This paper considers the nonlinear eigenvalue problems of boundary value problems for ordinary differential equations of the form (1) \(y'=t^{\alpha}A(t,\lambda)y\), \(1\leq t-1\), (2) \(B(\lambda)y(1)=0\), \((3)\quad y\in C([1,\infty]):\Leftrightarrow y\in C([1,\infty])\) and \(\lim_{t\to \infty}y(t)\) exists where y is an n- vector and A(t,\(\lambda)\) is an \(n\times n\) matrix. This type of eigenvalue problem on infinite intervals occurs in quantum mechanics and in fluid mechanics, when the stability of laminar flows over infinite media is investigated. It is shown that, under certain analyticity hypotheses, a domain in the complex plan can be identified, in which all eigenvalues are isolated. The technique to solve such problems is to cut the infinite interval at a finite point and to impose boundary conditions at this far end. In this paper suitable asymptotic boundary conditions are devised and the order of convergence is investigated. Some examples illustrate the theory.