Asymptotic properties of the half-linear difference equation (∗) ∆(an|∆xn| α sgn ∆xn) = bn|xn+1| α sgn xn+1 are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to (∗) are considered too. Our approach is based on a classification of solutions of (∗) and on some summation inequalities for double series, which can be used also in other different contexts.
Oscillatory properties of the second order nonlinear equation \[ (r(t)x^{\prime })^{\prime }+q(t)f(x)=0 \] are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.
We study solutions tending to nonzero constants for the third order differential equation with the damping term (a1(t)(a2(t)x ′ (t))′ ) ′ + q(t)x ′ (t) + r(t)f(x(ϕ(t))) = 0 in the case when the corresponding second order differential equation is oscillatory.