# On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

## TL;DR

This paper investigates the existence of homoclinic solutions in inhomogeneous second-order Lagrangian systems with superquadratic potentials, using periodic approximations and limit processes.

## Contribution

It establishes the existence of homoclinic solutions for a class of inhomogeneous Lagrangian systems with superquadratic potentials under certain conditions.

## Key findings

- Homoclinic solutions exist for the studied systems.
- Solutions are obtained as limits of periodic solutions.
- The approach uses approximation by periodic problems.

## Abstract

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$, $a\colon\mathbb{R}\to\mathbb{R}$ is a continuous positive bounded function, $G\colon\mathbb{R}^n\to\mathbb{R}$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb{R}\to\mathbb{R}^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.