On Lyapunov-type inequalities for nonlinear Hamiltonian-type systems

Abstract

In this paper, we state and prove new Lyapunov-type inequalities for nonlinear Hamiltonian-type system%
\begin{equation*}
\left\{
\begin{tabular}{l}
$x^{\prime }=p\left( t\right) x+\dfrac{\left\vert y\right\vert ^{p-2}y}{%
\Big( 1+\left\vert y\right\vert ^{p}\Big) ^{\frac{p-1}{p}}}$ \\
$y^{\prime }=-q(t)\dfrac{\left\vert x\right\vert ^{\beta -2}x}{\Big( %
1+\left\vert x\right\vert ^{\beta }\Big) ^{\frac{\beta -1}{\beta }}}-p\left(
t\right) y$%
\end{tabular}%
\ \right.
\end{equation*}%
involving the $p$-prescribed curvature operator%
\begin{equation*}
\Phi _{p}\left( v\right) =\dfrac{\left\vert v\right\vert ^{p-2}v}{\Big( %
1+\left\vert v\right\vert ^{p}\Big) ^{\frac{p-1}{p}}}\text{, }p>1\text{,}
\end{equation*}%
under Dirichlet boundary condition.