Parameter diagram for global asymptotic stability of damped half-linear oscillators
The authors consider a half-linear differential equation with an unbounded damping term \[ (\phi _p (x'))'+h(t)\phi _p (x')+\omega^p\phi _p(x)=0,\tag{1} \] where \(\omega > 0\), \(t \in \left[a,\infty\right),\) \[ \phi _p(z)=\begin{cases} |z|^{p-2}z,& \text{ if }z\neq 0, \\ 0, & \text{ if }z = 0 \end{cases} \] with \(p > 1\) and \(z \in\mathbb R\). The authors prove three new theorems which include certain new necessary and sufficient conditions to guarantee the global asymptotical stability of the equilibrium of equation (1). They also give parameter diagrams to clarify the relations between the index number \(p\) and the parameters. They extend and improve some results in the relevant literature.
scientific article; zbMATH DE number 6537864
scientific article