## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 9/10 (2009), 999-1018.

### Heteroclinics for non-autonomous second-order differential equations

A. Gavioli and L. Sanchez

#### Abstract

We investigate new conditions for the existence of heteroclinics connecting $\pm 1$ for a non-autonomous equation of the form \begin{equation} \label{hetero} \ddot u=a(t)f(u) \end{equation} where $a(t)$ is a bounded positive function and $f(\pm 1)=0$. Here $f=F'$, where $F$ is a $C^1$ non-negative function such that $F(-1)=F(1)=0$. We are interested mainly in the case where $a(t)$ approaches its positive limit, as $|t|\to\infty$, from above, but we allow also the (``asymptotically asymmetric") case where $|\lim _{t\to-\infty}a(t)-\lim _{t\to+\infty}a(t)|$ is a sufficiently small positive number. Variational methods are used in the proofs.

#### Article information

**Source**

Differential Integral Equations, Volume 22, Number 9/10 (2009), 999-1018.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019519

**Mathematical Reviews number (MathSciNet)**

MR2553067

**Zentralblatt MATH identifier**

1240.34227

**Subjects**

Primary: 34C37: Homoclinic and heteroclinic solutions

#### Citation

Gavioli, A.; Sanchez, L. Heteroclinics for non-autonomous second-order differential equations. Differential Integral Equations 22 (2009), no. 9/10, 999--1018. https://projecteuclid.org/euclid.die/1356019519