On a Second Order Nonlinear Oscillation Problem

Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations

“…Clearly condition (4) implies (9) for the function F. Also, by taking 9(x) = x112 and >p(x) = (x log x)1'2 in Theorem 6, we obtain Theorem 2 of [3] which is a generalization of earlier results of Kiguradze [8] and Nehari [13]. Note that in this case, since (<p2)" = 0, condition (83) is trivially satisfied.…”

“…In the sublinear case when y<\, Belohorec [2] has obtained results in both directions. For the more general equation (1), in the superlinear case, a study was initiated in Nehari [13] and continued in Coffman and Wong [3], but as far as we know the corresponding sublinear case has not been investigated.…”

“…In the Appendix, we further this approach by showing how it may be used in the study of the continuability problem and the uniqueness of the zero solution. This technique was first introduced by Coffman and Wong [3] for a special case of equation (1) and was suggested by some ingenious differential identities and inequalities used by Nehari in [13]. The main results, too detailed to describe here, include oscillation and nonoscillation theorems for both of the two classes of equation (1) and contain as special cases all of the results cited above.…”

“…These conditions suffice for local existence of solutions of the initial value problem (3) y(xx) = a, y'(xx) = b, xx > 0, for (1).…”

“…The first assertion, that is, the existence of oscillatory solutions of (1) under the hypothesis of Corollary 9, was given in [3,Theorem 1], An examination of the derivation of inequality (63) of Lemma 6 shows that in Theorem 4 the hypothesis (6) can be weakened to G(t, x)^KF(t, x), f¿M2<¡i(x). With a similar modification of the hypothesis, Corollary 9 in fact contains Theorem 1 of [3].…”