Oscillation theorems for fourth-order quasilinear ordinary differential equations

Abstract

Oscillation criteria for fourth-order quasilinear ordinary differential equations are obtained. An application to binary semilinear elliptic systems is also given.

Nonlinear oscillations of fourth order quasilinear ordinary differential equations

“…It is known that the asymptotic properties of equation (1.1) vary according to the convergence of the integrals appearing in (1.2) and/or (1.3) (see [1,5,7,8]). Roughly speaking, our assumptions (1.2) and (1.3) mean that the function p(t) becomes large near ∞.…”

“…Recently these results have been extended to the case α > 0, which is the so-called quasilinear case, for example in [1,3,5,7,8]. In [5,8], asymptotic properties of positive solutions of (1.1) have been discussed under the conditions…”

“…If there exists a positive solution of (1.1) of type (B +−+ ), then(3.3) ∞ tq(t) dt < ∞.The proof is essentially similar to that of[1, Theorem 4.2].…”

“…For the sub-homogeneous case, a necessary condition for the existence of a solution of type (D ++− ) follows, which can be proved by the same computations as used for [1,Theorem 4.5]. …”

On the structure of positive solutions of a class of fourth order nonlinear differential equations

“…Wu's theory has been enriched with information about the asymptotic behavior of nonoscillatory solutions of (A) provided by Naito and Wu [5] and Kamo and Usami [1,2] for the case where p(t) satisfies (1.2), or one of the following ∞ a t ( p(t)) −1/α dt = ∞, ∞ a t 1/α ( p(t)) −1/α dt < ∞,…”