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-u″+ρ²u=w(t)u ^s,0<s<1andρ≠0,u(0)=u(1),u′(0)=u′(1)on the parameter ρ as ρ → 0. We show that this solution u(t; ρ) is infinitely differentiable in ρ and has singularity in the order of ρ^-α with α = 2/(1 - s) as ρ → 0. Furthermore, the graph of u(t; ρ) has fluctuation in t only of the order O(ρ^-α+2). This will help us to determine the location of the solution in numerical computations to study the behavior of the solution when ρ is sufficiently close to 0.