We consider the problem of the existence of positive solutions u to the problem , (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.
Saddle connections and subharmonics are investigated for a class of forced second order differential equations which have a fixed saddle point. In these equations, which have linear damping and a nonlinear restoring term, the amplitude of the forcing term depends on displacement in the system. Saddle connections are significant in nonlinear systems since their appearance signals a homoclinic bifurcation. The approach uses a singular perturbation method which has a fairly broad application to saddle...
We extend the classical Leighton comparison theorem to a class of quasilinear forced second order differential equations where the endpoints , of the interval are allowed to be singular. Some applications of this statement in the oscillation theory of (*) are suggested.
We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class- estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class- estimate, exists if and only if the class- estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether...