DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SECOND-ORDER NONLINEAR OSCILLATIONS:
A CASE HISTORY
JAMES S. W. WONG
This paper gives an updated account on a nonlinear oscillation problem originated from
the earlier works of F. V. Atkinson and Z. Nehari.
Copyright © 2006 James S. W. Wong. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We are here concerned with the study of oscillatory behavior of solutions of second-order
Emden-Fowler equations:
y ′′ (x)+a(x) ∣∣ y(x) ∣ ∣ γ−1 y(x) = 0, γ>0, (1.1)
where a(x) is nonnegative and absolutely continuous on (0,∞). Under these conditions, it
is well known that every solution of (1.1) is uniquely continuable to the right throughout
(0,∞); see Hastings [17], Heidel [18], Coffman and Wong [10]. A solution y(x) of(1.1)
is said to be oscillatory if it has arbitrarily large zeros, that is, for any x 0 ∈ (0,∞), there
exists x 1 >x 0 such that y(x 1 ) = 0. Otherwise, the solution y(x) is said to be nonoscillatory
and it has only finitely many zeros on (0,∞), that is, there exists a last zero ̂x depending
on y(x)sothat|y(x)| > 0forallx>̂x.
Equation (1.1) issaidtobesuperlinearifγ>1 and sublinear if 0