Periodic boundary value problems for $n$th-order ordinary differential equations with $p$-laplacian

Abstract

We prove existence results for solutions of periodic boundary value problems concerning the $n$th-order differential equation with $p$-Laplacian ${\left[\phi \left(x^{\left(n-1\right)}\left(t\right)\right)\right]}^{\text{'}}=f\left(t,x\left(t\right),x\text{'}\left(t\right),\mathrm{...},x^{\left(n-1\right)}\left(t\right)\right)$ and the boundary value conditions $x^{\left(i\right)}\left(0\right)=x^{\left(i\right)}\left(T\right)$, $i=0,\mathrm{...},n-1$. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that $f$ may be a polynomial and the degree of some variables among $x_0,x_1,\mathrm{...},x_{n-1}$ in the function $f\left(t,x_0,x_1,\mathrm{...},x_{n-1}\right)$ is allowed to be greater than $1$.