STRONG OSCILLATIONS FOR SECOND ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we establish some strongly oscillation theorems for nonlinear second order functional differential equation x"(t) + p(t) f(x(t), x(g(t))) 0 without assuming that g(t) is retarded or advanced.

I. INTRODUCTION. equation We consider the second order nonlinear functional differential x"(t) + p(t) f(x(t) x(g(t))) 0 where p(t), g(t) e C([t ), R) g(t) as t and f(u,v) (R,R) and has the O sign of u and v when they have the same sign.We shall restrict our attention to solutions of (I.I) which exist on some positive half-line A nontrlvial solution x(t) is called oscillatory if x(t) has an unbounded set of zeros, and otherwise it is called nonoscillatory.Equation (I.I) is said to be oscillatory if every solution of (I.I) is oscillatory Oscillation theory for (I.I) has been developed by many authors.Bradley [I], Chiou [2], Erbe [3] Gollwitzer [4], Ladas [5], Travis  [6], Waltman [7], Wong [8] and references therein It is wellknown theorem of Wintner  [9] and Leighton [I0] that the linear equation x"(t) + p(t) and Waltman [7]  The purpose of this paper is to establish some strongly oscillation criteria for (1.1).We are primarily interested in the case when p(t) are satisfied.
Considering the equation x"(t) + %p(t) x(t)= 0, We shall call p(t) a strongly oscillatory coefficient if (1.3) is oscillatory for all positive %.If p(t) O, Nehari [12] shows that lira supt/t p(s)ds is a necessary and sufficient condition for p(t) to be a strongly oscillatory coefficient.In general, motivated by Nehari, we define as follows: Fzluatlon (I.I) is said to be strongly oscillatory if the related equation of (I.l) x"Ct) + Xp(t) fCx(t), x(g(t))) 0 (1.4) is osillatory for all positive 2. MAIN RESULTS.
For Equation (I.I) the following conditlons are assumed to hold throughout the paper: il) there exists m > 0 such that lul implies where (v) C'(R), v(v) > 0 and '(v) 0 for v # 0, and lira where I and 6 are constants.
We begin with a Lemma which needed in establishing our results.
PROOF.Assume that Equation (1.4)atI I has a nonoscillatory solution o x(t) > 0 for t ' t > 0. A similar proof will hold if x(t) < 0 for t ) t o o It is easy to verify that x"(t) < 0 and x'(t) >0 for all large t.Let w(t) x'(t)/(x(h(t))), then w'(t) -hoP(t) Since x'(t) > 0 for large t, llm x(t) exists either as a finite or infinite limit.

If llm x(t)
e is finite, then ==, then by (ii) we have that f(x(t), x(g(t))) ) (x( g (t) for large t.In either case, for sufficiently large t, we have that f.(x(t)_, x(g(t))) e where e rain (e e2). (x(g(t))) (2.2) Since x(t) is increasing, for large t we have that p(t) f(x(t), x(.g(t))) ) (t) f(x(t), x(g(t))) logP(t) and in view of x"(t) < 0 for large t and (ii) we see that , k6w2(t) OW2(t)  (2.6) 3) from t to t, then letting t we obtain that (2.1) holds for all large.
PROOF.Assume to the contrary that Equation (1.4)at X-X > 0, has a non- o oscillatory solution x(t) > 0 for t t 0. A similar argument holds when x(t) < 0 o for t t O. Let w(t) x'(t)/(x(h(t))).where k Go.However, by a well-known theorem of Wintner [13] this implies the o o equation y"(t) + goPo(t)y(t) 0 (2.11)   is nonosclllatory.This contradicts the fact that the condition (I I) implies Po(t) is a strongly oscillatory coefficient.
If m > I, integrating (2.10) from t to t we obtain (2.17) y"(t) + Ce_tPm_t is nonosctllatory.But this contradicts again the fact that the condition (I I) or (1 2 implies that equation (2.17) is oscillatory.The proof is thus complete.
(u)=u and m t.
THEOREM 2.3.For all positive constants and q assume that any one of following conditions is satisfied: (II I) there is a positive integer m such that qn(t,,n) is defined for n-l,2,..., m-l, but qm(t,,) does not exist; (II 2) qn(t,,n) is defined for n=l,2,..., but the function sequence (2.18) is not convergent for all large t.
PROOF.Assume that Equation (1.4)at k=k > 0, has a nonoscillatory solution o x'(t) x(t) > 0 for t t The case x(t) < 0 is handled similarly.Let w(t) o (x(h(t) By Lemmma 2.1, we find that (2.1) holds.
Suppose that (III I) holds, then, as proof of Theorem 2.3, or qm (t'o'no) o t q2(s'o' qo)dS + qo(t,o 'qo w(t) x' (t) x' (h(t) which is again a contradiction, and the proof of the theorem is complete.
Equation (I.i) is said to be strongly bounded oscillatory if all bounded solutions of Equation (1.4) for any %e(O, =) are oscillatory.

( 2
demonstrated that the equation As in the proof of Lemma 2.1, we can