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 $a<b$. We show the existence of at least one solution using: firstly, lower and upper solutions method; secondly, some sign conditions; and thirdly, a combination of lower and upper solutions and sign conditions. No Nagumo condition for the dependence of $f(\tau,u,v)$ with respect to $v$ is required.

This work is licensed under a Creative Commons Attribution 4.0 License.