Oscillation criteria for second-order damped nonlinear differential equations with p-Laplacian

This paper is concerned with the oscillation problem for the nonlinear differential equation with a damping term,(ϕp(x′))′+2(p−1)tϕp(x′)+a(t)g(x)=0, where p>1p>1 and ϕp(y)=|y|p−2yϕp(y)=|y|p−2y. Here a(t)a(t) is positive and continuous on (α,∞)(α,∞) for some α⩾0α⩾0; and g(x)g(x) is continuous on RR and satisfies the signum condition xg(x)>0xg(x)>0 if x≠0x≠0, but is not assumed to be monotone increasing. It is proved that under additional assumptions on a(t)a(t), all solutions tend to zero as t→∞t→∞. By means of this fact together with Riccati technique, sufficient conditions are given for all nontrivial solutions to be oscillatory. Sufficient conditions are also obtained for all nontrivial solutions to be nonoscillatory. Finally, a more general equation is discussed as an application to the main results.