Document Type : Research Paper
Author
- Mohsen Timoumi ^{} ^{}
Department of Mathematics, Faculty of Sciences, University of Monastir, P.O.Box 5019, Monastir, Tunisia.
Abstract
This article concerns the existence of fast homoclinic solutions for the following damped vibration system
\begin{equation*}
\frac{d}{dt}(P(t)\dot{u}(t))+q(t)P(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,
\end{equation*}
where $P,L\in C\left(\mathbb{R},\mathbb{R}^{N^{2}}\right)$ are symmetric and positive definite matrices, $q\in C\left(\mathbb{R},\mathbb{R}\right)$ and $W\in C^{1}\left(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\right)$. Applying the Fountain Theorem and Dual Fountain Theorem, we prove the above system possesses two different sequences of fast homoclinic solutions when $L$ satisfies a new coercive condition and the potential $W(t,x)$ is combined nonlinearity.
Keywords
- Damped vibration systems
- Fast homoclinic solutions
- Variational methods
- Fountain Theorem
- Dual Fountain Theorem
Main Subjects
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