation $$x'' + \beta \sin x = f(t) $$ has a subharmonic solution of order k for every large integer number k. This improves the well known result obtained with variational methods, where the existence when k is a (large) prime number is ensured.