Parametric perturbation problems in ordinary differential equations

Abstract

The asymptotic behavior of solutions of a nonlinear differential equation that arises through a nonlinear parametric perturbation of a linear system of differential equations is discussed. Fundamental hypotheses include the admissibility of a pair of Banach spaces for the linear system. Conclusions about the behavior of the perturbed system evolve through the behavior of certain manifolds of solutions of the unperturbed linear system. Asymptotic representations are found on a semi-infinite axis R + {R_ + } and on the real line R. The bifurcation condition, which is shown to be trivial on R + {R_ + } , plays an essential role for the perturbation problem on R. Illustrations and examples, primarily on the space L ∞ {{\text {L}}^\infty } , of the theoretical results are presented.