Global asymptotic stability for damped half-linear oscillators

A necessary and sufficient condition is established for the equilibrium of the oscillator of half-linear type with a damping term, (ϕp(x′))′+h(t)ϕp(x′)+ϕp(x)=0(ϕp(x′))′+h(t)ϕp(x′)+ϕp(x)=0 to be globally asymptotically stable. The obtained criterion is given by the form of a certain growth condition of the damping coefficient h(t)h(t) and it can be applied to not only the cases of large damping and small damping but also the case of fluctuating damping. The presented result is new even in the linear cases (p=2p=2). It is also discussed whether a solution of the half-linear differential equation (r(t)ϕp(x′))′+c(t)ϕp(x)=0(r(t)ϕp(x′))′+c(t)ϕp(x)=0 that converges to a non-zero value exists or not. Some suitable examples are included to illustrate the results in the present paper.