Periodic perturbations of a class of resonant problems

We study the existence of solution for nonlinear problems at resonance under Dirichlet boundary conditions. We deal with PDE's as well as systems of ODE's. The nonlinear terms considered are periodic functions: in particular, the problem is strongly resonant at infinity. By means of variational methods, we prove nondegeneracy under some hypotheses on the nonlinearities. Received: 31 October 2003, Accepted: 12 July 2004, Published online: 8 February 2005 Mathematics Subject Classification (2000): 34B15, 35B34, 35J20 The authors have been supported by the Ministry of Science and Technology of Spain (BFM2002-02649), and by J. Andalucía (FQM 116)     

Stability for a damped nonlinear oscillator

The stability of the null solution of Eq. (1.1) below is discussed. Under some unusual assumptions, we obtain new stability results for this classical oscillator equation. Our approach allows extensions to both the vector case and the case of the whole real line.     

Relaxation of nonlinear oscillator

Institute of Nuclear Research, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 3, pp. 367–374, March, 1991.     

Singular perturbations for a class of nonlinear hyperbolic-hyperbolic problems

We show how, for nonlinear equations satisfying certain symmetry conditions, the determining equations reduce for solutions taken in some invariant subspace.     

Multivalued perturbations for a class of nonlinear evolution equations

We study in this paper the initial value problem for the multivalued differential equation wheref is in MS(2) andG is a multifunction fromC([0, T];) into the closed subsets of L²(0, Y;), satisfying suitable regularity assumptions. As an application we prove a local existence result for the problem

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.     

Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations

We establish existence and uniqueness theorems for the impulsive operator differential equation {fx177-01}. The operator A can be noninvertible. The obtained results are applied to differential-algebraic equations and partial differential equations of the non-Kovalevskaya type. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 155–166, February, 2008.     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

Periodic solutions of a class of nonlinear evolution equations

Summary In this paper we prove the existence of periodic solutions of abstract evolution equations which are modelled after parabolic problems. More precisely we prove that existence results follow from degree type hypotheses on the «projection» of the problem onto a suitable finite dimensional space.     

Relative controllability of perturbations of nonlinear systems

Sufficient conditions are derived for the relative controllability of perturbations of nonlinear systems with distributed delays in the control variables. The results are a generalization of previous results and are obtained by using Schauder's fixed-point theorem.     

Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1/r$1/r$, plus an anisotropic term, b/(r²[1+?cos²θ])^β/2$b/{\left(r^2[1+?{cos}^2\theta ]\right)}^{\beta /2}$. Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion.