Oscillation of solutions to second-order nonlinear differential equations

On a Second Order Nonlinear Oscillation Problem

“…Repeating the same argument, but "going backwards" we see that if x0 is the last zero of the eventually positive solution y, then for X!>^x0, and when (11) holds at x = Xi we must have 4>{xi) < B-Using (4), (6), and (9) as before we conclude that </S(x) < B for all large x. If y{x) < 0 for all large x, then the above argument applied to the solution -y{x) gives (8). This completes the proof.…”

“…(Extension of Atkinson's result to more general equations similar to (1) have been given by Macki and Wong [8].) It would be tempting to conjecture that a necessary and sufficient condition for equation (2) to be nonoscillatory iŝ x(y + 1)l2p{x)dx < co.…”

“…Let y{x) be a nonoscillatory solution of {I). Then there exists a positive constant B, independent of the initial values of y{x) and y'{x) such that (8) lim sup \xll2y'{x)-$x-ll2y{x)\ è B.…”

Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations

“…The former problem arises primarily in the superlinear case, the latter primarily in the sublinear case. In fact, as we shall prove in the Appendix, in the sublinear case, under assumption (5) or under the weaker assumption (10), all solutions of (1) will have an unbounded right maximal interval of existence. This is probably not true if we assume only (7) and is certainly false in the superlinear case, as is well known.…”

Nonoscillatory solutions of second order nonlinear differential equations

“…T%βw [α, δ] is an interval of disconjugacy for equation (4). That is, no nontrival solution of (4) has more than one zero on [α, 6].…”