This paper presents a sufficient condition for the stability of periodic solutions of a newtonian equation. This condition depends on the third order approximation and does not involve small parameters. An application to an equation with cubic potential is given. 1996 Academic Press, Inc.
We consider semilinear elliptic problems of the form ∆u + g(u) = f (x) with Neumann boundary conditions or ∆u + λ 1 u + g(u) = f (x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained .
This paper studies the existence of bounded solutions of a forced non-linear differential equation of arbitrary order. Necessary and sufficient conditions for the existence of such solutions are obtained. These results are inspired by classical results on the periodic problem, both in the resonant and non-resonant cases.
The method of upper and lower solutions is a classical tool in the theory of periodic differential equations of the second order. We show that this method does not have a direct extension to almost periodic equations. To do this we construct equations of this type without almost periodic solutions but having two constants as ordered upper and lower… (More)
Results of the Landesman–Lazer type provide necessary and sufficient conditions for the existence of periodic solutions of certain nonlinear differential equations with forcing. Typically, they deal with scalar problems. This paper presents a discussion of possible extensions to systems. The emphasis is placed on the new phenomena produced by the increase… (More)
Motivated by the problem of the existence of a solution of the nonlinear telegraph equation wt + clll-u,, + h(t. c, u) = 0, such that u(t, ,) satisfies suitable boundary conditions over (0,~) ar;d Ilu(t,.)II is bounded over W for some function space norm 11 11: we prove the existence of bounded solutions over R of semilinear evolution equations in a Hilbert… (More)
A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.
We consider a forced harmonic oscillator at resonance with a nonlinear perturbation and obtain a sharp condition for the existence of unbounded motions. Such a condition is extended to the case of a semilinear vibrating string.