Uniform global asymptotic stability for oscillators with superlinear damping

The present paper deals with the damped superlinear oscillatorx″+h(t)ϕq(x′)+ω2x=0,x″+h(t)ϕq(x′)+ω2x=0, where ω>0ω>0 and ϕq(z)=|z|q−2zϕq(z)=|z|q−2z with q≥2q≥2. The origin (x,x′)=(0,0)(x,x′)=(0,0) is the only equilibrium of this oscillator. We herein establish a sufficient condition for the equilibrium to be uniformly globally asymptotically stable. We conclude that under the assumption that the damping coefficient h(t)h(t) is integrally positive, if the integral from σ   to t+σt+σ of a particular solution of the first-order nonlinear differential equationu′+h(t)ϕq(u)+1=0u′+h(t)ϕq(u)+1=0 diverges to negative infinity uniformly with respect to σ  , then the equilibrium is uniformly globally asymptotically stable. The above-mentioned result is expressed by an implicit condition. We examine when the implicit condition is satisfied and when it is not satisfied. We also give explicit sufficient conditions which assure that the equilibrium is uniformly globally asymptotically stable. Using the obtained result, we present an example of which the equilibrium is uniformly globally asymptotically stable even if h(t)h(t) is unbounded.