## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 1-2 (2004), 215-226.

### Three-term spectral asymptotics for the perturbed simple pendulum problems

#### Abstract

This paper is concerned with the perturbed simple pendulum problem $$ -u''(t) + g(u(t)) = \lambda \sin u(t), \ \ u(t) > 0, \ \ t \in I := (-T, T), \ \ u(\pm T) = 0, $$ where $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution develops the boundary layers. We adopt a new parameter $\epsilon \in (0, T)$ which characterizes both the boundary layers and the height of the solution and parametrize a solution pair $(\lambda, u)$ by $\epsilon$, namely, $(\lambda, u) = (\lambda(\epsilon), u_\epsilon),$ and establish the three-term asymptotics for $\lambda(\epsilon)$ as $\epsilon \to 0$.

#### Article information

**Source**

Differential Integral Equations Volume 17, Number 1-2 (2004), 215-226.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060480

**Mathematical Reviews number (MathSciNet)**

MR2035503

**Zentralblatt MATH identifier**

1164.34450

**Subjects**

Primary: 34B15: Nonlinear boundary value problems

#### Citation

Shibata, Tetsutaro. Three-term spectral asymptotics for the perturbed simple pendulum problems. Differential Integral Equations 17 (2004), no. 1-2, 215--226. https://projecteuclid.org/euclid.die/1356060480.