OSCILLATION AND NON-OSCILLATION OF SOME NEUTRAL DIFFERENTIAL EQUATIONS OF ODD

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of some nth order equations with nonlinearity in the neutral term.


INTRODUCTION.
In this paper we consider the first order neutral differential equations of the form (x(t)-cx(t-r)y-p(t)x(g(t))= o, > o (1.1} (x(t) + cx(t r))' p(t)x(g(t)) f(t) (1.2) and nonlinear neutral differential equation of nth order of the form (z(t)-h(t)f(z(r(t)))) (n) + p(t) f2(z(g(t))) O, >_ o (1.3) Most of the works on oscillation theory for neutral equations deal with stable type equations.
There are a few papers (see e.g.[2], [3], [5]) where nonoscillation of unstable type equations of order larger than one is discussed.In Section 2 we establish a result for the existence of an unbounded solution of Eq. (1.1) which tends to infinity exponentially.Some' results on oscillation and nonoscillation for Eq.(1.3) are given in Section 3. To the best of our knowledge this is the first time that a differential equation with nonlinearity in the neutral term is being studied.As pointed out by Hale [1] it is useful to study neutral nonlinear differential equations of the form (x(t)-a(t,x(t-r))y H(t,x(t-r)) As usual a solution x(t) of Eq. (1.j), j 1,2, is said to be oscillatory on [t0,oo if the set of zeros of z(t) is unbounded, otherwise it is called nonoscillatory.In Section 3 we need the following lemma: LEMMA.(e [6]).Lt X b BausCh p, r boded od d o,x ubt o X, A, B be maps on F to X such that Ax + By F for every pair x, y q r.If A is a contraction and B is completely continuous then the equation Ax + bz z has a solution in F.

RESULTS FOR EQUATIONS (1.1} AND (1.2).
We assume that and the functions p,g,F are continuous on [t0,oo), o > 0. In case p(t)=p, g(t)=t-a, a>O, from the analysis of the characteristic equation of Eq. (1.1) we know that Eq. (1.1) has always an unbounded solution z(t) Ae at er > 0 The question arises whether Eq.(1.1) has always an unbounded solution x(t) which tends to infinity exponentially as tends to infinity.We explore that possibility.
For c > 0, let x(t) be a positive solution of Eq. (1.1).We put z(t)= x(t)-cx(t- Then z'(t) > 0, and therefore two possibilities exist" (i) z(t) > 0, eventually, or (ii) z(t) < O, eventually.Consequently, the nonoscillatory solution x(t) must satisfy one of the following type of asymptotic behavior: () We prove the following: THEOREM 2.1.Based on the value of c we have the conclusions: (i) If c _> 0, Eq. (1.1) has always a positive solution x(t) satisfying (b) or (c); (ii) If c > 1, Eq. (1.1) has always an unbounded solution x(t) satisfying (c); (iii) If c > 1, Eq. (1.1) has always an unbounded solution x(t) which tends to infinity exponentially; (iv) If 0 _< c < 1, and /p(t)dt oo, T >_ O, Eq. (1.1) has always an unbounded positive solution, and every bounded solution of Eq. (1.1) either oscillates or tends to zero as tends to infinity.
PROOF.For a given continuous function p there exists a continuou function H(t)> 0 such that Let BC([to, oO ,R) be a Banach space of bounded and continuous functions y-[to, oO ---, R. Define a subset fl of BC as follows: ft={y_BC'O<y(t)<l, to<t<c }.
Clearly fl is a bounded, closed and a convex subset of BC.Now we define a mapping S on fl as follows: where T is chosen sufficiently large so that t-r > o y(t) > o z(t) >_ 1, and z(t-) [ c z(t) +(t) p(s)z((s))ds <_ , T for > T. Using (2.1) and (2.2) one finds that f p(s)z(g(s))ds (t-,') o z(t) .-,0,and z(t) 4oo, as t--.oo, (2.4) which shows that (2.4) is possible.Thus we have Sgl C f Let Yl and Y2 be elements of f.Then which shows that S is a contraction on ft.Hence, there is an element y Eft such that Sy y.
or x(t) > z(to)e p(t-t), for >_ o where p > 0, which shows that (iii) is true.In order to prove the second part of (iv) we let x(t) to be a bounded positive solution of Eq. (1.1).Put u(t) (t)-(t-) The u'(t) > 0 and limtou(t ests.Let limtu(t I.If > 0 then x(g(t)) I. Consequently, u(t)-u(T1)= / p(s)x(g(s))d, I T1 T1 since TlP(S)ds t we have a contradiction to the boundedness of x(t).In view of (2.7) we c sume that c0.Now for 0<c<1 we cnot have the ce that l<0.Thus limtu(t 0 d hence we have limtx(t 0. This completes the prf of the threm.EXAMPLE 2.1.Consider the equation which satisfies the sumptions of threm (2.1(i).In fact, (2.8) e solution: () 1 + .
OPEN PROBLEM.What is a criterion for the existence of oscillatory solutions for Eq.(1.2)?
THEOREM 2.2.Consider the Eq.(1.2) and assume that there exists a function f such that F(t) f'(t) and lira sup f(t) + o f(O=-o.(.) Then every bounded solution of Eq. (1.2) is oscillatory.
PROOF.Let o be sufficiently large so that 2" rain {t f0 r(t), inf g(t)} t>_T 0 As before, BC([T, oo)) denotes the Banach space of all bounded and continuous real valued functions defined on IT, oo).Let II be a subset of BC as defined in Se. 2. Define operators $1 and S 2 on 12 as follows: h($)e at f l(y('r(t)e-ar(t) for T _< t _< -u (n-1)!I (s-t)(n-1)p(s).f2(Y(9(sl)e-ag(e))ds, (sl(O r (s/(./+ ( -/, if t > t o for T _< t _< o By (iv), for every x,y E 12 we have SlX -k S2y E f. Condition (iv) implies that ,S' is a contraction on 12.It is easy to see that tt (S2Y)(t)[ < MI for E where M is a positive constant.From this it follows that S 2 is completely continuous.By Lemma there exists a y E f such that That is, (S + S2)y y h(')ea'fl(Y(r)(t))e-ar('))+ (n-1)' I (s-t)("-1)P(s)f2(Y(g(s))e-ag(s)) if > o o (t0) + ( ), for T < < o It is easy to see that y(t) > 0 for _> T. Set x(t)= y(t)e -at Then oo z(t) h(t) fl(z(r(t))) + (n 1 I (s t) n-l[(s)f2(x(g(s)))ds t>t o or (x(t) h(t) fl(x(r(t)))) n + p(t)f2(x(g(t)) 0, t _> t O This completes the proof.for all large values of t.In our notation h(t) 4 1fl(x) x 3 L 3 f2(x) x 1/3 Ogviously the hypotheses of theorem 3.1 are satisfied.Therefore Eq.(3.1) has a solution z(t) which tends to zero exponentially as too.In fact, x(t) e -t is such a solution of (3.1).
MARK.We first prove a lemma which we nd in the prf of the threm.LEMMA 3.2.Let x(t) eventuMly positive solution of Eq. (1.3).Set z(t) z(t) h(t) f (x(r(t))).