Christian Fabry

Periodic solutions of nonlinear differential equations with double resonance

SummaryWe prove the existence of periodic solutions of a second order nonlinear ordinary differential equation whose nonlinearity is at resonance with two successive eigenvalues of the associated linear operator and satisfies some Landesman-Laser type conditions at both of them.

Structure of the Fucik spectrum and existence of solutions for equations with asymmetric nonlinearities

Let L : dom L ⊂ L2(Ω) → L2(Ω) be a self-adjoint operator, Ω being open and bounded in RN. We give a description of the Fucik spectrum of L away from the essential spectrum. Let λ be a point in the discrete spectrum of L; provided that some non-degeneracy conditions are satisfied, we prove that the Fucik spectrum consists locally of a finite number of curves crossing at (λ, λ). Each of these curves can be associated to a critical point of the function H : x xs21A6 〈|x|,x〉L2 restricted to the unit sphere in ker(L – λI). The corresponding critical values determine the slopes of these curves. We also give global results describing the Fucik spectrum, and existence results for semilinear equations, by performing degree computations between the Fucik curves.

Unbounded motions of perturbed isochronous Hamiltonian systems at resonance

Abstract Large amplitude solutions of asymptotically positively homogeneous perturbations of hamiltonians systems at resonance can be unbounded, either in the past, or in the future. We present conditions for boundedness or unboundedness, generalizing in particular the results obtained by Alonso and Ortega [1] for scalar second order equations with asymmetric nonlinearities.

Nonlinear equations at resonance and generalized eigenvalue problems

IN A 1967 PAPER, Loud [25] obtained sharp nonresonance conditions for the second order differential equation -xn = g(x) + h(t). (1) More precisely, assuming g to be an odd function of class C’ and h to be a continuous T-periodic function, h being even and odd-harmonic, he proved the existence (and uniqueness) of a T-periodic solution to (l), whenever the range of the derivative of g does not interact with the set of eigenvalues ((21~i/T)~, i = 0, 1 , . . .) of the differential operator, i.e. for some n E N,

Asymmetric Nonlinear Oscillators

We review some results on large-amplitude periodic or almost periodic solutions of second order differential equations with asymmetric nonlinearities, when the system is close to “nonlinear resonance”.