Boundary blow-up for differential equations with indefinite weight

We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0⁺ and t→1⁻. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1].     

Periodic Motion of a System of Two or Three Charged Particles

We provide necessary and sufficient conditions for the existence of T-periodic solutions of a system of second-order ordinary differential equations that models the motion of two or three collinear charged particles of the same sign.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

A continuation theorem for periodic boundary value problems with oscillatory nonlinearities

We prove a continuation theorem for the solvability of the coincidence equationLx=Nx in normed spaces. Applications are given to the periodic boundary value problem for second order ordinary differential equations. Dealing, in particular, with the periodically forced Duffing equation

Existence results for forced nonlinear periodic BVPs at resonance

Sunto In questo lavoro si presentano alcuni risultati riguardanti l'esistenza di soluzioni p-periodiche per sistemi di equazioni differenziali non lineari in risonanza, del tipo x+ Dx + + Ag(t, x)=h(t), ove D ed A sono matrici m×m, con D di tipo diagonale, h è un termine forzante p-periodico e g è un campo vettoriale, non necessariamente limitato. In particolare, viene esteso ai sistemi, in ipotesi più generali, un classico teorema dovuto a Lazer e Leach. Le dimostrazioni sono basate sull'uso del grado topologico (teorema di continuazione di Mawhin).     

Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

We study the second-order nonlinear differential equation \(u'' + a(t) g(u) = 0\), where \(g\) is a continuously differentiable function of constant sign defined on an open interval \(I\subseteq {\mathbb R}\) and \(a(t)\) is a sign-changing weight function. We look for solutions \(u(t)\) of the differential equation such that \(u(t)\in I,\) satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for \(I = {\mathbb R}^+_0\) and \(g(u) \sim - u^{-\sigma },\) as well as the case of exponential nonlinearities, for \(I = {\mathbb R}\) and \(g(u) \sim \exp (u)\). The proofs are obtained by passing to an equivalent equation of the form \(x'' = f(x)(x')^2 + a(t)\).     

Chaotic dynamics in a vertically driven planar pendulum

Using elementary phase-plane analysis, combined with results from the theory of topological horseshoes and linked twist maps, we prove the presence of chaos-like dynamics for a vertically driven planar pendulum and other, more general, related equations.