Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x ) + b(t)g(x) = 0, where g(x) satisfies the signum condition xg(x) > 0 if x = 0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari.

We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations. These improve the work of Elbert and Schneider [6] in which connections are found between halflinear differential equations and linear differential equations. Our comparison theorems are not of Sturm type or Hille-Wintner type which are very famous. We can apply the main results in combination with Sturm's or Hille-Wintner's comparison theorem to a half-linear differential equation of the general form (|x | α−1 x ) + a(t)|x| α−1 x = 0.

In this paper, oscillation criteria are established for all solutions of second-order nonlinear differential equations of the formHere φ p (y) is the one-dimensional p-Laplacian operator, and g(x) satisfies the signum condition xg(x) > 0 if x = 0 but is not assumed to be monotone. The equation naturally includes the famous Euler differential equation and half-linear differential equations. The main purpose is to examine the influence of certain growth conditions of the nonlinear term g(x) on the oscillation of solutions. The conditions are shown to be sharp. Some of differential inequalities play important roles to prove our results. A simple example is included to illustrate the main result. A conjecture on the inverse problem is also given. Finally, elliptic equations with p-Laplacian operator are discussed as an application to our results.  2004 Elsevier Inc. All rights reserved.

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equationHere g(x) satisfied the signum condition xg(x) > 0 if x = 0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.