Dependence of the unique solution of a periodic boundary value problem on the parameter

In this paper, we study the dependence of the unique positive solution of the periodic boundary value problem u00 þ qu 1⁄4 wðtÞu; 0 < s < 1 and q – 0; uð0Þ 1⁄4 uð1Þ; u0ð0Þ 1⁄4 u0ð1Þ on the parameter q as q ? 0. We show that this solution u(t;q) is infinitely differentiable in q and has singularity in the order of q a with a = 2/(1 s) as q ? 0. Furthermore, the graph of u(t;q) has fluctuation in t only of the order O(q a+2). This will help us to determine the location of the solution in numerical computations to study the behavior of the solution when q is sufficiently close to 0. 2011 Elsevier Inc. All rights reserved.