Asymptotic expansion of the boundary layers of the perturbed simple pendulum problems

We consider the nonlinear eigenvalue problem on an interval −u (t )+ g u(t ) = λsin u(t ), u(t ) > 0, t ∈ I := (−T,T ), u(±T ) = 0, where λ>0 is a parameter andT >0 is a constant. It is known that if λ 1, then the corresponding solution uλ has boundary layers. In this paper, we establish an asymptotic expansion of the width of the boundary layers of uλ when λ 1, which is explicitly represented by g and completely different from that of the case where g ≡ 0.  2003 Elsevier Inc. All rights reserved