Analysis of Boundary Value Problems on Infinite Intervals

Abstract

In this paper boundary value problems of ordinary differential equations on infinite intervals are analysed. There is a theory for problems of this kind which requires the fundamental matrix of the system of differential equations to have certain decay properties near infinity. The aim of this paper is to establish a theory which holds under weaker assumptions. The analysis for linear problems is done by determining the fundamental matrix of the system of differential equations asymptotically. For inhomogeneous problems a suitable particular solution having a “nice” asymptotic behaviour is chosen and so global existence and uniqueness theorems are established in the linear case. The asymptotic behaviour of this solution follows immediately. Nonlinear problems are treated by using perturbation techniques meaning linearization near infinity and by using the methods for the linear case. Moreover, some problems from fluid dynamics and thermodynamics are dealt with and they illustrate the power of the asymptotic methods used.