Growth conditions for uniform asymptotic stability of damped oscillators

The present paper is devoted to an investigation on the uniform asymptotic stability for the linear differential equation with a damping term, x″+h(t)x′+ω2x=0x″+h(t)x′+ω2x=0 and its generalization (ϕp(x′))′+h(t)ϕp(x′)+ωpϕp(x)=0,(ϕp(x′))′+h(t)ϕp(x′)+ωpϕp(x)=0, where ω>0ω>0 and ϕp(z)=|z|p−2zϕp(z)=|z|p−2z with p>1p>1. Sufficient conditions are obtained for the equilibrium (x,x′)=(0,0)(x,x′)=(0,0) to be uniformly asymptotically stable under the assumption that the damping coefficient h(t)h(t) is integrally positive. The obtained condition for the damped linear differential equation is given by the form of a certain uniform growth condition on h(t)h(t). Another representation which is equivalent to this uniform growth condition is also given. Our results assert that the equilibrium can be uniformly asymptotically stable even if h(t)h(t) is unbounded. An example is attached to show this fact. In addition, easy-to-use conditions are given to guarantee that the uniform growth condition is satisfied. Moreover, a sufficient condition expressed by an infinite series is presented. The relation between the representation of an infinite series and the uniform growth condition is also clarified. Finally, our results are extended to be able to apply to the above-mentioned nonlinear differential equation.