We first establish a integral inequality for compact maximal space-like subman-ifolds in pseudo-Riemannian manifolds N^np+p. Then, we investigate compact space-like sub-manifolds and hupersurfaces with parallel second fundamental form in N^np+p and some other N^np+pambient spaces. We obtain some distribution theorems for the square norm of the secondfundamental form.
Several oscillation criteria are given for the second order nonlinear differentialequation with damped term of the form [a(t)(y′(t))^σ]′+p(t)(y′(t))^σ+q(t)f(y(t)) = 0, wherea E C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f E C^1(R, R) suchthat xf(x) > 0 for x ≠ 0. Our results improve and extend some known oscillation criteria.Examples are inserted to illustrate our results.
