Nonlinear Techniques for Linear Oscillation Problems

Abstract

ABSTRACT. It is shown that for differential equations of the form .y'"' + py = 0 there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case n = 2. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation.If p is of constant sign, the coefficients of the "Riccati systems" are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.1. One of the classical techniques employed in the discussion of oscillation problems for linear second-order differential equations consists in replacing the study of the original equation by that of the equivalent Riccati equation. While the linearity of the original problem is sacrificed in this process, the formal structure of the Riccati equation permits-indeed invites-analytic manipulations which were not possible, or at least not apparent, in the original formulation. In fact, as pointed out in E. Hille's fundamental paper on the subject [4], this seeming detour via the Riccati equation may in certain respects be regarded as the "natural" approach to the linear oscillation problem.In a recent paper [14] it was shown that a similar approach is available for differential equations of the form x' = ^4x, where x is an «-dimensional vector function and A a continuous n x n matrix. There exists a "Riccati system", i.e., a system of nonlinear differential equations which is equivalent to the original equation if and only if the latter is nonoscillatory in a certain sense, and this connection between the linear and nonlinear systems leads to effective ways of attacking various oscillation problems associated with the original equation. These Riccati systems, incidentally, are not related to the "matrix Riccati equations" [15], [2] which are used in the treatment of second-order vector-matrix equations and which represent a natural extension of the classical formalism.Since linear nth order differential equations are special'cases of the equation x' = Ax, they can be treated by the methods described in [14]. In fact, this treatment can be considerably simplified because of the particular character of the