On a second order nonlinear oscillation problem

Abstract

(1) y+yF(y2, x) = 0, where the function F(t, x) is continuous and nonnegative for t E [0, oo), x E (0, oo). It will be tacitly assumed here that every locally defined solution of (1) is continuously extendable throughout the entire nonnegative real axis. This will be the case if for example one requires that for fixed t, F(t, x) satisfies a uniform Lipschitz condition in some neighborhood of every x E [0, oo). (See Hastings [3], and Coffman and Ullrich [2].) Actually, this tacit assumption can easily be removed, see the remarks at the end of the paper. A nontrivial solution of (1) is said to be nonoscillatory if for every a > 0 the number of its zeros in [a, oo) is finite, and it is said to be oscillatory otherwise. Different from the linear equation, when F(t, x) is independent of t, the nonlinear equation may possess solutions of either kind. In view of this, one is led to consider the following types of oscillation and nonoscillation conditions; namely, those which guarantee all solutions of (1) oscillate and its converse, i.e. the existence of one nonoscillatory solution, and those which guarantee all solutions of (1) do not oscillate and its converse, i.e. the existence of one oscillatory solution. The first type of oscillation and nonoscillation conditions have been the centre of considerable amount of research and there are a number of results available for equation (1) or similar equations. For an expository account on this subject, we refer the reader to Wong [12], where other references may be found. An excellent discussion on the nature of oscillatory and nonoscillatory solutions may also be found in the papers by Moore and Nehari [9], and Nehari [10]. The second type of oscillation and nonoscillation conditions have received little attention until recently. The prototype of equation (1) is the following generalized Emden-Fowler equation: (2) y +p(x)y = 0,