Abstract

We study the existence of solutions of equation  \[(\phi(w'(\tau)))'= f(\tau,w(\tau),w'(\tau)),\quad  \tau\in [0,\ell]\] submitted to periodic boundary conditions on $[0,\ell]$. Where $f:[0,\ell]\time \mathbb{R}^{2}\rightarrow \mathbb{R}$ is a continuous function and $\phi:Dom(\phi)\subset\mathbb{R}\rightarrow \mathbb{R}$ is considered as a continuous function on $Dom(\phi)\subse \mathbb{R}$ and strictly increasing on $[a,b]\subset Dom(\phi)$ with $0\in[a,b]$ and $a

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