The Asymptotic Behavior Of The Half Line For A Class Of One-dimensional Problems A Unique Solution

In this article, we use perturbation method, nonlinear transform and Karamata regularly varying principle, construct sub-supersolution, get the asymptotic behavior of the unique solution at zero and infinity, when b, g satisfy appropriate structural conditions.The one-dimensional singular boundary value problem on the half-lineWhere, g, b∈C~1(0,∞), and they are positive and non-decreasing functions on(0,∞). The two given structural conditions imply that g regularly varying with index q (q＞1) or rapidly varying at infinity (rapidly toward infinity); g regularly varying with index p (p＞1) or rapidly varying at zero(rapidly toward infinity). This article investigate its asymptotic behavior of the unique solution in two parts.