Singular perturbations on the infinite interval

where e is a small positive parameter arise frequently in applications. Here/and x ate teal /c-dimensional vectors with components/= (/i,••-,/*) and x = (x¡, •••,xL), respectively, and g and y ate teal /dimensional vectors with components g = (S¡r--,gj) and y = (y\,---,y¡), respectively. The purpose of this paper is to investigate the behavior of solutions of (PE) as e -+ 0 + for í0 ^ í < co. In studying the behavior of solutions of (F£) for small positive e it is convenient to make use of two systems which are associated with (Pe). The first associated system, called the degenerate system, is obtained by formally setting e = 0 in (Pe): This gives