A necessary and sufficient condition for the global asymptotic stability of damped half-linear oscillators

We are concerned with the global asymptotic stability of the equilibrium of the half-linear differential equation with a damped term $$\left(\phi_p(x')\right)' + h(t)\phi_p(x') + (p-1)\phi_p(x) = 0. $$ A necessary and sufficient growth condition on h(t) is obtained by using the generalized Prüfer transformation and the generalized Riccati transformation. Equivalent planar systems to the above-mentioned equation are also considered in the proof of our main result.