OSCILLATION CRITERIA FOR CERTAIN NONLINEAR FOURTH ORDER EQUATIONS

This work investigates the behavior of solutions of certain nonlinear fourth order differential equations. An example is given showing that these equations can have both oscillatory and nonoscillatory solutions simultaneously. Finally, several criteria for the existence of oscillator solutions are established.

We will confine ourselves to those solutions y of (I) or (2) which are defined on some half-line [to,m), t o O, and are not identically zero on any subinterval of [to,m).Such a solution is termed oscillatory if it has a zero on every half-line [tl,m), t I a t o amd nonoscillatory otherwise.
The main objective if this work is to investigate the solutions of (i.i) and (1.2) relative to their asymptotic behavior and oscillation properties.An example showing that (i.i) can have both oscillatory and nonoscillatory solutions is given after which the nonoscillatory solutions of (1.2) are examined and several oscillation criteria are derived for (1.2).The techniques used herein are similar to those used by Heidel [i]   and Waltman [2] in their investigations of nonlinear third order equations, and also has points of contact with articles by this author [3] and the recent work of Lovelady [2] on nonlinear fourth order equations.
Thus F[y(t)] is decreasing on [0,=o) since y is not identically zero on any half-line.
No solution of (|.i) can have more than one multiple zero since this would imply F could be zero at two points, contradicting the decreasing nature of F.
PROOF.Suppose without loss of generality that y(t) > 0 on [at ) for some a 0 and assue F[y(t)] < 0 on [a,(R)).We will show that the latter assuptlon leads to a ontradlction.
where b >-a and y(t) is increasing on [b,oo).
Let < 0 be a number satisfying Q(t) < B on [b,oo).Then y"(t) < By(t) < y < 0 (2.1) for some 7 < O.Such a y exists because y(t) is increasing.But (2.1) implies y'(t) which is clearly impossible and the result follows.
COROLLARY.Suppose y(t) is a solution of (i.i) for which F[y(t)] < 0 on soe half- line [a,oo).o.The result follows by examining N[y(t)] along the ex of y"(t) which are the extrema of y'(t).
We now turn our attention to the forced monllnear equation y(4) + p(t)y' + q(t)f(y) r(t) where we assume that p,q,r, and f satisfy conditions (i) and (li) listed above and he additional hypothesis: (iv) y" t hence-#-(--is decreasing and eventually of one sign.Thus y(t)y"(t)y'(t) # 0 for all t on some half-line [tl,=).Assuming without loss of generality that y(t) > 0 on [tl,=), the following cases must be considered: (a) y(t) > 0, y'(t) > O, y"(t Case (d) is clearly Impossible.So let us suppose that (c) holds.Then p(t)y2(t) + 2y(t)y"'(t) 2y'(t) < H 0 1 for all t -> t I and we conclude that y(t)y"'(t) < H 0 on [tl,).
Since y(t) is decreasing it is easy to see that y"' (t) < H 0 2y(tl) which implies that y"(t) a contradiction.So (c) is impossible.Similarly, (b) is impossible and the result follows.
Our oscillation criteria is based on theorem 4.
THEOREM 5. Suppose t q(t)dt =, llm sup I r(s)ds < and f(y) is nondecreasing.
Then any solution y(t) of (1.2) satisfying H[y(t 0)] < O, for some t o is oscillatory.
PROOF.Let y(t) be a nonoscillatory solution of (1.2) which satisfies H[y(t0)] < 0, for some t o Then according to our Theorem, there exists t I t O such that sgn y(t) sgn y' (t) sgn y"(t) for all t > t I. Assume without loss of generality that y(t) > 0 on It l,=].Then from (1.2) we have (4) y (t) < r(t) f(y(t))q(t).
Integrating from t I to t we get t t y"'(t) < / r(s)ds f(Y(tl)) / q(s)ds + y"'(tl).(2.5) t But based on the boundedness of ( r(s)ds, (2.5) implies y"' (t) as t but this t 1 would force y(t) to eventually become negative, a contradiction.Hence (1.2) cannot have a nonoscillatory solution y(t) satisfying H[y(t 0)] < 0, for some t o and the proof is complete.
PROOF.Suppose that y(t) is a positive nonoscillatory solution of (1.2) with H[Y(t0)] < 0 for some t 0. Since f(y) m and y'(t) > 0 on some half-line [tl,) it y follows that t t y":(t) _< y"'(tl)+ / r(s)ds-m[ q(s)y(s)ds.This completes the proof of our Theorem.
Finally we have (2.7) [ p(t)dt and y(t) is a solution of (1.2) such that H[y(t 0)] < O, 7. Suppose 0 for some tO, then y(t) is oscillatory.
We omit the proof of Theorem 7 because of its similarity to the above proofs.
It appears from the above Theorems that any condition that implies oscillation in (1.2) also implies oscillation in (i.i), thus it is natural to ask whether the oscillation of (1.2) implies the oscillation of (I.i).We shall leave this as an open question although the "feeling" that one gets from linear examples is that the answer is probably a negative one.
Then y(t) is oscillatory.In particular, any solution having a multiple zero is oscillatory.
e is a solution of(2.2).It follows from the corollary that (2.2) also has oscillatory solutions, e.g., and nontrivial solution satisfying y(a) y' (a) 0 at some a > 0. THEORE 3. Suppose y(t) is an oscillatory solution of (1.1) satisfying Fly(c)] < 0 for some c > 0. Then y'(t) is unbounded.
Since y' (t) is increasing it is easily verified that y(t) > y'(tl)(tt I) for all t >-t I.